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## Meet the editor

Dr. Tuong T. Truong is a graduate of the "Ecole Centrale des Arts et Manufactures", Paris, France and has a Ph.D in theoretical physics from Columbia University, New York, USA. He has served as an assistant professor of physics at the Free University of Berlin (Germany) and a professor of physics at the University François Rabelais in Tours (France) before moving to the University of Cergy-Pontoise (France), where he is presently an

emeritus professor. Throughout his career, he has worked successively in quantum mechanics, integrable quantum field theory in low dimensions, exactly soluble models in two-dimensional statistical mechanics, quantum superintegrable models, and integral geometry for imaging science. He has published around 100 research papers in peer-reviewed journals.

Contents

**Section 1**

**Section 2**

*by Tuong T. Truong*

*by Nicolae A. Enaki*

Particle System

Systems

**Section 3**

*by Viktor I. Gerasimenko*

**Preface III**

Introduction **1**

**Chapter 1 3**

Many Particle Quantum Features **7**

**Chapter 2 9**

**Chapter 3 31**

**Chapter 4 49**

Structural Quantum Developments **73**

**Chapter 5 75**

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

Processes of Creation and Propagation of Correlations in Large Quantum

Recent Progresses in *Ab Initio* Electronic Structure Calculation toward Understandings of Functional Mechanisms of Biological Macromolecular

Supersymmetric Quantum Mechanics: Two Factorization Schemes and

*by José Socorro García Díaz, Marco A. Reyes, Carlos Villaseñor Mora* 

*by Jiyoung Kang, Takuya Sumi and Masaru Tateno*

Quasi-Exactly Solvable Potentials

*and Edgar Condori Pozo*

Introductory Chapter: Panorama of Contemporary Quantum

Mechanics - Concepts and Applications

### Contents


Preface

Tremendous technological progress has been witnessed in the last few decades and this progress is mainly due to advances made in quantum theory (or quantum mechanics as it was known). Quantum theory was created at the beginning of the twentieth century to decipher the growing number of atomic phenomena. However, quantum theory has introduced many unconventional and non-intuitive concepts, which cannot directly be exploited for technological applications. Over the years, thanks to the contributions of many scientists, the understanding of quantum theory has vastly improved and this has led to many of the present-day discoveries. The aim of this book is to describe some of the development aspects of quantum theory, which may incite or generate further useful applications. After an introductory chapter, we have focused on particular topics in quantum theory, which are

Chapter 2, by Nicolae A. Enaki, concerns the cooperative spontaneous lasing mechanism and its possible quantum retardation effects. The author considers the effects of cooperative scattering and two-photon resonances on the decay of threelevel systems involving non-linear dipole type interactions. These effects occurs in hydrogen- (or helium-) like atoms with cascade transitions, in which scattering is in concurrence with resonance via dipole-forbidden transitions. Also discussed are interferences between single and two quantum collective transitions of inverted radiative emissions, two particle collective decay rates, time dependence of kinetic processes, correlations between radiative emission sources as well as their behaviors

Chapter 3, by V. I. Gerasimenko, discusses new approaches to the evolution of states of large quantum particle systems by means of marginal correlation operators. It is shown that they are governed by the nonlinear quantum BBGKY hierarchy of equations in the Von Neumann dynamical framework of correlation functions. The non-perturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in large quantum particle systems. Described in detail is the collective behavior of quantum many particle systems by means of a marginal one-particle correlation operator, which is a solution of the generalized quantum kinetic equa-

Chapter 4, by Jiyoung Kang, Takuya Sumi and Masaru Tateno, aims to explain the functional mechanisms of biological macromolecular systems. These mechanisms are due to the electron transfer responsible for the redox regulations and the catalytic reactions for hydrogen metabolism and hydrolysis. Electron transfer induces dramatic rearrangements of electronic structures as well as internal threedimensional structures, which are crucial for biological functions. Two distinct

types of rearrangement triggers are found by two distinct approaches:



discussed in the following chapters of this book.

at short and long periods under retardation effects.

tion with initial conditions, in particular as condensed states.

## Preface

Tremendous technological progress has been witnessed in the last few decades and this progress is mainly due to advances made in quantum theory (or quantum mechanics as it was known). Quantum theory was created at the beginning of the twentieth century to decipher the growing number of atomic phenomena. However, quantum theory has introduced many unconventional and non-intuitive concepts, which cannot directly be exploited for technological applications. Over the years, thanks to the contributions of many scientists, the understanding of quantum theory has vastly improved and this has led to many of the present-day discoveries. The aim of this book is to describe some of the development aspects of quantum theory, which may incite or generate further useful applications. After an introductory chapter, we have focused on particular topics in quantum theory, which are discussed in the following chapters of this book.

Chapter 2, by Nicolae A. Enaki, concerns the cooperative spontaneous lasing mechanism and its possible quantum retardation effects. The author considers the effects of cooperative scattering and two-photon resonances on the decay of threelevel systems involving non-linear dipole type interactions. These effects occurs in hydrogen- (or helium-) like atoms with cascade transitions, in which scattering is in concurrence with resonance via dipole-forbidden transitions. Also discussed are interferences between single and two quantum collective transitions of inverted radiative emissions, two particle collective decay rates, time dependence of kinetic processes, correlations between radiative emission sources as well as their behaviors at short and long periods under retardation effects.

Chapter 3, by V. I. Gerasimenko, discusses new approaches to the evolution of states of large quantum particle systems by means of marginal correlation operators. It is shown that they are governed by the nonlinear quantum BBGKY hierarchy of equations in the Von Neumann dynamical framework of correlation functions. The non-perturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in large quantum particle systems. Described in detail is the collective behavior of quantum many particle systems by means of a marginal one-particle correlation operator, which is a solution of the generalized quantum kinetic equation with initial conditions, in particular as condensed states.

Chapter 4, by Jiyoung Kang, Takuya Sumi and Masaru Tateno, aims to explain the functional mechanisms of biological macromolecular systems. These mechanisms are due to the electron transfer responsible for the redox regulations and the catalytic reactions for hydrogen metabolism and hydrolysis. Electron transfer induces dramatic rearrangements of electronic structures as well as internal threedimensional structures, which are crucial for biological functions. Two distinct types of rearrangement triggers are found by two distinct approaches:


First, redox regulations and catalytic reactions for hydrogen metabolism and hydrolysis by electronic transfer, which are catalyzed by transition metal (4Fe-3S) clusters, are obtained using the first approach. Second, dynamic rearrangements of the electronic structures occurring in the catalytic reaction of RNA-protein complexes have instead emerged from the second approach on hyper parallel supercomputer simulations. Such features are characteristic of the electronic structures in biological macromolecular systems.

Chapter 5, by J Socorro, Marco A. Reyes, Carlos Villasen or Mora and Condori Pozo, presents new aspects of supersymmetric quantum mechanics. This is a theoretic extension of the conventional one-dimensional Schrödinger equation of quantum mechanics *via* the so-called factorization method. Supersymmetry incorporates the fermion-boson symmetry into the theory through the so-called supercharge operators. Although many properties are known, the authors have brought up some features that are susceptible to new insights.

I would like to thank the authors for devoting time and effort to providing high quality texts, which are beneficial to a wide audience in perceiving the realm of quantum theory and its appearance in technology. My appreciations go to Ms Kristina Kardum for her patience and help in guiding this book project through the many steps.

> **Tuong T. Truong** Laboratoire de Physique Théorique et Modélisation, University of Cergy-Pontoise, Cergy-Pontoise, France

> > **1**

Section 1

Introduction

Section 1 Introduction

**3**

**Chapter 1**

Applications

*Tuong T. Truong*

**1. Introduction**

decades.

Introductory Chapter: Panorama

Quantum mechanics has been around for more than a century. Since its birth at the beginning of the twentieth century, it has undergone a tremendous growth. But, it is only now that quantum mechanics has emerged in our daily life. This is just a normal evolution for any branch of physics. Take for example, the electromagnetic theory. It came into existence with the stunning work of James Clerk Maxwell in 1865 [1], which predicted the existence of radio waves, and this has led to the tremendous development of electronics throughout the twentieth century in the fields

of communication, detection, and transmission of information and data. Quantum mechanics is the physics of subatomic phenomena, which has remained a mysterious domain for a long time. Its laws have bewildered many because they are quite counter-intuitive. Its development has started with the very concept of "quantization," which entails the absorption as well as the emission of energy in discrete amounts and not continuous as it is usually perceived in classical physics. This milestone principle, established by Max Planck toward the end of the nineteenth century, has started a golden age of discoveries during almost three

**2. The present role of quantum mechanics in technologies**

engineering (quantum electronics/spintronics).

Since then, quantum principles have been at the foundations of our day-to-day technologies, such as the transistor, computer chip, LASER, GPS, NMR imaging system, LED lamps, solar cells, etc. to name a few. The working of transistors is based simultaneously on the quantum description of matter, namely, the waveparticle duality and the Heisenberg uncertainty principle, which is inherent to quantum evolution equations. In recent years, with the appearance of Big data, massive exchange of information, and the ensuing cryptography challenges, it becomes necessary to turn to quantum engineering to find a way out to manage these problems. A full array of "quantum" technologies has been initiated since the seminal paper of R. P. Feynman "Simulating Physics with Computers" [2], in which the notion of quantum computer was introduced as it is built on new quantum

There is no doubt that quantum principles are here to stay and will continue to be at the origin of new technological innovations in coming years. This book

of Contemporary Quantum

Mechanics - Concepts and

#### **Chapter 1**

## Introductory Chapter: Panorama of Contemporary Quantum Mechanics - Concepts and Applications

*Tuong T. Truong*

#### **1. Introduction**

Quantum mechanics has been around for more than a century. Since its birth at the beginning of the twentieth century, it has undergone a tremendous growth. But, it is only now that quantum mechanics has emerged in our daily life. This is just a normal evolution for any branch of physics. Take for example, the electromagnetic theory. It came into existence with the stunning work of James Clerk Maxwell in 1865 [1], which predicted the existence of radio waves, and this has led to the tremendous development of electronics throughout the twentieth century in the fields of communication, detection, and transmission of information and data.

Quantum mechanics is the physics of subatomic phenomena, which has remained a mysterious domain for a long time. Its laws have bewildered many because they are quite counter-intuitive. Its development has started with the very concept of "quantization," which entails the absorption as well as the emission of energy in discrete amounts and not continuous as it is usually perceived in classical physics. This milestone principle, established by Max Planck toward the end of the nineteenth century, has started a golden age of discoveries during almost three decades.

#### **2. The present role of quantum mechanics in technologies**

Since then, quantum principles have been at the foundations of our day-to-day technologies, such as the transistor, computer chip, LASER, GPS, NMR imaging system, LED lamps, solar cells, etc. to name a few. The working of transistors is based simultaneously on the quantum description of matter, namely, the waveparticle duality and the Heisenberg uncertainty principle, which is inherent to quantum evolution equations. In recent years, with the appearance of Big data, massive exchange of information, and the ensuing cryptography challenges, it becomes necessary to turn to quantum engineering to find a way out to manage these problems. A full array of "quantum" technologies has been initiated since the seminal paper of R. P. Feynman "Simulating Physics with Computers" [2], in which the notion of quantum computer was introduced as it is built on new quantum engineering (quantum electronics/spintronics).

There is no doubt that quantum principles are here to stay and will continue to be at the origin of new technological innovations in coming years. This book

is intended to give a first glimpse on a few topics of this fascinating development perspective on the future of the real world and to stimulate research in order to meet the futures challenges. Most urgent is the investigation into the quantum behavior of large systems such as populations of photons or atoms in the regime of Bose condensates in which unexpected properties may arise. In particular, some macroscopic effects may be explained from microscopic levels, thanks to quantum mechanics which governs the evolution rules at the atomic level.

One of the most salient features of modern physics is the inherent existence of hidden symmetry in nature. The discovery of such symmetries is often very fruitful in the sense that it leads to further discoveries and predictions. Strangely enough, even symmetry breaking can also be a source of new phenomena occurrence. This is why since decades, one has sought to make quantum mechanics supersymmetric and the pursuit of supersymmetry in elementary particles is still ongoing these days in large particle accelerators.

Finally, it should be mentioned that quantum mechanics has ushered mankind into the area of fictional reality with the search for the realization and exploitation of the quantum concept of entanglement. In 1935, analyzing the possible outcome of a Gedankenexperiment following the rules of quantum mechanics by Einstein, Podolski, and Rosen has arrived at a paradoxical conclusion, known for a long time as the EPR Paradox [3]. This is because one can predict the value of a dynamical quantity of a system, which has classically nothing to do with a companion system on which measurements are performed. Quantum mechanics coins these systems as "entangled." Nowadays, quantum entanglement has been experimentally demonstrated and considered to be the main ingredient in the working of a quantum computer.

As computing is an exponentially growing activity in science and technology as well as in economics, finance, and management, "classical" computers have reached their limits as far as performance and costs are concerned. Quantum computers which are based on totally new quantum concepts (superposition and entanglement) with their revolutionary capacity for data storage and speed of calculation seem to be the ideal solution to the previous problem. Basic logic gates and circuits are cast and discussed in the language of the so-called quantum bits or q-bits (instead of classical bits) which are their building blocks. They open the way to mastering cyber security in data transfer, Big data mining, and the like.

Thus, our future appeared to be structured by quantum principles via quantum technology and engineering, and this book offers a view of what might be the coming reality we will have to deal with.

#### **Author details**

Tuong T. Truong University of Cergy-Pontoise, Cergy-Pontoise, France

\*Address all correspondence to: tuong.truong@u-cergy.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**5**

*Introductory Chapter: Panorama of Contemporary Quantum Mechanics - Concepts…*

*DOI: http://dx.doi.org/10.5772/intechopen.87971*

[1] Maxwell JC. A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of

[2] Feynman RP. Stimulating physics with computers. International Journal of Theoretical Physics.

[3] Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Physical Review. 1935;**47**(10):777-780

London. 1865;**155**:459-512

1982;**21**(6/7):467-488

**References**

*Introductory Chapter: Panorama of Contemporary Quantum Mechanics - Concepts… DOI: http://dx.doi.org/10.5772/intechopen.87971*

#### **References**

*Panorama of Contemporary Quantum Mechanics - Concepts and Applications*

mechanics which governs the evolution rules at the atomic level.

is intended to give a first glimpse on a few topics of this fascinating development perspective on the future of the real world and to stimulate research in order to meet the futures challenges. Most urgent is the investigation into the quantum behavior of large systems such as populations of photons or atoms in the regime of Bose condensates in which unexpected properties may arise. In particular, some macroscopic effects may be explained from microscopic levels, thanks to quantum

One of the most salient features of modern physics is the inherent existence of hidden symmetry in nature. The discovery of such symmetries is often very fruitful in the sense that it leads to further discoveries and predictions. Strangely enough, even symmetry breaking can also be a source of new phenomena occurrence. This is why since decades, one has sought to make quantum mechanics supersymmetric and the pursuit of supersymmetry in elementary particles is still ongoing these days

Finally, it should be mentioned that quantum mechanics has ushered mankind into the area of fictional reality with the search for the realization and exploitation of the quantum concept of entanglement. In 1935, analyzing the possible outcome of a Gedankenexperiment following the rules of quantum mechanics by Einstein, Podolski, and Rosen has arrived at a paradoxical conclusion, known for a long time as the EPR Paradox [3]. This is because one can predict the value of a dynamical quantity of a system, which has classically nothing to do with a companion system on which measurements are performed. Quantum mechanics coins these systems as "entangled." Nowadays, quantum entanglement has been experimentally demonstrated and considered to be the main ingredient in the working of a quantum

As computing is an exponentially growing activity in science and technology as well as in economics, finance, and management, "classical" computers have reached their limits as far as performance and costs are concerned. Quantum computers which are based on totally new quantum concepts (superposition and entanglement) with their revolutionary capacity for data storage and speed of calculation seem to be the ideal solution to the previous problem. Basic logic gates and circuits are cast and discussed in the language of the so-called quantum bits or q-bits (instead of classical bits) which are their building blocks. They open the way to

Thus, our future appeared to be structured by quantum principles via quantum technology and engineering, and this book offers a view of what might be the com-

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

mastering cyber security in data transfer, Big data mining, and the like.

**4**

**Author details**

ing reality we will have to deal with.

computer.

in large particle accelerators.

Tuong T. Truong

University of Cergy-Pontoise, Cergy-Pontoise, France

provided the original work is properly cited.

\*Address all correspondence to: tuong.truong@u-cergy.fr

[1] Maxwell JC. A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London. 1865;**155**:459-512

[2] Feynman RP. Stimulating physics with computers. International Journal of Theoretical Physics. 1982;**21**(6/7):467-488

[3] Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete? Physical Review. 1935;**47**(10):777-780

Section 2

Many Particle Quantum

Features

7

Section 2

## Many Particle Quantum Features

Chapter 2

Effects

Abstract

Nicolae A. Enaki

tion time between them is studied.

quantum statistical methods

81R15

9

1. Introduction

Cooperative Spontaneous Lasing

and Possible Quantum Retardation

The collective decay effects between the dipole-active three-level subsystems in

the nonlinear interaction with dipole-forbidden transitions, like 2S 1S of hydrogen-like radiators, are proposed, taking into consideration the cooperative exchanges between two species of atoms through the vacuum field in the scattering and the two-photon resonance processes. One of them corresponds to the situation when the total energy of the emitted two photons by the three-level radiator in the cascade configuration enters into the two-photon resonance with another type of dipole-forbidden transitions of hydrogen-like (or helium-like) atoms. The similar situation appears in the cooperative scattering between two species of quantum emitters when the difference of the excited energies of the two dipole-active transitions of the three-level radiators is in the resonance with the dipole-forbidden transitions of the Hydrogen-like radiators. These effects are accompanied by the interference between single- and two-quantum collective transitions of the inverted radiators from the ensemble. The two-particle collective decay rate is defined in the description of the atomic correlation functions taking into consideration the phase retardation between them. The kinetic equations which describe the cooperative processes as the function of time and correlation are obtained. The behavior of the system of radiators at short and long time intervals in comparison with the retarda-

Keywords: 42.50.Fx Cooperative phenomena in quantum optical systems, 32.80.Qk Coherent control of atomic interactions with photons, 03.65.Ud Entanglement and quantum nonlocality, 03.65.Yz Decoherence, open systems,

2000 AMS Subject Classification: Primary 82C10, 81Q15; Secondary 20G42,

The single-photon cooperative emission of the inverted system of radiators proposed by Dicke [1] opens the new possibilities of this phenomenon in the description of decay processes in the multilevel system [2] and multi-photon interaction of radiators with EMF (see, e.g., [3, 4]). The experimental possibilities [3, 4] of nonlinear cooperative interaction of radiators with vacuum field remain in the center of attention of many theoretical models proposed in the last time [5, 6].

#### Chapter 2

## Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

Nicolae A. Enaki

#### Abstract

The collective decay effects between the dipole-active three-level subsystems in the nonlinear interaction with dipole-forbidden transitions, like 2S 1S of hydrogen-like radiators, are proposed, taking into consideration the cooperative exchanges between two species of atoms through the vacuum field in the scattering and the two-photon resonance processes. One of them corresponds to the situation when the total energy of the emitted two photons by the three-level radiator in the cascade configuration enters into the two-photon resonance with another type of dipole-forbidden transitions of hydrogen-like (or helium-like) atoms. The similar situation appears in the cooperative scattering between two species of quantum emitters when the difference of the excited energies of the two dipole-active transitions of the three-level radiators is in the resonance with the dipole-forbidden transitions of the Hydrogen-like radiators. These effects are accompanied by the interference between single- and two-quantum collective transitions of the inverted radiators from the ensemble. The two-particle collective decay rate is defined in the description of the atomic correlation functions taking into consideration the phase retardation between them. The kinetic equations which describe the cooperative processes as the function of time and correlation are obtained. The behavior of the system of radiators at short and long time intervals in comparison with the retardation time between them is studied.

Keywords: 42.50.Fx Cooperative phenomena in quantum optical systems, 32.80.Qk Coherent control of atomic interactions with photons, 03.65.Ud Entanglement and quantum nonlocality, 03.65.Yz Decoherence, open systems, quantum statistical methods

2000 AMS Subject Classification: Primary 82C10, 81Q15; Secondary 20G42, 81R15

#### 1. Introduction

The single-photon cooperative emission of the inverted system of radiators proposed by Dicke [1] opens the new possibilities of this phenomenon in the description of decay processes in the multilevel system [2] and multi-photon interaction of radiators with EMF (see, e.g., [3, 4]). The experimental possibilities [3, 4] of nonlinear cooperative interaction of radiators with vacuum field remain in the center of attention of many theoretical models proposed in the last time [5, 6].

For example, using the classical and quantum approaches in Refs. [7–10], it is given the quantitative description of two-color super-fluorescence, observed in [2]. In the recent experiment [11], the cooperative emission of excited atomic oxygen relatively the transition 3p3<sup>P</sup> ! <sup>3</sup><sup>s</sup> <sup>3</sup>S at wavelength 845nm as a result of two-photon photolysis of atmospheric O<sup>2</sup> followed by two-photon excitation of atomic oxygen by a laser pulse at 226nm is demonstrated.

photon resonance and scattering processes between the dipole-forbidden subsystem D and dipole-active subsystems of Ξ, Λ, and V, respectively. Here, the product of two vacuum polarizations of the atom Ξ (or V, Λ) comes into resonance with the

polarization of the dipole-forbidden transitions of the D atom.

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

DOI: http://dx.doi.org/10.5772/intechopen.83013

Using two small parameters in Section 2, we propose the projection operator method of elimination of the EMF operators from the generalized equation of atomic subsystems in single- and two-photon resonances. The possibilities of two-photon cooperative resonance between three-level radiators situated at a distance compared with the emission wavelength are demonstrated. Following this description the resonance interaction of a dipole-forbidden atom and three-level dipole-active radiator in the cascade configuration is described by the cooperative rate and the exchange integral (13). The similar expression (16)

is obtained in the scattering process of three-level system in V or Λ—

2. Master equation of cooperative exchange between three-level

radiators in two-quantum exchanges

between them.

radiator (see Figure 1A).

atomic and field operators:

11

configurations with dipole-forbidden D subsystem represented in Figure 1. In Section 3 the spontaneous emission for the two radiators in the cascade or scattering resonances is given without the de-correlation of the atomic correlation functions

Let us consider the interaction of three-level subsystems of radiators in V and Ξ configuration with D dipole-forbidden two-level ensemble through the vacuum of EMF. The Ξ three-level subsystem in cascade configuration, prepared in excited state ∣2ξi, can pass into the Dicke super-radiance regime [1] relatively the dipoleactive transitions 2<sup>ξ</sup> ! ιξ ! 1<sup>ξ</sup> at frequencies ω<sup>2</sup> and ω<sup>1</sup> (Figure 1A). According to Figure 1A, the excited D atom relatively the dipole-forbidden transition 2<sup>d</sup> ! 1<sup>d</sup> passes in the ground state ∣1di simultaneously generating two quanta under the influence of cooperative decay of the Ξ three-level subsystem. Two-photon transition of the D-atom takes place through the virtual levels represented by the notations ∣3di with opposite parity relative to the ground ∣1di and excited ∣2di states, respectively. This case corresponds to the situation when the emission frequencies of the dipole-active Ξ radiators and D dipole-forbidden radiators satisfy the resonance condition ω<sup>1</sup> þ ω<sup>2</sup> ¼ 2ω0. Here ω<sup>1</sup> and ω<sup>2</sup> are the transition frequencies of the Ξ dipole-active radiators in Ξ, and ℏω<sup>d</sup> ¼ 2ℏω<sup>0</sup> is the energy distance between the ground ∣nSi and excited ∣ð Þ n þ 1 Si states of the dipole-forbidden transitions of D

The similar cooperative emissions can be observed in the two-quantum resonance interactions between the V (or Λ) three-level radiator in two quanta scattering interactions and the dipole-forbidden transitions of D atoms through the vacuum field (see Figure 1B,C). In this situation, we consider that the dipole-active transitions of the three-level radiator in the V (or Λ) configuration satisfy the scattering condition ω<sup>a</sup> � ω<sup>s</sup> ¼ ω<sup>d</sup> in interaction with the D subsystem. As it is represented in Figure 1B, the cone of the transition energies of the V or Λ dipoleactive three-level atoms must be larger than the dipole-forbidden transition

∣ð Þ n þ 1 Si � ∣nSi of atoms D, so that two-photon resonance between the two dipoleactive transitions of V atom enters in the exact scattering resonance, ω<sup>a</sup> � ω<sup>s</sup> ¼ ωd, with D atom. This nonlinear transition increases with the decreasing of the detuning

The Hamiltonian of the system consists of the free and interaction parts <sup>H</sup> <sup>¼</sup> <sup>H</sup>^ <sup>0</sup> <sup>þ</sup> <sup>H</sup>^ <sup>I</sup>. Here the free part of this Hamiltonian is represented through the

from resonance with virtual ∣3di states of the D two-level system.

Combining single- and two-photon processes, this chapter aims to investigate the cooperative emission of the inverted system of radiators taking into account the resonance between one- and two-photon cooperative transitions of two three-level atomic subsystems represented in Figure 1. In this approach, the two dipole-active species of radiators studied in Refs. [12, 13] are replaced with one three-level atomic subsystem Ξ (or V) inverted relative to the single-photon emission in the resonance with 2S - 1S dipole-forbidden transitions of hydrogen (or He)-like sub-ensemble. This new cooperative effect between two species of radiators occurs when two three-level emitters enter into two-quantum resonances with other emitters of the second ensemble inverted relative dipole-forbidden transition. Similar collectivization processes can amplify (or inhibit) the collective spontaneous emission rate of each atomic sub-ensemble. The sign of exchange integral between the two atoms from different sub-ensembles depends on the retardation time and distance between them. This problem is connected to the possibilities of amplifying of entangled quanta and established the coherence between photon pairs. For this, the cooperative interaction of three-radiator subsystems is proposed in which one of them is inverted relative to the dipole-forbidden transitions, but another inverted dipole-active three-level system ignites this transition.

Taking into consideration the elementary acts of two-photon resonance between radiators, we have demonstrated the increasing of two-photon emission rate in one of the radiator subsystem comparison with traditional two-photon superfluorescence [5]. The mutual influence of two- and single-photon super-fluorescent processes on the two-photon cooperative emission of the inverted subsystem relatively dipole-forbidden transition depends on the position of atoms in the exchange potential. Two possibilities of two- and three-particle exchanges through the vacuum field are represented in Figure 1A–C, taking into consideration the two-

Figure 1.

The resonances between the two-photon transitions of D atomic subsystem and the three-level dipole-active systems in Ξ (A), V (B), and Λ (C) configurations. The three-level atoms are situated at relative distances rdξ, rdλ, and rrv. The exchange energies between the D subsystem in the two-photon resonance ω<sup>0</sup> ¼ ω<sup>1</sup> þ ω<sup>2</sup> with the Ξ subsystem (A) and the scattering resonance 2ω<sup>0</sup> ¼ ω<sup>a</sup> � ω<sup>s</sup> with V subsystem (B) are given by the expressions (14) and (17).

#### Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

photon resonance and scattering processes between the dipole-forbidden subsystem D and dipole-active subsystems of Ξ, Λ, and V, respectively. Here, the product of two vacuum polarizations of the atom Ξ (or V, Λ) comes into resonance with the polarization of the dipole-forbidden transitions of the D atom.

Using two small parameters in Section 2, we propose the projection operator method of elimination of the EMF operators from the generalized equation of atomic subsystems in single- and two-photon resonances. The possibilities of two-photon cooperative resonance between three-level radiators situated at a distance compared with the emission wavelength are demonstrated. Following this description the resonance interaction of a dipole-forbidden atom and three-level dipole-active radiator in the cascade configuration is described by the cooperative rate and the exchange integral (13). The similar expression (16) is obtained in the scattering process of three-level system in V or Λ configurations with dipole-forbidden D subsystem represented in Figure 1. In Section 3 the spontaneous emission for the two radiators in the cascade or scattering resonances is given without the de-correlation of the atomic correlation functions between them.

#### 2. Master equation of cooperative exchange between three-level radiators in two-quantum exchanges

Let us consider the interaction of three-level subsystems of radiators in V and Ξ configuration with D dipole-forbidden two-level ensemble through the vacuum of EMF. The Ξ three-level subsystem in cascade configuration, prepared in excited state ∣2ξi, can pass into the Dicke super-radiance regime [1] relatively the dipoleactive transitions 2<sup>ξ</sup> ! ιξ ! 1<sup>ξ</sup> at frequencies ω<sup>2</sup> and ω<sup>1</sup> (Figure 1A). According to Figure 1A, the excited D atom relatively the dipole-forbidden transition 2<sup>d</sup> ! 1<sup>d</sup> passes in the ground state ∣1di simultaneously generating two quanta under the influence of cooperative decay of the Ξ three-level subsystem. Two-photon transition of the D-atom takes place through the virtual levels represented by the notations ∣3di with opposite parity relative to the ground ∣1di and excited ∣2di states, respectively. This case corresponds to the situation when the emission frequencies of the dipole-active Ξ radiators and D dipole-forbidden radiators satisfy the resonance condition ω<sup>1</sup> þ ω<sup>2</sup> ¼ 2ω0. Here ω<sup>1</sup> and ω<sup>2</sup> are the transition frequencies of the Ξ dipole-active radiators in Ξ, and ℏω<sup>d</sup> ¼ 2ℏω<sup>0</sup> is the energy distance between the ground ∣nSi and excited ∣ð Þ n þ 1 Si states of the dipole-forbidden transitions of D radiator (see Figure 1A).

The similar cooperative emissions can be observed in the two-quantum resonance interactions between the V (or Λ) three-level radiator in two quanta scattering interactions and the dipole-forbidden transitions of D atoms through the vacuum field (see Figure 1B,C). In this situation, we consider that the dipole-active transitions of the three-level radiator in the V (or Λ) configuration satisfy the scattering condition ω<sup>a</sup> � ω<sup>s</sup> ¼ ω<sup>d</sup> in interaction with the D subsystem. As it is represented in Figure 1B, the cone of the transition energies of the V or Λ dipoleactive three-level atoms must be larger than the dipole-forbidden transition ∣ð Þ n þ 1 Si � ∣nSi of atoms D, so that two-photon resonance between the two dipoleactive transitions of V atom enters in the exact scattering resonance, ω<sup>a</sup> � ω<sup>s</sup> ¼ ωd, with D atom. This nonlinear transition increases with the decreasing of the detuning from resonance with virtual ∣3di states of the D two-level system.

The Hamiltonian of the system consists of the free and interaction parts <sup>H</sup> <sup>¼</sup> <sup>H</sup>^ <sup>0</sup> <sup>þ</sup> <sup>H</sup>^ <sup>I</sup>. Here the free part of this Hamiltonian is represented through the atomic and field operators:

For example, using the classical and quantum approaches in Refs. [7–10], it is given the quantitative description of two-color super-fluorescence, observed in [2]. In the recent experiment [11], the cooperative emission of excited atomic oxygen rela-

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

photolysis of atmospheric O<sup>2</sup> followed by two-photon excitation of atomic oxygen

Combining single- and two-photon processes, this chapter aims to investigate the cooperative emission of the inverted system of radiators taking into account the resonance between one- and two-photon cooperative transitions of two three-level atomic subsystems represented in Figure 1. In this approach, the two dipole-active species of radiators studied in Refs. [12, 13] are replaced with one three-level atomic subsystem Ξ (or V) inverted relative to the single-photon emission in the resonance with 2S - 1S dipole-forbidden transitions of hydrogen (or He)-like sub-ensemble. This new cooperative effect between two species of radiators occurs when two three-level emitters enter into two-quantum resonances with other emitters of the second ensemble inverted relative dipole-forbidden transition. Similar collectivization processes can amplify (or inhibit) the collective spontaneous emission rate of each atomic sub-ensemble. The sign of exchange integral between the two atoms from different sub-ensembles depends on the retardation time and distance between them. This problem is connected to the possibilities of amplifying of entangled quanta and established the coherence between photon pairs. For this, the cooperative interaction of three-radiator subsystems is proposed in which one of them is inverted relative to the dipole-forbidden transitions, but another inverted

Taking into consideration the elementary acts of two-photon resonance between radiators, we have demonstrated the increasing of two-photon emission rate in one

fluorescence [5]. The mutual influence of two- and single-photon super-fluorescent processes on the two-photon cooperative emission of the inverted subsystem relatively dipole-forbidden transition depends on the position of atoms in the exchange potential. Two possibilities of two- and three-particle exchanges through the vacuum field are represented in Figure 1A–C, taking into consideration the two-

The resonances between the two-photon transitions of D atomic subsystem and the three-level dipole-active systems in Ξ (A), V (B), and Λ (C) configurations. The three-level atoms are situated at relative distances rdξ, rdλ, and rrv. The exchange energies between the D subsystem in the two-photon resonance ω<sup>0</sup> ¼ ω<sup>1</sup> þ ω<sup>2</sup> with the Ξ subsystem (A) and the scattering resonance 2ω<sup>0</sup> ¼ ω<sup>a</sup> � ω<sup>s</sup> with V subsystem (B) are given by the expressions

of the radiator subsystem comparison with traditional two-photon super-

<sup>3</sup>S at wavelength 845nm as a result of two-photon

tively the transition 3p3<sup>P</sup> ! <sup>3</sup><sup>s</sup>

Figure 1.

10

(14) and (17).

by a laser pulse at 226nm is demonstrated.

dipole-active three-level system ignites this transition.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

$$\begin{split} \hat{H}\_{0} &= \sum\_{k} \hbar o\_{k} \hat{a}\_{k}^{\dagger} \hat{a}\_{k} + \hbar \sum\_{m=1}^{N} o\_{d} \hat{D}\_{xm} - \sum\_{a=s\_{1}}^{2} \sum\_{d=1}^{N\_{\downarrow}} \hbar o\_{a} \hat{\Lambda}\_{al}^{a} \\ &+ \sum\_{a=s\_{1}}^{2} \sum\_{d=1}^{N\_{\downarrow}} \hbar o\_{a} \hat{V}\_{al}^{a} + \sum\_{a=1,} \sum\_{j=1}^{N\_{\downarrow}} \hbar (-1)^{a} o\_{a} \hat{\Xi}\_{aj}^{a}, \end{split} \tag{1}$$

This interaction is expressed by two-photon emission terms ε2H^ <sup>b</sup><sup>þ</sup>

and possible scattering of an emitted photon by the Ξ and V subsystems

βj ; U^ <sup>β</sup><sup>0</sup> α0 l h i <sup>¼</sup> <sup>δ</sup>l,j <sup>U</sup>^ <sup>α</sup>

parison with single-photon interaction of Ξ and V atoms with vacuum field

� �, the nonlinear interaction of <sup>D</sup> two-level subsystem with EMF in twophoton and scattering interaction is described by the interaction constants and

� �

� �

levels of the D atom. In the definition of the interaction parts of the Hamiltonian (2) and (3), we introduced the fictive small parameters ε<sup>1</sup> and ε<sup>2</sup> which will help us to establish the contributions of the second and third orders in two-photon decay

In this section the conditions for which the pure super-fluorescence of the small number of radiators [14, 15] in the subsystems Ξ, V, and D enters into interaction during the delay time of cooperative spontaneous emission of each subsystem are considered, so that inhomogeneous broadening of excited atomic states can be neglected, τ<sup>i</sup> ≪ T2,i. Here τ<sup>i</sup> ¼ τ0=Ni is the collective time for which the polarization

<sup>2</sup><sup>ℏ</sup> <sup>ω</sup><sup>32</sup> <sup>þ</sup> <sup>ω</sup><sup>k</sup><sup>1</sup> ð Þ <sup>þ</sup>

<sup>ℏ</sup> <sup>ω</sup><sup>32</sup> � <sup>ω</sup><sup>k</sup><sup>1</sup> ð Þ <sup>þ</sup>

of the i subsystem becomes macroscopic; T2,i is the de-phasing time of the

achieved using laser cooling method [17, 18] for three atomic ensembles represented in Figure 1A,B. Let us suppose that delay time of the super-radiant pulse is less than T2,i; we will drop the terms connected with de-phasing time T2,i from the kinetic equations. In order to estimate the three-particle cooperative interaction, we will examine the situation in which one- and two-quantum interactions with the EMF bath are taken into account simultaneously. In this case it is necessary to eliminate from the density matrix equation the boson operators of EMF in nonlinear interaction with atomic subsystem. In comparison with the paper [12], here we will take into consideration the two-quantum effects connected with the influence of three-level atomic systems V and Ξ on the two-photon spontaneous emission of dipole-forbidden D subsystem. In this case instead of two dipole-active atoms, we can take into consideration only one three-level atom in two-photon

subsystem i, which includes the reciprocal inhomogeneous and Doppler-broadened line-width, i � Ξ, V, and D (see, e.g., the papers [15, 16]). These conditions can be

Let P be the projection operator for the complete density matrix �ρð Þt on the vector basis of a free EMF subsystem ρsðÞ¼ t P�ρð Þt and �ρbðÞ¼ t P�ρð Þt , where �ρ<sup>s</sup>

and �ρbð Þt are slower and rapidly oscillating parts of the density matrix, respectively,

<sup>β</sup><sup>j</sup> is equivalent with <sup>V</sup> and <sup>Ξ</sup> operators, <sup>V</sup>^ <sup>α</sup>

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

d31; g<sup>k</sup><sup>1</sup> � � d32; <sup>g</sup><sup>k</sup><sup>2</sup>

d31; g<sup>k</sup><sup>1</sup> � � d32; <sup>g</sup><sup>k</sup><sup>2</sup>

� h i ¼ �D^ <sup>l</sup>

inversion D^ lz together with lowering and exciting D^ <sup>j</sup>

active three-level subsystems are described by the operators of Uð Þ3 algebra, which

�

a^k<sup>1</sup> . The excitation and lowering operators of V, Λ, and Ξ dipole-

δl,j and D^ <sup>þ</sup>

d31; g<sup>k</sup><sup>2</sup> � � d32; <sup>g</sup><sup>k</sup><sup>1</sup>

d31; g<sup>k</sup><sup>2</sup> � � d32; <sup>g</sup><sup>k</sup><sup>1</sup>

2πℏωk=V p and di,j is dipole momentum transitions between the

α0 j

<sup>β</sup><sup>j</sup> and Ξ^ <sup>α</sup> βj

> <sup>l</sup> ; <sup>D</sup>^ <sup>þ</sup> m

δβ, <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>U</sup>^ <sup>β</sup>

β0 j δα,α<sup>0</sup> n o. Here the

� operators of D subsystem

� �

� �

<sup>2</sup><sup>ℏ</sup> <sup>ω</sup><sup>31</sup> � <sup>ω</sup><sup>k</sup><sup>2</sup> ð Þ ,

<sup>ℏ</sup> <sup>ω</sup><sup>31</sup> <sup>þ</sup> <sup>ω</sup><sup>k</sup><sup>1</sup> ð Þ ,

h i <sup>¼</sup> <sup>2</sup>δl,mD^ lz. In com-

<sup>ε</sup>2H� <sup>I</sup>2s� � <sup>D</sup>^ <sup>∓</sup>

operator U^ <sup>α</sup>

second order:

μi,j ; g<sup>k</sup>

rates.

13

ma^† k2

satisfy the commutation relations <sup>U</sup>^ <sup>α</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83013

belongs to SUð Þ<sup>2</sup> algebra: <sup>D</sup>^ lz; <sup>D</sup>^ <sup>j</sup>

qbð Þ¼ k1; k2

ð Þ¼ k1; k2

resonance with dipole-forbidden system.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qs

where g<sup>k</sup> ¼ ελ

<sup>I</sup><sup>2</sup> � <sup>D</sup>^ �

, respectively. The

ma^† k2 a^† k1

ð Þt

where N, Nξ, Nλ, and Nv are the number of atoms in the D, Ξ, Λ, and V subsystems, respectively; the energies of first and second levels of the Ξ, Λ, and V three-level subsystems are measured from the third intermediate state ∣ιi. The operators Ξ<sup>1</sup> 1,j , Ξ<sup>ι</sup> ι,j , and Ξ<sup>2</sup> <sup>2</sup>,j describe the population of the ground, intermediary, and excited states of the Ξ atom. The population operators of two excited and ground states V^ <sup>2</sup> 2,j , V^ <sup>1</sup> 1,j , and V^ <sup>ι</sup> <sup>ι</sup>,j can be introduced for the three-level atom in V configuration too. The similar expressions for two ground and one excited state can be introduced for Λ three-level atomic configuration Λ^ <sup>2</sup> 2,j , Λ^ <sup>1</sup> 1,j , and Λ^ <sup>ι</sup> ι,j , respectively. The D atoms are considered as a two-level system, the state energy positions of which are measured from the middle point between the excited and ground states, respectively, Dz,j <sup>¼</sup> <sup>D</sup><sup>2</sup> <sup>2</sup>,j � <sup>D</sup><sup>1</sup> 1,j � �=2. The first term of the Hamiltonian describes the free energy of EMF, the k � k, λ modes of which is initially considered in the vacuum state <sup>∣</sup>0ki. Here <sup>a</sup>^<sup>k</sup> and <sup>a</sup>^† <sup>k</sup> are annihilation and creation operators of EMF photons with wave vector k, polarization ελ, and the frequency ωk, which satisfy the commutation relation a^† <sup>k</sup>; <sup>a</sup>^† k 0 h i <sup>¼</sup> <sup>δ</sup>k,k 0 .

Taking into consideration the conservation energy laws, ℏð Þ¼ ω<sup>1</sup> þ ω<sup>2</sup> 2ℏω<sup>0</sup> and ℏð Þ¼ ω<sup>a</sup> � ω<sup>s</sup> 2ℏω<sup>0</sup> (according to Figure 1A–C, respectively), we introduce the interaction Hamiltonian <sup>H</sup>^ <sup>I</sup> <sup>¼</sup> <sup>H</sup>^ <sup>I</sup><sup>1</sup> <sup>þ</sup> <sup>H</sup>^ <sup>I</sup><sup>2</sup> of the <sup>Ξ</sup>, <sup>Λ</sup>, <sup>V</sup>, and <sup>D</sup> subsystems with free EMF. Here H^ <sup>I</sup><sup>1</sup> describes the single-photon interaction of three-level atoms in the Ξ, V, and Λ configurations with a vacuum of EMF:

$$\begin{split} \hat{H}\_{I1} &= -\sum\_{k} \sum\_{j=1}^{N\_{\boldsymbol{l}}} \Big[ \left( \mu\_{\boldsymbol{1}t}, \mathbf{g}\_{k} \right) \hat{\Xi}\_{\mathbf{j}\boldsymbol{l}}^{\boldsymbol{l}} + \left( \mu\_{\boldsymbol{2}t}, \mathbf{g}\_{k} \right) \hat{\Xi}\_{\mathbf{j}\boldsymbol{l}}^{2} \Big] \hat{a}\_{k} \exp\left[ \boldsymbol{i} \left( \mathbf{k}, \mathbf{r}\_{\boldsymbol{l}} \right) \right] \\ &- \sum\_{k} \sum\_{l=1}^{N\_{\boldsymbol{l}}} \Big[ \left( \mu\_{\boldsymbol{1}t}, \mathbf{g}\_{k} \right) \hat{\Lambda}\_{\mathbf{il}}^{\boldsymbol{l}} + \left( \mu\_{\boldsymbol{2}t}, \mathbf{g}\_{k} \right) \hat{\Lambda}\_{\mathbf{il}}^{\boldsymbol{l}} \Big] \hat{a}\_{k} \exp\left[ \boldsymbol{i} \left( \mathbf{k}, \mathbf{r}\_{\boldsymbol{l}} \right) \right] \\ &- \sum\_{k} \sum\_{l=1}^{N\_{\boldsymbol{r}}} \Big[ \left( \mu\_{\boldsymbol{r}1}, \mathbf{g}\_{k} \right) \hat{\mathbf{V}}\_{\mathbf{il}}^{\boldsymbol{1}} + \left( \mu\_{\boldsymbol{r}2}, \mathbf{g}\_{k} \right) \hat{\mathbf{V}}\_{\mathbf{il}}^{2} \Big] \hat{a}\_{k} \exp\left[ \boldsymbol{i} \left( \mathbf{k}, \mathbf{r}\_{\boldsymbol{l}} \right) \right] + H.c. \end{split} \tag{2}$$

where ε1H^ <sup>Ξ</sup>1� <sup>I</sup><sup>1</sup> � <sup>Ξ</sup>^<sup>ι</sup> 1j a^<sup>k</sup> and ε1H^ <sup>Ξ</sup>2<sup>¼</sup> <sup>I</sup><sup>1</sup> � <sup>Ξ</sup>^<sup>2</sup> ιj a^<sup>k</sup> represent the two-photon cascade excitation of <sup>Ξ</sup> atom through the intermediary state <sup>∣</sup>ιi; <sup>ε</sup>1H^ <sup>S</sup>� <sup>I</sup><sup>1</sup> � <sup>Λ</sup>^ <sup>ι</sup> 2,j a^<sup>k</sup> (or ε1H^ <sup>S</sup>� <sup>I</sup><sup>1</sup> � <sup>V</sup>^ <sup>2</sup> ι,j a^k) and ε1H^ <sup>A</sup>� <sup>I</sup><sup>1</sup> � <sup>Λ</sup>^<sup>ι</sup> 1,j a^<sup>k</sup> (ε1H^ <sup>A</sup>� <sup>I</sup><sup>1</sup> � <sup>V</sup>^ <sup>1</sup> ι,j a^k) describe the excitation of Λ (or V) atom with the absorption of the photons with the energies ℏω<sup>s</sup> and ℏωa, respectively. μi,j is dipole momentum transitions between the i and j states of the atoms. The second part of interaction Hamiltonian, H^ <sup>I</sup>2, describes the nonlinear interaction of the dipole-forbidden transition of D two-level system with vacuum field:

$$\begin{split} \hat{H}\_{I2} &= \sum\_{k\_1,k\_2} \sum\_{m=1}^{N} \left[ q\_s(\mathbf{k}\_1,\mathbf{k}\_2) \hat{D}\_m^- \hat{a}\_{k\_2}^\dagger \hat{a}\_{k\_1} (\mathbf{1} - \delta\_{k\_1,k\_2}) \exp\left[i(\mathbf{k}\_1 - \mathbf{k}\_2, \mathbf{r}\_m)\right] \right] \\ &- q\_b(\mathbf{k}\_1,\mathbf{k}\_2) \hat{D}\_m^+ \hat{a}\_{k\_2} \hat{a}\_{k\_1} \exp\left[i(\mathbf{k}\_1 + \mathbf{k}\_2, \mathbf{r}\_m)\right] + H.c. \end{split} \tag{3}$$

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

This interaction is expressed by two-photon emission terms ε2H^ <sup>b</sup><sup>þ</sup> <sup>I</sup><sup>2</sup> � <sup>D</sup>^ � ma^† k2 a^† k1 and possible scattering of an emitted photon by the Ξ and V subsystems <sup>ε</sup>2H� <sup>I</sup>2s� � <sup>D</sup>^ <sup>∓</sup> ma^† k2 a^k<sup>1</sup> . The excitation and lowering operators of V, Λ, and Ξ dipoleactive three-level subsystems are described by the operators of Uð Þ3 algebra, which satisfy the commutation relations <sup>U</sup>^ <sup>α</sup> βj ; U^ <sup>β</sup><sup>0</sup> α0 l h i <sup>¼</sup> <sup>δ</sup>l,j <sup>U</sup>^ <sup>α</sup> α0 j δβ, <sup>β</sup><sup>0</sup> <sup>þ</sup> <sup>U</sup>^ <sup>β</sup> β0 j δα,α<sup>0</sup> n o. Here the operator U^ <sup>α</sup> <sup>β</sup><sup>j</sup> is equivalent with <sup>V</sup> and <sup>Ξ</sup> operators, <sup>V</sup>^ <sup>α</sup> <sup>β</sup><sup>j</sup> and Ξ^ <sup>α</sup> βj , respectively. The inversion D^ lz together with lowering and exciting D^ <sup>j</sup> � operators of D subsystem belongs to SUð Þ<sup>2</sup> algebra: <sup>D</sup>^ lz; <sup>D</sup>^ <sup>j</sup> � h i ¼ �D^ <sup>l</sup> � δl,j and D^ <sup>þ</sup> <sup>l</sup> ; <sup>D</sup>^ <sup>þ</sup> m h i <sup>¼</sup> <sup>2</sup>δl,mD^ lz. In comparison with single-photon interaction of Ξ and V atoms with vacuum field μi,j ; g<sup>k</sup> � �, the nonlinear interaction of <sup>D</sup> two-level subsystem with EMF in twophoton and scattering interaction is described by the interaction constants and second order:

$$q\_{b}(\mathbf{k}\_{1},\mathbf{k}\_{2}) = \frac{\left(\mathbf{d}\_{31},\mathbf{g}\_{\mathbf{k}\_{1}}\right)\left(\mathbf{d}\_{32},\mathbf{g}\_{\mathbf{k}\_{2}}\right)}{2\hbar(\alpha\_{32}+\alpha\_{\mathbf{k}\_{1}})} + \frac{\left(\mathbf{d}\_{31},\mathbf{g}\_{\mathbf{k}\_{2}}\right)\left(\mathbf{d}\_{32},\mathbf{g}\_{\mathbf{k}\_{1}}\right)}{2\hbar(\alpha\_{31}-\alpha\_{\mathbf{k}\_{2}})},$$

$$q\_{\iota}(\mathbf{k}\_{1},\mathbf{k}\_{2}) = \frac{\left(\mathbf{d}\_{31},\mathbf{g}\_{\mathbf{k}\_{1}}\right)\left(\mathbf{d}\_{32},\mathbf{g}\_{\mathbf{k}\_{2}}\right)}{\hbar(\alpha\_{32}-\alpha\_{\mathbf{k}\_{1}})} + \frac{\left(\mathbf{d}\_{31},\mathbf{g}\_{\mathbf{k}\_{2}}\right)\left(\mathbf{d}\_{32},\mathbf{g}\_{\mathbf{k}\_{1}}\right)}{\hbar(\alpha\_{31}+\alpha\_{\mathbf{k}\_{1}})},$$

where g<sup>k</sup> ¼ ελ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πℏωk=V p and di,j is dipole momentum transitions between the levels of the D atom. In the definition of the interaction parts of the Hamiltonian (2) and (3), we introduced the fictive small parameters ε<sup>1</sup> and ε<sup>2</sup> which will help us to establish the contributions of the second and third orders in two-photon decay rates.

In this section the conditions for which the pure super-fluorescence of the small number of radiators [14, 15] in the subsystems Ξ, V, and D enters into interaction during the delay time of cooperative spontaneous emission of each subsystem are considered, so that inhomogeneous broadening of excited atomic states can be neglected, τ<sup>i</sup> ≪ T2,i. Here τ<sup>i</sup> ¼ τ0=Ni is the collective time for which the polarization of the i subsystem becomes macroscopic; T2,i is the de-phasing time of the subsystem i, which includes the reciprocal inhomogeneous and Doppler-broadened line-width, i � Ξ, V, and D (see, e.g., the papers [15, 16]). These conditions can be achieved using laser cooling method [17, 18] for three atomic ensembles represented in Figure 1A,B. Let us suppose that delay time of the super-radiant pulse is less than T2,i; we will drop the terms connected with de-phasing time T2,i from the kinetic equations. In order to estimate the three-particle cooperative interaction, we will examine the situation in which one- and two-quantum interactions with the EMF bath are taken into account simultaneously. In this case it is necessary to eliminate from the density matrix equation the boson operators of EMF in nonlinear interaction with atomic subsystem. In comparison with the paper [12], here we will take into consideration the two-quantum effects connected with the influence of three-level atomic systems V and Ξ on the two-photon spontaneous emission of dipole-forbidden D subsystem. In this case instead of two dipole-active atoms, we can take into consideration only one three-level atom in two-photon resonance with dipole-forbidden system.

Let P be the projection operator for the complete density matrix �ρð Þt on the vector basis of a free EMF subsystem ρsðÞ¼ t P�ρð Þt and �ρbðÞ¼ t P�ρð Þt , where �ρ<sup>s</sup> ð Þt and �ρbð Þt are slower and rapidly oscillating parts of the density matrix, respectively,

<sup>H</sup>^ <sup>0</sup> <sup>¼</sup> <sup>∑</sup> k

, and Ξ<sup>2</sup>

, and V^ <sup>ι</sup>

be introduced for Λ three-level atomic configuration Λ^ <sup>2</sup>

operators Ξ<sup>1</sup>

ground states V^ <sup>2</sup>

1,j , Ξ<sup>ι</sup> ι,j

> 2,j , V^ <sup>1</sup> 1,j

states, respectively, Dz,j <sup>¼</sup> <sup>D</sup><sup>2</sup>

<sup>H</sup>^ <sup>I</sup><sup>1</sup> ¼ � <sup>∑</sup>

where ε1H^ <sup>Ξ</sup>1�

<sup>H</sup>^ <sup>I</sup><sup>2</sup> <sup>¼</sup> <sup>∑</sup> <sup>k</sup>1, <sup>k</sup><sup>2</sup> ∑ N m¼1 ½qs

<sup>I</sup><sup>1</sup> � <sup>V</sup>^ <sup>2</sup> ι,j

ε1H^ <sup>S</sup>�

field:

12

k ∑ N<sup>ξ</sup>

� ∑ k ∑ N<sup>λ</sup> l¼1

� ∑ k ∑ Nv l¼1

<sup>I</sup><sup>1</sup> � <sup>Ξ</sup>^<sup>ι</sup> 1j

a^k) and ε1H^ <sup>A</sup>�

� qbð Þ <sup>k</sup>1; <sup>k</sup><sup>2</sup> <sup>D</sup>^ <sup>þ</sup>

j¼1

in the vacuum state <sup>∣</sup>0ki. Here <sup>a</sup>^<sup>k</sup> and <sup>a</sup>^†

V, and Λ configurations with a vacuum of EMF:

μ1ι; g<sup>k</sup> � �Ξ^<sup>ι</sup>

μι1; g<sup>k</sup> � �Λ^<sup>ι</sup>

μι1; g<sup>k</sup> � �V^ <sup>1</sup>

a^<sup>k</sup> and ε1H^ <sup>Ξ</sup>2<sup>¼</sup>

<sup>I</sup><sup>1</sup> � <sup>Λ</sup>^<sup>ι</sup> 1,j

ð Þ k1; k2 <sup>D</sup>^ �

ma^† k2

excitation of <sup>Ξ</sup> atom through the intermediary state <sup>∣</sup>ιi; <sup>ε</sup>1H^ <sup>S</sup>�

satisfy the commutation relation a^†

þ ∑ 2 <sup>α</sup>¼s, <sup>a</sup> ∑ Nv l¼1

ℏωka^†

<sup>k</sup>a^<sup>k</sup> þ ℏ ∑ N m¼1

> <sup>α</sup><sup>l</sup> þ ∑ <sup>α</sup>¼1, <sup>2</sup> ∑ N<sup>ξ</sup>

where N, Nξ, Nλ, and Nv are the number of atoms in the D, Ξ, Λ, and V subsystems, respectively; the energies of first and second levels of the Ξ, Λ, and V three-level subsystems are measured from the third intermediate state ∣ιi. The

and excited states of the Ξ atom. The population operators of two excited and

configuration too. The similar expressions for two ground and one excited state can

tively. The D atoms are considered as a two-level system, the state energy positions of which are measured from the middle point between the excited and ground

describes the free energy of EMF, the k � k, λ modes of which is initially considered

¼ δk,k 0 .

ιj

2l

ιl

<sup>I</sup><sup>1</sup> � <sup>V</sup>^ <sup>1</sup> ι,j

<sup>m</sup>a^<sup>k</sup>2a^<sup>k</sup><sup>1</sup> exp ½ið Þ k1 þ k2;r<sup>m</sup> �� þ H:c:

a^<sup>k</sup> exp i k;r<sup>j</sup> � � � �

a^<sup>k</sup> exp ið Þ k;r<sup>l</sup> ½ �

a^<sup>k</sup><sup>1</sup> 1 � δ<sup>k</sup>1,k<sup>2</sup> ð Þ exp ½ � ið Þ k1 � k2;r<sup>m</sup>

a^<sup>k</sup> exp ið Þ k;r<sup>l</sup> ½ �þ H:c:,

a^<sup>k</sup> represent the two-photon cascade

<sup>I</sup><sup>1</sup> � <sup>Λ</sup>^ <sup>ι</sup> 2,j a^<sup>k</sup> (or

a^k) describe the excitation of Λ

Taking into consideration the conservation energy laws, ℏð Þ¼ ω<sup>1</sup> þ ω<sup>2</sup> 2ℏω<sup>0</sup> and

EMF photons with wave vector k, polarization ελ, and the frequency ωk, which

ℏð Þ¼ ω<sup>a</sup> � ω<sup>s</sup> 2ℏω<sup>0</sup> (according to Figure 1A–C, respectively), we introduce the interaction Hamiltonian <sup>H</sup>^ <sup>I</sup> <sup>¼</sup> <sup>H</sup>^ <sup>I</sup><sup>1</sup> <sup>þ</sup> <sup>H</sup>^ <sup>I</sup><sup>2</sup> of the <sup>Ξ</sup>, <sup>Λ</sup>, <sup>V</sup>, and <sup>D</sup> subsystems with free EMF. Here H^ <sup>I</sup><sup>1</sup> describes the single-photon interaction of three-level atoms in the Ξ,

<sup>k</sup>; <sup>a</sup>^† k 0 h i

<sup>1</sup><sup>j</sup> þ μ2ι; g<sup>k</sup> � �Ξ^<sup>2</sup>

<sup>1</sup><sup>l</sup> þ μι2; g<sup>k</sup> � �Λ^<sup>ι</sup>

<sup>ι</sup><sup>l</sup> þ μι2; g<sup>k</sup> � �V^ <sup>2</sup>

<sup>I</sup><sup>1</sup> � <sup>Ξ</sup>^<sup>2</sup> ιj

a^<sup>k</sup> (ε1H^ <sup>A</sup>�

(or V) atom with the absorption of the photons with the energies ℏω<sup>s</sup> and ℏωa, respectively. μi,j is dipole momentum transitions between the i and j states of the atoms. The second part of interaction Hamiltonian, H^ <sup>I</sup>2, describes the nonlinear interaction of the dipole-forbidden transition of D two-level system with vacuum

h i

h i

h i

ℏωαV^ <sup>α</sup>

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

<sup>2</sup>,j � <sup>D</sup><sup>1</sup> 1,j � � <sup>ω</sup>dD^ zm � <sup>∑</sup>

j¼1

2 <sup>α</sup>¼s, <sup>a</sup> ∑ N<sup>λ</sup> l¼1

> ωαΞ^<sup>α</sup> αj ,

<sup>ℏ</sup>ð Þ �<sup>1</sup> <sup>α</sup>

<sup>2</sup>,j describe the population of the ground, intermediary,

<sup>ι</sup>,j can be introduced for the three-level atom in V

2,j , Λ^ <sup>1</sup> 1,j

=2. The first term of the Hamiltonian

<sup>k</sup> are annihilation and creation operators of

, and Λ^ <sup>ι</sup> ι,j

, respec-

(2)

(3)

ℏωαΛ^ <sup>α</sup> αl

(1)

<sup>P</sup> <sup>¼</sup> <sup>1</sup> � <sup>P</sup>. It can be shown that <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup> and PP <sup>¼</sup> 0. Recognizing that for <sup>t</sup> <sup>¼</sup> <sup>0</sup> an electronic subsystem does not interact with the EMF, we define the projection operator P ¼ �ρphð Þ 0 ⊗Trphf g ⋯ , where the trace is taking over the photon states and �ρphð Þ¼ 0 j i 0 h j 0 represents the density matrix of the vacuum of EMF. In this case one can represent the slow part of density matrix through the density matrix W t � ðÞ¼ Trphf g �ρð Þ<sup>t</sup> of the atomic subsystem �ρ<sup>s</sup> ðÞ¼ <sup>t</sup> �ρph⊗W t � ð Þ, where <sup>W</sup>� ð Þ¼ <sup>0</sup> Trph�ρð Þ¼ <sup>0</sup> �ρrð Þ <sup>0</sup> is the density matrix of the prepared state of the atomic subsystem. The equations for the matrix �ρ<sup>s</sup> ð Þt and �ρbð Þt are

$$\frac{\partial \check{\rho}\_s(t)}{\partial t} = -i\mathcal{P}L\_I(t)\{\check{\rho}\_s(t) + \check{\rho}\_b(t)\},\tag{4}$$

that the projection of the operator product ε<sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83013

boson operators, Trph �ρ0a�†

by the following diagrams:

Here LΞ1�

<sup>I</sup><sup>2</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>b</sup>�

where Ls�

15

Lb�

Δρ<sup>b</sup> <sup>3</sup> <sup>¼</sup> <sup>i</sup>λ<sup>3</sup>

the operator product <sup>P</sup>H� <sup>I</sup>1H� <sup>I</sup>2<sup>P</sup> <sup>¼</sup> 0.

L^I<sup>1</sup> and L^I<sup>2</sup> must be found from the terms like PL^I1L^I2L^I<sup>1</sup>

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

k1 a�k2 a�k3 †

ðt

0 dτ<sup>1</sup> ðτ1

<sup>þ</sup>PL^<sup>R</sup>�

þPL^<sup>Ξ</sup>2�

þPL^<sup>Ξ</sup>1�

þPL^<sup>b</sup><sup>þ</sup>

<sup>þ</sup>PL^<sup>b</sup><sup>þ</sup>

tors in the single- and two-quantum interactions.

0 dτ<sup>1</sup> ðτ1

0

<sup>þ</sup>PLS<sup>þ</sup>

<sup>þ</sup>PLA�

<sup>þ</sup>PLS<sup>þ</sup>

<sup>þ</sup>PLs<sup>þ</sup>

<sup>þ</sup>PLs<sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þt ; …

scarpering process of D atomic subsystem and L<sup>S</sup>�

dτ<sup>2</sup> PLA�

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> LA�

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> <sup>L</sup><sup>s</sup><sup>þ</sup>

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> Ls<sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ<sup>t</sup> <sup>L</sup><sup>A</sup>�

<sup>I</sup><sup>2</sup> ð Þ<sup>t</sup> <sup>L</sup><sup>S</sup><sup>þ</sup>

<sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>Ξ</sup>1�

<sup>I</sup><sup>2</sup> ð Þt ; …

Δρ<sup>s</sup> <sup>3</sup> ¼ i ðt

<sup>I</sup><sup>2</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>s</sup>�

0

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> <sup>L</sup>^<sup>S</sup>�

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> <sup>L</sup>^<sup>b</sup><sup>þ</sup>

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> <sup>L</sup>^<sup>b</sup><sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ<sup>t</sup> <sup>L</sup>^<sup>Ξ</sup>1�

<sup>I</sup><sup>2</sup> ð Þ<sup>t</sup> <sup>L</sup>^<sup>Ξ</sup>2�

<sup>I</sup><sup>1</sup> ð Þt ; … h i=ℏ, <sup>L</sup>Ξ2�

<sup>d</sup>τ<sup>2</sup> <sup>P</sup>L^<sup>Ξ</sup>1�

n

In the third order of the small parameters εi, the contribution of Liouville operator

Following this procedure of calculation of mean value of boson operators, it is observed that the two-photon resonance represented in Figure 1A can be described

> <sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> ^ L^<sup>Ξ</sup>2�

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>b</sup><sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>Ξ</sup>1�

<sup>I</sup><sup>2</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>Ξ</sup>2�

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>Ξ</sup>2�

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>Ξ</sup>1�

the Ξ and D atoms expressed through EMF annihilation and atomic exciting opera-

<sup>I</sup><sup>1</sup> ð Þ<sup>t</sup> <sup>L</sup><sup>S</sup><sup>þ</sup>

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> Ls<sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> LS<sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> <sup>L</sup><sup>A</sup>�

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> LS<sup>þ</sup>

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> <sup>L</sup><sup>A</sup>�

h i=<sup>ℏ</sup> is the Liouville parts for two-photon

The scattering resonance can be represented by the diagrams in which the conservation law ω<sup>a</sup> � ω<sup>s</sup> ¼ 2ω<sup>0</sup> must take place as represented in Figure 1B:

n o <sup>¼</sup> 0, which corresponds to the projection of

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> <sup>L</sup>^<sup>b</sup><sup>þ</sup>

<sup>I</sup><sup>2</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ <sup>t</sup> � <sup>τ</sup><sup>2</sup> Ls<sup>þ</sup> <sup>I</sup><sup>2</sup> <sup>ð</sup>t� � <sup>τ</sup>1Þρsð Þ<sup>t</sup>

<sup>I</sup><sup>2</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>2</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>2</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>2</sup> ρsð Þt

<sup>I</sup><sup>1</sup> ð Þ t � τ<sup>2</sup> ρsð Þg þ t H:c:,

<sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>S</sup>�

<sup>I</sup><sup>1</sup> ð Þt ; … h i=<sup>ℏ</sup> and

<sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>Ξ</sup>2�

h i=<sup>ℏ</sup> represent the Liouville operators of the interaction part of

o

<sup>I</sup><sup>2</sup> ð Þt ; … h i=ℏ, and

þ H:c:

two-photon resonances between the single- and two-photon transitions in the three-level atomic systems described by the Hamiltonian part (2) and (3), respectively. It is not difficult to observe that second-order decomposition on the interaction Hamiltonian gives zero contributions in the correlations between the Ξ, V, and D subsystems. This follows from the zero value of the trace of the odd number of

<sup>2</sup>ε1PH� <sup>I</sup>1H� <sup>I</sup>2H� <sup>I</sup><sup>2</sup> takes the zero value too.

<sup>I</sup><sup>2</sup> ð Þ t � τ<sup>1</sup> ρsð Þt

(8)

(9)

^�ρsð Þ<sup>t</sup> , which corresponds to

$$\frac{\partial \check{\rho}\_b(t)}{\partial t} = -i\overline{\mathcal{P}}L\_l(\mathbf{t})\{\check{\rho}\_s(\mathbf{t}) + \check{\rho}\_b(\mathbf{t})\},\tag{5}$$

where <sup>L</sup>^IðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>I</sup>1ð Þ<sup>t</sup> ; … � �=<sup>ℏ</sup> <sup>+</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>I</sup>2ð Þ<sup>t</sup> ; … � �=<sup>ℏ</sup> is the interaction part of Liouville operator. Following the known procedure of elimination of the rapidly oscillating part of the density matrix, we integrate Eq. (5) with respect to �ρbð Þt and substitute the resulting solution in Eq. (4). After this procedure we obtain the expression

$$\frac{\partial \check{\rho}\_s(t)}{\partial t} = -P \int\_0^t d\tau L\_I(t) U(t, t - \tau) L\_I(t - \tau) \rho\_s(t - \tau), \tag{6}$$

where the two-time evolution operator is represented by the T product U t � ð Þ¼ ; <sup>t</sup> � <sup>τ</sup> <sup>T</sup> exp �i<sup>P</sup> <sup>Ð</sup><sup>t</sup> t�τ dτ1LIð Þτ � �. In comparison with well-known procedure of the decomposition on the small parameter ε of the right-hand site of expression (6), here we have two parameters ε<sup>1</sup> and ε2. The quantum correlation between the singleand two-photon interactions of atoms through the vacuum of the EMF can be found in the third order of the expansion on the small parameter product ε<sup>2</sup> <sup>1</sup>ε<sup>2</sup> of the right-hand side of Eq. (6). Indeed considering the second and third order of the expansion on the small parameters <sup>ε</sup><sup>1</sup> and <sup>ε</sup>2, we represent the evolution operators U t � ð Þ ; <sup>t</sup> � <sup>τ</sup> and �ρs ð Þ <sup>t</sup> � <sup>τ</sup> in the following approximate form U t � ð Þ ; <sup>t</sup> � <sup>τ</sup> <sup>≈</sup><sup>1</sup> � <sup>i</sup><sup>P</sup> <sup>Ð</sup><sup>t</sup> t�τ dτ1Lið Þ τ<sup>1</sup> and �ρs ð Þ¼ t � τ �ρ<sup>s</sup> ðÞþ<sup>t</sup> <sup>P</sup> <sup>Ð</sup> τ <sup>d</sup>τ1L^ið Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> Ðt�τ1 <sup>d</sup>τ2L^ið Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> � <sup>τ</sup><sup>2</sup> �ρ<sup>s</sup> ð Þ t � τ<sup>1</sup> � τ<sup>2</sup> . Upon

 $\text{If } \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot, \cdot$  substitution of this expression in Eq. (6), in the third order of small parameter  $\lambda$ , the equation for  $\rho\_{\boldsymbol{\epsilon}}(t)$  becomes

$$\frac{\partial}{\partial t}\check{\rho}\_{\varepsilon}(t) = -\mathcal{P}\left\{d\tau\_{1}\hat{L}\_{i}(t)\left\{\hat{L}\_{i}(t-\tau\_{1}) - i\int\_{t-\tau\_{1}}^{t}d\tau\_{2}\hat{L}\_{i}(\tau\_{2})\right\}\hat{L}\_{i}(t-\tau\_{1})\check{\rho}\_{\varepsilon}(t). \tag{7}$$

Representing the Liouville operator, <sup>L</sup>^Ið Þ<sup>t</sup> , through single-, LI1ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>I</sup>1ð Þ<sup>t</sup> ; … � �=ℏ, and two-photon, <sup>λ</sup>LI2ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>I</sup>2ð Þ<sup>t</sup> ; … � �=ℏ, interaction parts, we can observe that in the third order on the decomposition on interaction Hamiltonian, the main contribution to the right-hand site of Eq. (7) gives the terms proportional to the ε<sup>2</sup> <sup>1</sup>ε2. Indeed, taking into consideration that the trace of an odd number of boson operator is zero, Trph <sup>ρ</sup>0a�† k1 a�k2 †a�<sup>k</sup><sup>3</sup> a�k4 a�k5 n o <sup>¼</sup> 0, it is not difficult to observe

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

that the projection of the operator product ε<sup>2</sup> <sup>2</sup>ε1PH� <sup>I</sup>1H� <sup>I</sup>2H� <sup>I</sup><sup>2</sup> takes the zero value too. In the third order of the small parameters εi, the contribution of Liouville operator L^I<sup>1</sup> and L^I<sup>2</sup> must be found from the terms like PL^I1L^I2L^I<sup>1</sup> ^�ρsð Þ<sup>t</sup> , which corresponds to two-photon resonances between the single- and two-photon transitions in the three-level atomic systems described by the Hamiltonian part (2) and (3), respectively. It is not difficult to observe that second-order decomposition on the interaction Hamiltonian gives zero contributions in the correlations between the Ξ, V, and D subsystems. This follows from the zero value of the trace of the odd number of boson operators, Trph �ρ0a�† k1 a�k2 a�k3 † n o <sup>¼</sup> 0, which corresponds to the projection of the operator product <sup>P</sup>H� <sup>I</sup>1H� <sup>I</sup>2<sup>P</sup> <sup>¼</sup> 0.

Following this procedure of calculation of mean value of boson operators, it is observed that the two-photon resonance represented in Figure 1A can be described by the following diagrams:

$$\begin{split} \Delta \rho\_{3}^{b} &= i \hat{\boldsymbol{\beta}}^{b} \Big| \int\_{0}^{\tau\_{1}} \hat{\boldsymbol{\sigma}}\_{1} \Big| \mathcal{D} \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{12}^{b+}(t-\tau\_{1}) \rho\_{i}(t) \\ &+ \mathcal{P} \hat{\boldsymbol{L}}\_{11}^{\Sigma\_{1}-}(t) \hat{\boldsymbol{L}}\_{11}^{S\_{1}-}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{12}^{b+}(t-\tau\_{1}) \rho\_{i}(t) \\ &+ \mathcal{P} \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t) \hat{\boldsymbol{L}}\_{12}^{b+}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{1}) \rho\_{i}(t) \\ &+ \mathcal{P} \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t) \hat{\boldsymbol{L}}\_{12}^{b+}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{1}) \rho\_{i}(t) \\ &+ \mathcal{P} \hat{\boldsymbol{L}}\_{12}^{b+}(t) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{1}) \rho\_{i}(t) \\ &+ \mathcal{P} \hat{\boldsymbol{L}}\_{12}^{b+}(t) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{2}) \hat{\boldsymbol{L}}\_{11}^{\Xi\_{1}-}(t-\tau\_{1}) \rho\_{i}(t) \Big$$

Here LΞ1� <sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>Ξ</sup>1� <sup>I</sup><sup>1</sup> ð Þt ; … h i=ℏ, <sup>L</sup>Ξ2� <sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>Ξ</sup>2� <sup>I</sup><sup>2</sup> ð Þt ; … h i=ℏ, and Lb� <sup>I</sup><sup>2</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>b</sup>� <sup>I</sup><sup>2</sup> ð Þt ; … h i=<sup>ℏ</sup> represent the Liouville operators of the interaction part of the Ξ and D atoms expressed through EMF annihilation and atomic exciting operators in the single- and two-quantum interactions.

The scattering resonance can be represented by the diagrams in which the conservation law ω<sup>a</sup> � ω<sup>s</sup> ¼ 2ω<sup>0</sup> must take place as represented in Figure 1B:

$$\begin{split} \Delta \rho\_{3}^{\*} &= i \left[ d\tau\_{1} \right]\_{0}^{\tau\_{1}} d\tau\_{2} \Big/ \langle \mathcal{P} L\_{11}^{A-}(t) L\_{11}^{S+}(t-\tau\_{2}) L\_{12}^{s+}(t-\tau\_{1}) \rho\_{s}(t) \Big| \\ &+ \mathcal{P} L\_{11}^{S+}(t) L\_{11}^{A-}(t-\tau\_{2}) L\_{12}^{s+}(t-\tau\_{1}) \rho\_{s}(t) \\ &+ \mathcal{P} L\_{11}^{A-}(t) L\_{12}^{s+}(t-\tau\_{1}) L\_{11}^{S+}(t-\tau\_{2}) \rho\_{s}(t) \\ &+ \mathcal{P} L\_{11}^{S+}(t) L\_{12}^{s+}(t-\tau\_{1}) L\_{11}^{A-}(t-\tau\_{2}) \rho\_{s}(t) \\ &+ \mathcal{P} L\_{12}^{S+}(t) L\_{11}^{A-}(t-\tau\_{1}) L\_{11}^{S+}(t-\tau\_{2}) \rho\_{s}(t) \\ &+ \mathcal{P} L\_{12}^{s+}(t) L\_{11}^{S+}(t-\tau\_{1}) L\_{11}^{A-}(t-\tau\_{2}) \rho\_{s}(t) \rangle + H.c., \end{split} \tag{9}$$

where Ls� <sup>I</sup><sup>2</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>s</sup>� <sup>I</sup><sup>2</sup> ð Þt ; … h i=<sup>ℏ</sup> is the Liouville parts for two-photon scarpering process of D atomic subsystem and L<sup>S</sup>� <sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>S</sup>� <sup>I</sup><sup>1</sup> ð Þt ; … h i=<sup>ℏ</sup> and

<sup>P</sup> <sup>¼</sup> <sup>1</sup> � <sup>P</sup>. It can be shown that <sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>P</sup> and PP <sup>¼</sup> 0. Recognizing that for <sup>t</sup> <sup>¼</sup> <sup>0</sup> an electronic subsystem does not interact with the EMF, we define the projection operator P ¼ �ρphð Þ 0 ⊗Trphf g ⋯ , where the trace is taking over the photon states and �ρphð Þ¼ 0 j i 0 h j 0 represents the density matrix of the vacuum of EMF. In this case one can represent the slow part of density matrix through the density matrix

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

<sup>W</sup>� ð Þ¼ <sup>0</sup> Trph�ρð Þ¼ <sup>0</sup> �ρrð Þ <sup>0</sup> is the density matrix of the prepared state of the atomic

<sup>∂</sup><sup>t</sup> ¼ �iPLIð Þ<sup>t</sup> �ρ<sup>s</sup>

<sup>∂</sup><sup>t</sup> ¼ �iPLIð Þ<sup>t</sup> �ρ<sup>s</sup>

where <sup>L</sup>^IðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>I</sup>1ð Þ<sup>t</sup> ; … � �=<sup>ℏ</sup> <sup>+</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>I</sup>2ð Þ<sup>t</sup> ; … � �=<sup>ℏ</sup> is the interaction part of Liouville operator. Following the known procedure of elimination of the rapidly oscillating part of the density matrix, we integrate Eq. (5) with respect to �ρbð Þt and substitute the resulting solution in Eq. (4). After this procedure we obtain the

where the two-time evolution operator is represented by the T product

of the decomposition on the small parameter ε of the right-hand site of expression (6), here we have two parameters ε<sup>1</sup> and ε2. The quantum correlation between the singleand two-photon interactions of atoms through the vacuum of the EMF can be found in

side of Eq. (6). Indeed considering the second and third order of the expansion on

substitution of this expression in Eq. (6), in the third order of small parameter λ, the

ðÞ¼ <sup>t</sup> �ρph⊗W t � ð Þ, where

dτLIð Þt U tð Þ ; t � τ LIð Þ t � τ ρsð Þ t � τ , (6)

� ð Þ ; <sup>t</sup> � <sup>τ</sup> <sup>≈</sup><sup>1</sup> � <sup>i</sup><sup>P</sup> <sup>Ð</sup><sup>t</sup>

<sup>d</sup>τ2L^ið Þ <sup>τ</sup><sup>2</sup>

9 = ;

<sup>d</sup>τ2L^ið Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> � <sup>τ</sup><sup>2</sup> �ρ<sup>s</sup>

ðt

t�τ<sup>1</sup>

<sup>1</sup>ε2. Indeed, taking into consideration that the trace of an odd number of

. In comparison with well-known procedure

t�τ

ð Þ t � τ<sup>1</sup> � τ<sup>2</sup> . Upon

¼ 0, it is not difficult to observe

<sup>L</sup>^ið Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup> �ρsð Þ<sup>t</sup> : (7)

<sup>1</sup>ε<sup>2</sup> of the right-hand

� ð Þ ; <sup>t</sup> � <sup>τ</sup> and

dτ1Lið Þ τ<sup>1</sup> and

ðÞþ<sup>t</sup> �ρbð Þ<sup>t</sup> � �, (4)

ð Þþ<sup>t</sup> �ρbð Þ<sup>t</sup> � �, (5)

ð Þt and �ρbð Þt are

W t � ðÞ¼ Trphf g �ρð Þ<sup>t</sup> of the atomic subsystem �ρ<sup>s</sup>

∂�ρs ð Þt

<sup>∂</sup>�ρbð Þ<sup>t</sup>

ðt

0

dτ1LIð Þτ � �

the third order of the expansion on the small parameter product ε<sup>2</sup>

<sup>d</sup>τ1L^ið Þ<sup>t</sup> <sup>L</sup>^ið Þ� <sup>t</sup> � <sup>τ</sup><sup>1</sup> <sup>i</sup>

k1 a�k2 †a�<sup>k</sup><sup>3</sup> a�k4 a�k5

8 < :

the small parameters ε<sup>1</sup> and ε2, we represent the evolution operators U t

Ðt�τ1 0

Representing the Liouville operator, <sup>L</sup>^Ið Þ<sup>t</sup> , through single-, LI1ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>I</sup>1ð Þ<sup>t</sup> ; … � �=ℏ, and two-photon, <sup>λ</sup>LI2ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>2</sup> <sup>H</sup>� <sup>I</sup>2ð Þ<sup>t</sup> ; … � �=ℏ, interaction parts, we can observe that in the third order on the decomposition on interaction Hamiltonian, the main contribution to the right-hand site of Eq. (7) gives the terms proportional

n o

t�τ

ð Þ t � τ in the following approximate form U t

<sup>d</sup>τ1L^ið Þ <sup>t</sup> � <sup>τ</sup><sup>1</sup>

τ 0

ðt

0

boson operator is zero, Trph <sup>ρ</sup>0a�†

ðÞþ<sup>t</sup> <sup>P</sup> <sup>Ð</sup>

equation for ρsð Þt becomes

ðÞ¼� t P

subsystem. The equations for the matrix �ρ<sup>s</sup>

∂�ρs ð Þt <sup>∂</sup><sup>t</sup> ¼ �<sup>P</sup>

� ð Þ¼ ; <sup>t</sup> � <sup>τ</sup> <sup>T</sup> exp �i<sup>P</sup> <sup>Ð</sup><sup>t</sup>

expression

U t

�ρs

�ρs

ð Þ¼ t � τ �ρ<sup>s</sup>

∂ ∂t �ρs

to the ε<sup>2</sup>

14

LA� <sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>A</sup>� <sup>I</sup><sup>1</sup> ð Þt ; … h i=<sup>ℏ</sup> correspond to the single-photon transitions in <sup>Ξ</sup> atomic subsystem described by the Hamiltonian parts (3) and (2), respectively.

So that after the trace on the EMF variables, we obtain Tr ρ^pha�k<sup>1</sup> a�k3 a�k2 †a�k<sup>4</sup> † n o ¼ δk1k<sup>2</sup> δk3,k<sup>4</sup> þ δk1,k<sup>4</sup> δk3,k<sup>2</sup> � �, Tr <sup>ρ</sup>^pha�k<sup>1</sup> a�k2 † n o <sup>¼</sup> <sup>δ</sup>k1k<sup>2</sup> , and Tr <sup>ρ</sup>pha�† k2 a�k4 †a�k<sup>1</sup> a�k3 n o <sup>¼</sup> 0. We found the correlations between Ξ, V, and D atomic subsystem represented in Figure 1.

We found the correlations between Ξ, V, and D atomic subsystem represented in the Figure 1. Following projection technique procedures developed in Refs. [5, 13, 19], we find the terms of in the right-hand side of the master equation (7)–(9) for three species of radiators in interaction

$$\frac{d\check{W}(t)}{dt} = \frac{d\check{W}\_0(t)}{dt} + \frac{d\check{W}\_{21b}(t)}{dt} + \frac{d\check{W}\_{21s}(t)}{dt}.\tag{10}$$

dW <sup>21</sup>bð Þ<sup>t</sup>

dt ¼ � <sup>i</sup>

4τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83013

� <sup>D</sup> �

<sup>þ</sup> <sup>U</sup> <sup>∗</sup>

þ ½<sup>Ξ</sup><sup>ι</sup> 1,j ; <sup>½</sup><sup>Ξ</sup><sup>2</sup> ι,l

� <sup>D</sup> �

þ H:c:

<sup>¼</sup> <sup>4</sup> 3 � �<sup>2</sup> ω<sup>3</sup>

1 τb 12d

Here 1=τ<sup>b</sup>

D subsystem.

Figure 2.

17

l¼1 ∑ N<sup>ξ</sup>

<sup>m</sup>;W t ð Þ<sup>Ξ</sup><sup>2</sup>

<sup>b</sup> ð Þ <sup>j</sup>; <sup>l</sup>; <sup>m</sup> <sup>Ξ</sup><sup>2</sup>

mW t ð Þ<sup>Ξ</sup><sup>2</sup> ιl ; Ξι 1,j h i <sup>þ</sup> <sup>D</sup> �

Ubð Þ¼ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> exp �iω1rmj=<sup>c</sup> � �Vbð Þ <sup>j</sup>; <sup>l</sup>; <sup>m</sup> :

Here for ω<sup>s</sup> ≃ωr, we have found the following integrals:

<sup>s</sup>ð Þ <sup>ω</sup><sup>r</sup> <sup>3</sup>

2ℏ<sup>2</sup> c6

Vbð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>≃</sup> <sup>c</sup><sup>2</sup>½ � exp ½ �� �iω2rml=<sup>c</sup> <sup>1</sup> exp <sup>i</sup>ω1rjm=<sup>c</sup> � � � <sup>1</sup> � �

j¼1

ι,l Ξι 1,j h i <sup>þ</sup> <sup>D</sup> �

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

ι,l ; <sup>Ξ</sup><sup>ι</sup> 1,j ; <sup>D</sup> <sup>m</sup>�W t ð Þ h i h i

; <sup>D</sup> <sup>m</sup>�W t ð Þ��<sup>o</sup>

fUbð Þ m; l

n o h i

n o h i

μι2μι1d23d<sup>31</sup>

ω1ω2rjmrml

subsystems and one atom from D ensemble situated at the relatively small distance rjl ≪ λs rð Þ. Vbð Þ j; m; l is the exchange integral which describes the influence of the m atom from D ensemble on the single-photon transitions of the j and l radiators from the Ξ subsystem. Ubð Þ j; m; l is the inverse process of the cooperative action of j and l radiators from the Ξ ensemble on the two-photon transitions of m radiator from the

The real part of exchange integral Vb, defined in expression (14), is plotted as a function of relative distance

between radiators, X ¼ ω0r=c, and relative displacement, Δ ¼ ð Þ ω<sup>1</sup> � ω<sup>0</sup> =ω0.

<sup>12</sup><sup>d</sup> is the three-particle cooperative emission rate of two atoms from Ξ

<sup>m</sup>;W t ð Þ<sup>Ξ</sup><sup>ι</sup>

mW t ð Þ<sup>Ξ</sup><sup>ι</sup>

� <sup>i</sup> 2τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup>

1,j Ξ2 ι,l

j¼1 ∑ N<sup>ξ</sup>

1,j ; <sup>Ξ</sup><sup>2</sup> ι,l

1 ω<sup>32</sup> þ ω<sup>2</sup>

l¼1

þ

� �,

Vbð Þ j; m; l

1 ω<sup>31</sup> þ ω<sup>1</sup>

,

(12)

(13)

First term describes the cooperative single- and two-photon effects in each subsystem, respectively. Second term describes the exchanges between the singlephoton processes of Ξ three-level subsystem and the two-photon transitions of the D radiators as this is represented in Figure 1A. The third term describes the scattering effect of the two radiators represented in Figure 1B.

All parameters and collective exchange integrals between the three-level radiators in V configuration and dipole-forbidden two-level system D are defined in the literature [1–12]:

dW� <sup>0</sup>ð Þ<sup>t</sup> dt <sup>¼</sup> <sup>1</sup> 2τι, <sup>1</sup> ∑ N<sup>ξ</sup> l,j¼<sup>1</sup> <sup>χ</sup>1ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>�<sup>1</sup> ι,j ;W t � ð ÞΞ�<sup>ι</sup> 1,l h i <sup>þ</sup> 1 2τι, <sup>2</sup> ∑ N<sup>ξ</sup> l,j¼<sup>1</sup> <sup>χ</sup>2ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>�<sup>ι</sup> 2,j ;W t � ð ÞΞ�<sup>2</sup> ι,l h i þ 1 2τι,a ∑ Nv l,j¼<sup>1</sup> <sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>� <sup>ι</sup> 1,j ;W t � ð ÞV� <sup>1</sup> ι,l h i <sup>þ</sup> 1 2τι,s ∑ Nv l,j¼<sup>1</sup> <sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>� <sup>ι</sup> 2,j ;W t � ð ÞV� <sup>2</sup> ι,l h i þ 1 2τι,s ∑ N<sup>λ</sup> l,j¼<sup>1</sup> <sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>Λ</sup>� <sup>2</sup> ι,j ;W t � ð ÞΛ� <sup>ι</sup> 2,l h i <sup>þ</sup> 1 2τι,a ∑ N<sup>λ</sup> l,j¼<sup>1</sup> <sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>Λ</sup>� <sup>2</sup> ι,j ;W t � ð ÞΛ� <sup>ι</sup> 2,l h i þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup> <sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>� � <sup>j</sup> ;W t � ð ÞD� <sup>l</sup> þ h i <sup>þ</sup> <sup>H</sup>:c:, (11)

where τι,<sup>α</sup> <sup>¼</sup> <sup>3</sup>ℏc<sup>3</sup><sup>=</sup> <sup>4</sup>μ<sup>2</sup> α,ι ω3 α � � is the spontaneous emission time of the dipoleactive transitions ∣αi ! ∣ιi of three-level atom in Ξ and V configurations and <sup>τ</sup><sup>d</sup> <sup>¼</sup> <sup>π</sup>3<sup>2</sup> ℏ2 c<sup>6</sup>= 4<sup>2</sup> ω7 0d2 23d<sup>2</sup> 31q<sup>2</sup> <sup>b</sup>ð Þ ω0;ω<sup>0</sup> � � is the two-photon spontaneous emission rate in the D atomic subsystem. This equation can be used for the description of interaction between the dipole-forbidden and dipole-active subsystems of radiators. For comparison of the real parts of the single- and two-photon exchange integrals, we can observe that the second decreases inversely proportional to the square distance rJl between two <sup>D</sup> radiators: Re χα ½ � ð Þ <sup>j</sup>; <sup>l</sup> = sin ωαrj,l=<sup>c</sup> � �<sup>=</sup> ωαrj,l=<sup>c</sup> � � and Re <sup>χ</sup><sup>d</sup> ½ �� ð Þ <sup>j</sup>; <sup>l</sup> sin <sup>2</sup> <sup>ω</sup>0rj,l=<sup>c</sup> � �<sup>=</sup> <sup>ω</sup>0rj,l=<sup>c</sup> � �<sup>2</sup> .

Following the two-parameter approach projection technique proposed in Ref. [13], HI1 � ε<sup>1</sup> and HI2 � ε2, we easily found the three-particle exchanges between the radiators represented in Figure 1A described by master equation

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

$$\begin{split} \frac{d\dot{W}\_{21b}(t)}{dt} &= -\frac{i}{4r\_{12d}^{b}} \sum\_{m=1}^{N} \sum\_{i=1}^{N} \{U\_b(m,l) \\ &\times \left\{ \left[ \breve{D}\_{m}^{-}, \breve{W}(t) \breve{\Xi}\_{\ast,l}^{2} \breve{\Xi}\_{\mathbb{1},j}^{\prime} \right] + \left[ \breve{D}\_{m}^{-}, \breve{W}(t) \breve{\Xi}\_{\mathbb{1},j}^{\prime} \breve{\Xi}\_{\ast,l}^{2} \right] \right\} \\ &+ U\_b^\*(j,l,m) \left[ \breve{\Xi}\_{\mathbb{1},l}^{2}, \left[ \breve{\Xi}\_{\mathbb{1},j}^{\prime}, \breve{D}\_{m}^{-} - \breve{W}(t) \right] \right] \\ &+ \left[ \breve{\Xi}\_{\mathbb{1},j}^{\prime}, \breve{\Xi}\_{\mathbb{1},l}^{2}, \breve{D}\_{m} - \breve{W}(t) \right] \left\{ -\frac{i}{2t\_{12d}^{b}} \sum\_{m=1}^{N} \sum\_{i=1}^{N} \sum\_{l} V\_{b}(j,m,l) \\ &\times \left\{ \left[ \breve{D}\_{m}^{-} \breve{W}(t) \breve{\Xi}\_{\mathbb{1},j}^{2}, \breve{\Xi}\_{\mathbb{1},j}^{\prime} \right] + \left[ \breve{D}\_{m}^{-} \breve{W}(t) \breve{\Xi}\_{\mathbb{1},j}^{\prime}, \breve{\Xi}\_{\mathbb{1},l}^{2} \right] \right\} \\ &+ H.c. \end{split} \tag{12}$$

Here for ω<sup>s</sup> ≃ωr, we have found the following integrals:

$$\frac{1}{\tau\_{12d}^b} = \left\{\frac{4}{3}\right\}^2 \frac{\alpha\_s^3 (\alpha\_r)^3 \mu\_{12} \mu\_{11} d\_{23} d\_{31}}{2\hbar^2 c^6} \left[\frac{1}{\alpha\_{32} + \alpha\_2} + \frac{1}{\alpha\_{31} + \alpha\_1}\right],$$

$$V\_b(j, m, l) \simeq \frac{c^2 \left[\exp\left[-i\alpha\_2 r\_{ml}/c\right] - 1\right] \left[\exp\left[i\alpha\_1 r\_{jm}/c\right] - 1\right]}{\alpha\_1 \alpha\_2 r\_{jm} r\_{ml}},\tag{13}$$

$$U\_b(j, m, l) = \exp\left[-i\alpha\_1 r\_{mj}/c\right] V\_b(j, l, m).$$

Here 1=τ<sup>b</sup> <sup>12</sup><sup>d</sup> is the three-particle cooperative emission rate of two atoms from Ξ subsystems and one atom from D ensemble situated at the relatively small distance rjl ≪ λs rð Þ. Vbð Þ j; m; l is the exchange integral which describes the influence of the m atom from D ensemble on the single-photon transitions of the j and l radiators from the Ξ subsystem. Ubð Þ j; m; l is the inverse process of the cooperative action of j and l radiators from the Ξ ensemble on the two-photon transitions of m radiator from the D subsystem.

#### Figure 2.

The real part of exchange integral Vb, defined in expression (14), is plotted as a function of relative distance between radiators, X ¼ ω0r=c, and relative displacement, Δ ¼ ð Þ ω<sup>1</sup> � ω<sup>0</sup> =ω0.

LA�

Figure 1.

<sup>I</sup><sup>1</sup> ðÞ¼ <sup>t</sup> <sup>ε</sup><sup>1</sup> <sup>H</sup>� <sup>A</sup>�

¼ δk1k<sup>2</sup> δk3,k<sup>4</sup> þ δk1,k<sup>4</sup> δk3,k<sup>2</sup>

the literature [1–12]:

2τι, <sup>1</sup>

þ 1 2τι,a

þ 1 2τι,s ∑ N<sup>λ</sup> l,j¼<sup>1</sup>

þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup>

where τι,<sup>α</sup> <sup>¼</sup> <sup>3</sup>ℏc<sup>3</sup><sup>=</sup> <sup>4</sup>μ<sup>2</sup>

Re <sup>χ</sup><sup>d</sup> ½ �� ð Þ <sup>j</sup>; <sup>l</sup> sin <sup>2</sup> <sup>ω</sup>0rj,l=<sup>c</sup> � �<sup>=</sup> <sup>ω</sup>0rj,l=<sup>c</sup> � �<sup>2</sup>

<sup>τ</sup><sup>d</sup> <sup>¼</sup> <sup>π</sup>3<sup>2</sup>

16

ℏ2 c<sup>6</sup>= 4<sup>2</sup> ω7 0d2 23d<sup>2</sup> 31q<sup>2</sup>

∑ N<sup>ξ</sup> l,j¼<sup>1</sup>

> ∑ Nv l,j¼<sup>1</sup>

dW� <sup>0</sup>ð Þ<sup>t</sup> dt <sup>¼</sup> <sup>1</sup>

<sup>I</sup><sup>1</sup> ð Þt ; … h i

� �, Tr <sup>ρ</sup>^pha�k<sup>1</sup>

(7)–(9) for three species of radiators in interaction

dW t � ð Þ

dt <sup>¼</sup> dW� <sup>0</sup>ð Þ<sup>t</sup>

scattering effect of the two radiators represented in Figure 1B.

<sup>χ</sup>1ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>�<sup>1</sup>

<sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>� <sup>ι</sup>

<sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>Λ</sup>� <sup>2</sup>

<sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>� �

<sup>b</sup>ð Þ ω0;ω<sup>0</sup>

α,ι ω3 α ι,j

dt þ

First term describes the cooperative single- and two-photon effects in each subsystem, respectively. Second term describes the exchanges between the singlephoton processes of Ξ three-level subsystem and the two-photon transitions of the D radiators as this is represented in Figure 1A. The third term describes the

All parameters and collective exchange integrals between the three-level radiators in V configuration and dipole-forbidden two-level system D are defined in

;W t � ð ÞΞ�<sup>ι</sup>

h i

1,j

ι,j

1,l

;W t � ð ÞV� <sup>1</sup> ι,l

h i

;W t � ð ÞΛ� <sup>ι</sup>

h i

<sup>j</sup> ;W t � ð ÞD� <sup>l</sup>

active transitions ∣αi ! ∣ιi of three-level atom in Ξ and V configurations and

rate in the D atomic subsystem. This equation can be used for the description of interaction between the dipole-forbidden and dipole-active subsystems of radiators. For comparison of the real parts of the single- and two-photon exchange integrals, we can observe that the second decreases inversely proportional to the square distance rJl between two <sup>D</sup> radiators: Re χα ½ � ð Þ <sup>j</sup>; <sup>l</sup> = sin ωαrj,l=<sup>c</sup> � �<sup>=</sup> ωαrj,l=<sup>c</sup> � � and

. Following the two-parameter approach projection technique proposed in Ref. [13], HI1 � ε<sup>1</sup> and HI2 � ε2, we easily found the three-particle exchanges between

the radiators represented in Figure 1A described by master equation

h i

2,l

þ

� � is the two-photon spontaneous emission

þ 1 2τι, <sup>2</sup>

> þ 1 2τι,s ∑ Nv l,j¼<sup>1</sup>

þ 1 2τι,a

þ H:c:,

� � is the spontaneous emission time of the dipole-

=ℏ correspond to the single-photon transitions in Ξ atomic

<sup>¼</sup> <sup>δ</sup>k1k<sup>2</sup> , and Tr <sup>ρ</sup>pha�†

dW� <sup>21</sup><sup>s</sup> ð Þt

∑ N<sup>ξ</sup> l,j¼<sup>1</sup>

> ∑ N<sup>λ</sup> l,j¼<sup>1</sup>

<sup>χ</sup>2ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>�<sup>ι</sup>

<sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>� <sup>ι</sup>

<sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>Λ</sup>� <sup>2</sup>

2,j

;W t � ð ÞΞ�<sup>2</sup> ι,l

> ;W t � ð ÞV� <sup>2</sup> ι,l

h i

;W t � ð ÞΛ� <sup>ι</sup>

h i

2,l

(11)

h i

2,j

ι,j

a�k3 a�k2 †a�k<sup>4</sup> †

n o

dt : (10)

k2 a�k4 †a�k<sup>1</sup> a�k3

n o

¼ 0.

subsystem described by the Hamiltonian parts (3) and (2), respectively. So that after the trace on the EMF variables, we obtain Tr ρ^pha�k<sup>1</sup>

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

a�k2 † n o

We found the correlations between Ξ, V, and D atomic subsystem represented in

We found the correlations between Ξ, V, and D atomic subsystem represented in the Figure 1. Following projection technique procedures developed in Refs. [5, 13, 19], we find the terms of in the right-hand side of the master equation

> dW� <sup>21</sup>bð Þ<sup>t</sup> dt þ

For two atoms represented in Figure 1A, the simple exchange integral between these radiators can be obtained from expression (13):

$$W\_b = \frac{\lambda\_1 \lambda\_2 [\exp\left(-2i\pi r/\lambda\_2\right) - 1][\exp\left(2i\pi r/\lambda\_1\right) - 1]}{\left(2\pi r\right)^2},\tag{14}$$

where

V� 1 <sup>ι</sup>,j ! <sup>Λ</sup>^ <sup>ι</sup> 1j : <sup>V</sup>� <sup>ι</sup>

sions (16).

ε<sup>2</sup> � qbð Þ k1; k<sup>2</sup> or qs

photon emission, ε<sup>2</sup>

ε2

ε2

19

1 τs sad

the influence of D subsystem.

<sup>2</sup>,l ! <sup>Λ</sup>� <sup>2</sup>

<sup>¼</sup> <sup>4</sup> 3 <sup>2</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83013

Usð Þ¼ j; m; l exp ½ � �iωsrml=c Vsð Þ j; m:l :

can be observed in the dependence of cooperative rate 1=τ<sup>s</sup>

cooperative diagrams of the kinetic equation ε<sup>2</sup>

magnitude as the second order <sup>ε</sup><sup>2</sup>N<sup>2</sup> � <sup>ε</sup><sup>3</sup>N<sup>3</sup>

μι2μι1d23d31ω<sup>3</sup>

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

c6ℏ<sup>2</sup>

Vsð Þ¼ <sup>j</sup>; <sup>m</sup>:<sup>l</sup> ð Þ exp ½ �� �iωsrml=<sup>c</sup> <sup>1</sup> exp <sup>i</sup>ωarmj=<sup>c</sup> � <sup>1</sup>

<sup>s</sup>ð Þ <sup>ω</sup><sup>a</sup> <sup>3</sup>

First term in Eq. (15) describes the transition of D atom under the influence of the scattering process of emitted photons of the atoms from V subsystem. This process is described by exchange integral Vsð Þ j; m:l . The last two terms in master Eq. (15) describe the scattering process of emitted photons by the V atoms under

The similar expression is obtained for the interaction of Λ three-level radiator with D atom represented in Figure 1C. In this case we must replace the operators of V subsystem in expression (15) with corresponding transition operators of Λ system

<sup>ι</sup>,l and their Hermit conjugated operators. For the two atoms, expression (16) was reduced to the simple representation

> Vs <sup>¼</sup> <sup>λ</sup>sλ<sup>a</sup> <sup>1</sup> � exp ð Þ �2iπr=λ<sup>s</sup> ½ �½ � <sup>1</sup> � exp 2ð Þ <sup>i</sup>πr=λ<sup>a</sup> ð Þ 2πr

Here the wavelength λ<sup>s</sup> (λa) corresponds to the emitted photons at Stokes or anti-Stokes frequencies represented in Figure 1. The numerical representation of the real part of the exchange integral (17) as the function of the relive distance between the atoms X ¼ ω0r=c and the relative Stokes frequency ωs=ω<sup>0</sup> is plotted in the Figure 3. It is observing the nonsignificant dependence of this exchange integral on the frequency ωs. The significant dependence on the detuning from resonance

In this section we obtained the correlations between dipole-active and dipoleforbidden subsystems of radiators, where the two-quantum exchange integral has the same magnitude as the two-photon quantum interaction between atoms of D subsystem. In the case of the big number of radiators in each subsystem, the correlated terms, expressions (12) and (15), give the cubic contribution in the

can archived the value proportional to the Dicke super-radiance [1] even for the same small parameters of each subsystem ε<sup>1</sup> ¼ ε2. In this case the number of atoms in each subsystem must achieve the value for which the third order has the same

decomposition on the small parameter ε can be regarded as a sum of single- and the two-photon transition amplitudes proportional to ε<sup>1</sup> and ε2, where ε<sup>1</sup> � μ1<sup>ι</sup>; g<sup>k</sup>

tude is smaller than the single-photon amplitude ε<sup>2</sup> < ε1, we conclude that beginning with the third-order term, the correlation diagrams (12) and (15), proportional to

1ε2, can play an important role in the two-quantum decay process even for the twoatomic system consisted from one atom of each subsystems: D and Ξ (or D and V). For example, in the situation when ε<sup>1</sup> ¼ 0:7 and ε<sup>2</sup> ¼ 0:25, the magnitude of two-

<sup>1</sup>ε<sup>2</sup> ¼ 0:1225Þ. In other words we can find the condition for which we can neglect

1ε2NN<sup>2</sup>

ð Þ k1; k<sup>2</sup> . Considering the situation when the two-photon ampli-

<sup>2</sup> ¼ 0:0625, becomes smaller than the cooperative magnitude

1 ω<sup>32</sup> � ω<sup>s</sup>

<sup>ω</sup>sωað Þ rml=<sup>c</sup> rmj=<sup>c</sup> ,

þ

,

1 ω<sup>31</sup> þ ω<sup>s</sup>

<sup>2</sup> : (17)

sad represented in expres-

<sup>ξ</sup>. When N ¼ N<sup>ξ</sup> these terms

and

. In conclusion we observe that the

(16)

where λ<sup>2</sup> and λ<sup>1</sup> are the emission wavelengths in cascade transition of the dipoleactive three radiators in Ξ configuration, situated at distance r. The real part of this function describes the three-particle decay rate of the system. The dependence of exchange integral (14) on the relative distance between the Ξ and D atoms (14), X ¼ ω0r=c and the displacement, Δ ¼ ð Þ ω<sup>1</sup> � ω<sup>0</sup> =ω<sup>0</sup> relatively the degenerate frequency ω0, is plotted in Figure 2. As follows from this dependence, the exchange integral achieved the maximal radius, when ω<sup>1</sup> ¼ ω2, which corresponds to the situation Δ ¼ 0.

The part of master Equation (10) for resonance scattering interaction between the absorbed and emitted photons by the dipole-active Λ and V subsystems and D dipole-forbidden radiators can be obtained from the third-order expansion on the smallest parameter λ. In this situation, the scattering part of the master equation represented by the scheme 1 B becomes

$$\begin{split} \frac{d\check{W}\_{21}(t)}{dt} &= \frac{i}{2\tau^{\prime}\_{\text{sad}}} \sum\_{m,j,l=1} \left\{ U\_{\boldsymbol{s}}(j,m,l)[\check{\boldsymbol{V}}^{\boldsymbol{1}}\_{\boldsymbol{\iota},j}, \check{\boldsymbol{V}}^{\boldsymbol{\iota}}\_{2,l}\check{\boldsymbol{D}}\_{m} - \check{\boldsymbol{W}}(t)] \right. \\ &+ U^{\ast}\_{\boldsymbol{s}}\left(j,m,l\right)[\check{\boldsymbol{V}}^{\boldsymbol{\iota}}\_{2,l}, \check{\boldsymbol{W}}(t)\check{\boldsymbol{V}}^{\boldsymbol{1}}\_{\boldsymbol{\iota},j}\check{\boldsymbol{D}}\_{m} - \boldsymbol{\big}] \\ &- \frac{i}{2\tau^{\prime}\_{\text{sad}}} \sum\_{m,j,l=1} V\_{\boldsymbol{s}}\left(j,m,l\right) \left[\check{\boldsymbol{D}}^{\boldsymbol{-}}\_{m}, \check{\boldsymbol{V}}^{\boldsymbol{\iota}}\_{2,l}\check{\boldsymbol{W}}(t)\check{\boldsymbol{V}}^{\boldsymbol{1}}\_{\boldsymbol{\iota},j}\right] \\ &+ H.c., \end{split} \tag{15}$$

The real part of the scattering exchange integrals Vs, defined in expressions (17), is plotted as a function of relative distance between radiators, X ¼ ω0r=c, and relative scattering frequency, ωs=ω0.

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

where

For two atoms represented in Figure 1A, the simple exchange integral between

where λ<sup>2</sup> and λ<sup>1</sup> are the emission wavelengths in cascade transition of the dipoleactive three radiators in Ξ configuration, situated at distance r. The real part of this function describes the three-particle decay rate of the system. The dependence of exchange integral (14) on the relative distance between the Ξ and D atoms (14), X ¼ ω0r=c and the displacement, Δ ¼ ð Þ ω<sup>1</sup> � ω<sup>0</sup> =ω<sup>0</sup> relatively the degenerate frequency ω0, is plotted in Figure 2. As follows from this dependence, the exchange integral achieved the maximal radius, when ω<sup>1</sup> ¼ ω2, which corresponds to the

The part of master Equation (10) for resonance scattering interaction between the absorbed and emitted photons by the dipole-active Λ and V subsystems and D dipole-forbidden radiators can be obtained from the third-order expansion on the smallest parameter λ. In this situation, the scattering part of the master equation

Usð Þ½ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>V</sup>� <sup>1</sup>

;W t � ð ÞV� <sup>1</sup> ι,j <sup>D</sup>� <sup>m</sup>�� o

Vsð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>D</sup>� �

ι,j ;V� ι 2,l

<sup>m</sup>;V� <sup>ι</sup> 2,l

∑ m,j,l¼<sup>1</sup>

<sup>s</sup> ð Þ½ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>V</sup>� <sup>ι</sup>

∑ m,j,l¼<sup>1</sup>

<sup>þ</sup><sup>U</sup> <sup>∗</sup>

� <sup>i</sup> 2τs sad

þH:c:,

n

2,l

The real part of the scattering exchange integrals Vs, defined in expressions (17), is plotted as a function of

relative distance between radiators, X ¼ ω0r=c, and relative scattering frequency, ωs=ω0.

<sup>2</sup> , (14)

<sup>D</sup>� <sup>m</sup>�W t ^ ð Þ�

(15)

W t � ð ÞV� <sup>1</sup> ι,j

h i

Vb <sup>¼</sup> <sup>λ</sup>1λ2½ � exp ð Þ� �2iπr=λ<sup>2</sup> <sup>1</sup> ½ � exp 2ð Þ� <sup>i</sup>πr=λ<sup>1</sup> <sup>1</sup> ð Þ 2πr

these radiators can be obtained from expression (13):

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

situation Δ ¼ 0.

Figure 3.

18

represented by the scheme 1 B becomes

dW� <sup>21</sup><sup>s</sup> ð Þt dt <sup>¼</sup> <sup>i</sup> 2τ <sup>s</sup> sad

$$\frac{1}{\tau\_{\text{sat}}^{\epsilon}} = \left(\frac{4}{3}\right)^2 \frac{\mu\_{i2}\mu\_{l1}d\_{31}o\_{s}^3(o\_d)^3}{c^6\hbar^2} \left[\frac{\mathbf{1}}{o\_{32} - o\_l} + \frac{\mathbf{1}}{o\_{31} + o\_l}\right],$$

$$V\_s(j, m, l) = \frac{\left(\exp\left[-io\_lr\_{ml}/c\right] - \mathbf{1}\right)\left(\exp\left[i o\_l r\_{mj}/c\right] - \mathbf{1}\right)}{o\_{3}o\_d(r\_{ml}/c)\left(r\_{mj}/c\right)},\tag{16}$$

$$U\_s(j, m, l) = \exp\left[-io\_lr\_{ml}/c\right]V\_s(j, m, l).$$

First term in Eq. (15) describes the transition of D atom under the influence of the scattering process of emitted photons of the atoms from V subsystem. This process is described by exchange integral Vsð Þ j; m:l . The last two terms in master Eq. (15) describe the scattering process of emitted photons by the V atoms under the influence of D subsystem.

The similar expression is obtained for the interaction of Λ three-level radiator with D atom represented in Figure 1C. In this case we must replace the operators of V subsystem in expression (15) with corresponding transition operators of Λ system V� 1 <sup>ι</sup>,j ! <sup>Λ</sup>^ <sup>ι</sup> 1j : <sup>V</sup>� <sup>ι</sup> <sup>2</sup>,l ! <sup>Λ</sup>� <sup>2</sup> <sup>ι</sup>,l and their Hermit conjugated operators. For the two atoms, expression (16) was reduced to the simple representation

$$V\_s = \frac{\lambda\_s \lambda\_d [1 - \exp\left(-2i\pi r/\lambda\_s\right)] [1 - \exp\left(2i\pi r/\lambda\_d\right)]}{\left(2\pi r\right)^2}. \tag{17}$$

Here the wavelength λ<sup>s</sup> (λa) corresponds to the emitted photons at Stokes or anti-Stokes frequencies represented in Figure 1. The numerical representation of the real part of the exchange integral (17) as the function of the relive distance between the atoms X ¼ ω0r=c and the relative Stokes frequency ωs=ω<sup>0</sup> is plotted in the Figure 3. It is observing the nonsignificant dependence of this exchange integral on the frequency ωs. The significant dependence on the detuning from resonance can be observed in the dependence of cooperative rate 1=τ<sup>s</sup> sad represented in expressions (16).

In this section we obtained the correlations between dipole-active and dipoleforbidden subsystems of radiators, where the two-quantum exchange integral has the same magnitude as the two-photon quantum interaction between atoms of D subsystem. In the case of the big number of radiators in each subsystem, the correlated terms, expressions (12) and (15), give the cubic contribution in the cooperative diagrams of the kinetic equation ε<sup>2</sup> 1ε2NN<sup>2</sup> <sup>ξ</sup>. When N ¼ N<sup>ξ</sup> these terms can archived the value proportional to the Dicke super-radiance [1] even for the same small parameters of each subsystem ε<sup>1</sup> ¼ ε2. In this case the number of atoms in each subsystem must achieve the value for which the third order has the same magnitude as the second order <sup>ε</sup><sup>2</sup>N<sup>2</sup> � <sup>ε</sup><sup>3</sup>N<sup>3</sup> . In conclusion we observe that the decomposition on the small parameter ε can be regarded as a sum of single- and the two-photon transition amplitudes proportional to ε<sup>1</sup> and ε2, where ε<sup>1</sup> � μ1<sup>ι</sup>; g<sup>k</sup> and ε<sup>2</sup> � qbð Þ k1; k<sup>2</sup> or qs ð Þ k1; k<sup>2</sup> . Considering the situation when the two-photon amplitude is smaller than the single-photon amplitude ε<sup>2</sup> < ε1, we conclude that beginning with the third-order term, the correlation diagrams (12) and (15), proportional to ε2 1ε2, can play an important role in the two-quantum decay process even for the twoatomic system consisted from one atom of each subsystems: D and Ξ (or D and V). For example, in the situation when ε<sup>1</sup> ¼ 0:7 and ε<sup>2</sup> ¼ 0:25, the magnitude of twophoton emission, ε<sup>2</sup> <sup>2</sup> ¼ 0:0625, becomes smaller than the cooperative magnitude ε2 <sup>1</sup>ε<sup>2</sup> ¼ 0:1225Þ. In other words we can find the condition for which we can neglect

the decay rate of two-photon emission of the D atom in comparison with the cooperative effect described by expressions (12) and (15). This possibility to control the two-photon decay process of D atom with the decay process of Ξ or V excited three-level atom is given in the next section.

�1 ¼ 2 1½ � � cosð Þ ω0r=c . According to this expression, the exchange integrals

In this case one can introduce the expression exchange rate 1=τ<sup>b</sup>

the distance between the dipole-active and dipole-forbidden subsystems:

<sup>2</sup> , U<sup>b</sup>

2ð1 � cos 2ð Þ πr=λ<sup>s</sup>

where λ<sup>0</sup> ¼ c=½ � 2πω<sup>0</sup> . Taking into account the above definitions and introducing

the correlation functions between the polarizations of <sup>Ξ</sup> and <sup>D</sup> atoms <sup>F</sup>^bð Þ <sup>t</sup>;<sup>r</sup> � �

þ

2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ <sup>x</sup> ,

1 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � <sup>þ</sup> sin ð Þ <sup>x</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � � ,

<sup>f</sup> cosð Þ <sup>x</sup> <sup>2</sup> <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � � h i <sup>N</sup>2ð Þ <sup>t</sup>; <sup>x</sup> �

� <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ðÞ�<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � � � <sup>þ</sup> 〈N^ <sup>d</sup>ð Þð <sup>t</sup> <sup>1</sup> � <sup>N</sup>^ <sup>2</sup>ðÞ�<sup>t</sup> <sup>2</sup>N^ <sup>i</sup>ð Þ<sup>t</sup> 〉�

� <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

� <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup>

τι, <sup>1</sup> þ 1 τd � �:

<sup>12</sup><sup>d</sup> ¼ exp ½ � �iω0r=c Vsrd:

ð Þ <sup>2</sup>πr=λ<sup>s</sup> <sup>2</sup> , (19)

<sup>1</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E <sup>þ</sup> <sup>D</sup>^ <sup>þ</sup>

cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � sin ð Þ <sup>x</sup> E t ^ð Þ ; <sup>x</sup> � � � � ,

cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � sin ð Þ <sup>x</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � �

<sup>½</sup> cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �

sin ð Þ½ <sup>x</sup> <sup>2</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �

srd as a function of

ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>1</sup> <sup>2</sup>ð Þt D E,

(20)

<sup>12</sup><sup>d</sup> <sup>¼</sup> <sup>2</sup>ð<sup>1</sup> � cos 2ð Þ <sup>π</sup>r=λ<sup>0</sup> ð Þ 2πr=λ<sup>0</sup>

> 1 τb

ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>1</sup> <sup>2</sup>ð Þt h i D E and <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>Ξ</sup>^<sup>2</sup>

τι, <sup>2</sup>

τι, <sup>2</sup>

τι, <sup>1</sup>

� <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

τd

1 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

2

2

τι, <sup>2</sup>

� <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

dt <sup>¼</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �

dt <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup>

dt <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �

<sup>12</sup>dð Þ<sup>r</sup> <sup>¼</sup> <sup>1</sup> τb srd

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

we obtain the closed system of equations from expression (18):

� <sup>1</sup> 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

� <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � τι, <sup>1</sup>

� sin ð Þ <sup>x</sup> E t ^ð Þ ; <sup>x</sup> � �� þ <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �

<sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �;

1 τd þ 1 τι,<sup>2</sup> � �

� <sup>2</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>2</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �g;

� <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ð Þ�<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � � � � <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>1</sup> � <sup>N</sup>^ <sup>2</sup>ðÞ�<sup>t</sup> <sup>2</sup>N^ <sup>i</sup>ð Þ<sup>t</sup> � � � � �;

> τι, <sup>2</sup> þ 1 τd � �,

1 τd þ 1 τι, <sup>2</sup> � �

� <sup>1</sup> 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x

become

<sup>¼</sup> <sup>i</sup> <sup>Ξ</sup>^<sup>2</sup>

d

d

d

d

d

d

d

21

<sup>1</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup>

dt <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �

dt <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> � �

dt <sup>N</sup>^ <sup>1</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> � �

dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; <sup>x</sup> � �

dt <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �

<sup>d</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �

þ

Vb

DOI: http://dx.doi.org/10.5772/intechopen.83013

#### 3. Two-photon energy transfer between the two three-level radiators

Master Eq. (10) can be used for the description of cooperative interaction between the dipole-forbidden and dipole-active radiators in two-photon exchanges. Indeed passing again from Schrodinger to Heisenberg pictures Tr W t ^ ð ÞO^ ð Þ <sup>0</sup> h i <sup>¼</sup> Tr <sup>W</sup>^ ð Þ <sup>0</sup> O t ^ ð Þ h i, we can obtain from this expression the equation of the arbitrary atomic operator O t ^ ð Þ. Let us firstly discuss the nonlinear interaction in which Ξ and D atoms enter in two-photon resonance as represented in Figure 1A. Studying the cooperative interaction between the dipole-forbidden and dipoleactive radiators, the closed system of equations for the correlation functions can be found in such approach. Considering that the numbers of atoms in the each subsystem are relatively small, we can obtain the following generalized equation for the arbitrary operator Ob:

d Oh i <sup>b</sup>ð Þt dt <sup>¼</sup> <sup>1</sup> 2τι, <sup>1</sup> ∑ N<sup>ξ</sup> l,j¼<sup>1</sup> <sup>χ</sup>1ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>^<sup>ι</sup> 1,l ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>1</sup> ι,j ð Þt D E h i þ 1 2τι, <sup>2</sup> ∑ N<sup>ξ</sup> l,j¼<sup>1</sup> <sup>χ</sup>2ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>^<sup>2</sup> ι,l ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>ι</sup> 2,j ð Þt D E h i þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup> <sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>^ <sup>þ</sup> <sup>l</sup> ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>j</sup> ð Þt D E h i � <sup>i</sup> 4τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup> l¼1 ∑ N<sup>ξ</sup> j¼1 ( Ubð Þ½ <sup>m</sup>; <sup>l</sup>; <sup>j</sup> <sup>Ξ</sup>^<sup>2</sup> ι,l ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>ι</sup> 1,j ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þt D E h i <sup>þ</sup> <sup>Ξ</sup>^<sup>ι</sup> 1j ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>2</sup> ιl ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þt D E h i � þ <sup>U</sup> <sup>∗</sup> <sup>b</sup> ð Þ j; l; m 〈 ( <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>ι</sup> 1,j ð Þt h i; <sup>Ξ</sup>^ 2 ι,l ð Þt � � <sup>þ</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>2</sup> ι,l ð Þt h i; <sup>Ξ</sup>^ ι 1,j ð Þt � �) D^ � m〉 ) � <sup>i</sup> 2τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup> j¼1 ∑ N<sup>ξ</sup> l¼1 Vbð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈Ξ^<sup>ι</sup> 1,j ð Þ½ <sup>t</sup> <sup>Ξ</sup>^<sup>2</sup> ι,l ð Þ<sup>t</sup> ; <sup>O</sup>^ <sup>b</sup>ð Þ� <sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉 n þ〈Ξ^<sup>2</sup> ι,l ð Þ½ <sup>t</sup> <sup>Ξ</sup>^<sup>ι</sup> 1,j ð Þ<sup>t</sup> ; <sup>O</sup>^ <sup>b</sup>ð Þ� <sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉 o þ H:c: (18)

In order to simplify this problem, we analyze below the situation in which we have only a single atom in each subsystem. In this case we can replace the operator Ob with the excitation numbers operators <sup>N</sup>^ <sup>α</sup> <sup>¼</sup> <sup>Ξ</sup>^<sup>α</sup> <sup>α</sup>ð Þ<sup>t</sup> and <sup>N</sup>^ <sup>d</sup> <sup>¼</sup> <sup>D</sup>^ <sup>z</sup> <sup>þ</sup> <sup>0</sup>:5 of <sup>Ξ</sup> and D atoms, respectively. Here α ¼ 1, 2 and ι. When emission frequencies of the onephoton radiators coincide with ω<sup>1</sup> ≃ω<sup>2</sup> ≃ω0, the dependence (14) becomes real and positive defined function Ξ and D radiators. Here exp ½ � ½ �� iω0r=c 1 ½ exp ½ � �iω0r=c

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

�1 ¼ 2 1½ � � cosð Þ ω0r=c . According to this expression, the exchange integrals become

$$V\_{12d}^b = \frac{2(1 - \cos\left(2\pi r/\lambda\_0\right))}{\left(2\pi r/\lambda\_0\right)^2}, \quad U\_{12d}^b = \exp\left[-i\alpha v/c\right]V\_{ml}A$$

In this case one can introduce the expression exchange rate 1=τ<sup>b</sup> srd as a function of the distance between the dipole-active and dipole-forbidden subsystems:

$$\frac{1}{\pi\_{12d}^b(r)} = \frac{1}{\pi\_{srl}^b} \frac{2(1 - \cos\left(2\pi r/\lambda\_s\right))}{\left(2\pi r/\lambda\_s\right)^2},\tag{19}$$

where λ<sup>0</sup> ¼ c=½ � 2πω<sup>0</sup> . Taking into account the above definitions and introducing the correlation functions between the polarizations of <sup>Ξ</sup> and <sup>D</sup> atoms <sup>F</sup>^bð Þ <sup>t</sup>;<sup>r</sup> � � <sup>¼</sup> <sup>i</sup> <sup>Ξ</sup>^<sup>2</sup> <sup>1</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup> ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>1</sup> <sup>2</sup>ð Þt h i D E and <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>Ξ</sup>^<sup>2</sup> <sup>1</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E <sup>þ</sup> <sup>D</sup>^ <sup>þ</sup> ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>1</sup> <sup>2</sup>ð Þt D E, we obtain the closed system of equations from expression (18):

d dt <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � τι, <sup>2</sup> � <sup>1</sup> 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � sin ð Þ <sup>x</sup> E t ^ð Þ ; <sup>x</sup> � � � � , d dt <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � τι, <sup>2</sup> � <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � τι, <sup>1</sup> þ 1 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x <sup>½</sup> cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � sin ð Þ <sup>x</sup> E t ^ð Þ ; <sup>x</sup> � �� þ <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ <sup>x</sup> , d dt <sup>N</sup>^ <sup>1</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> � � τι, <sup>1</sup> � <sup>1</sup> 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � sin ð Þ <sup>x</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � � � <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � �; d dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; <sup>x</sup> � � τd þ 1 4τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x cosð Þ <sup>x</sup> <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � <sup>þ</sup> sin ð Þ <sup>x</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � � � , d dt <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � ¼ � <sup>F</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � 2 1 τd þ 1 τι,<sup>2</sup> � � � <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x <sup>f</sup> cosð Þ <sup>x</sup> <sup>2</sup> <sup>N</sup>^ <sup>2</sup>ð Þ <sup>t</sup>; <sup>x</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � � h i <sup>N</sup>2ð Þ <sup>t</sup>; <sup>x</sup> � � <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ðÞ�<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � � � <sup>þ</sup> 〈N^ <sup>d</sup>ð Þð <sup>t</sup> <sup>1</sup> � <sup>N</sup>^ <sup>2</sup>ðÞ�<sup>t</sup> <sup>2</sup>N^ <sup>i</sup>ð Þ<sup>t</sup> 〉� � <sup>2</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>2</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �g; <sup>d</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � dt <sup>¼</sup> <sup>E</sup>^bð Þ <sup>t</sup>; <sup>x</sup> � � 2 1 τd þ 1 τι, <sup>2</sup> � � � <sup>1</sup> 2τ<sup>b</sup> <sup>12</sup><sup>d</sup>ð Þ x sin ð Þ½ <sup>x</sup> <sup>2</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>i</sup>ð Þ�<sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � � � � <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> <sup>1</sup> � <sup>N</sup>^ <sup>2</sup>ðÞ�<sup>t</sup> <sup>2</sup>N^ <sup>i</sup>ð Þ<sup>t</sup> � � � � �; d dt <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup> τι, <sup>2</sup> þ 1 τd � �, d dt <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � τι, <sup>2</sup> � <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup> τι, <sup>1</sup> þ 1 τd � �: (20)

the decay rate of two-photon emission of the D atom in comparison with the cooperative effect described by expressions (12) and (15). This possibility to control the two-photon decay process of D atom with the decay process of Ξ or V excited

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

3. Two-photon energy transfer between the two three-level radiators

Master Eq. (10) can be used for the description of cooperative interaction between the dipole-forbidden and dipole-active radiators in two-photon exchanges.

in which Ξ and D atoms enter in two-photon resonance as represented in Figure 1A. Studying the cooperative interaction between the dipole-forbidden and dipoleactive radiators, the closed system of equations for the correlation functions can be

subsystem are relatively small, we can obtain the following generalized equation for

ι,j ð Þt

> 2,j ð Þt

<sup>j</sup> ð Þt

ι,l ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>ι</sup> 1,j

� þ <sup>U</sup> <sup>∗</sup>

D^ � m〉 )

> 1,j ð Þ½ <sup>t</sup> <sup>Ξ</sup>^<sup>2</sup> ι,l

> > þ H:c:

found in such approach. Considering that the numbers of atoms in the each

ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>1</sup>

D E h i

ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>ι</sup>

<sup>l</sup> ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ �

Ubð Þ½ <sup>m</sup>; <sup>l</sup>; <sup>j</sup> <sup>Ξ</sup>^<sup>2</sup>

<sup>m</sup>ð Þt

Vbð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈Ξ^<sup>ι</sup>

n

<sup>m</sup>ð Þt 〉 o

In order to simplify this problem, we analyze below the situation in which we have only a single atom in each subsystem. In this case we can replace the operator

D atoms, respectively. Here α ¼ 1, 2 and ι. When emission frequencies of the onephoton radiators coincide with ω<sup>1</sup> ≃ω<sup>2</sup> ≃ω0, the dependence (14) becomes real and positive defined function Ξ and D radiators. Here exp ½ � ½ �� iω0r=c 1 ½ exp ½ � �iω0r=c

D E h i

D E h i

, we can obtain from this expression the equation

^ ð Þ. Let us firstly discuss the nonlinear interaction

ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ �

(

ð Þ<sup>t</sup> ; <sup>O</sup>^ <sup>b</sup>ð Þ� <sup>t</sup> <sup>D</sup>^ �

D E h i

<sup>b</sup> ð Þ j; l; m 〈

<sup>m</sup>ð Þt

<sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>ι</sup>

<sup>m</sup>ð Þt 〉

<sup>α</sup>ð Þ<sup>t</sup> and <sup>N</sup>^ <sup>d</sup> <sup>¼</sup> <sup>D</sup>^ <sup>z</sup> <sup>þ</sup> <sup>0</sup>:5 of <sup>Ξ</sup> and

h i

1,j ð Þt

� �

; Ξ^ 2 ι,l ð Þt

(18)

Indeed passing again from Schrodinger to Heisenberg pictures

^ ð Þ h i

three-level atom is given in the next section.

<sup>¼</sup> Tr <sup>W</sup>^ ð Þ <sup>0</sup> O t

of the arbitrary atomic operator O t

the arbitrary operator Ob:

þ 1 2τι, <sup>2</sup>

þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup>

� <sup>i</sup> 4τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup>

<sup>þ</sup> <sup>Ξ</sup>^<sup>ι</sup> 1j ð Þ<sup>t</sup> <sup>Ξ</sup>^<sup>2</sup> ιl

� <sup>i</sup> 2τ<sup>b</sup> 12d ∑ N m¼1 ∑ N<sup>ξ</sup>

þ〈Ξ^<sup>2</sup> ι,l ð Þ½ <sup>t</sup> <sup>Ξ</sup>^<sup>ι</sup> 1,j

20

<sup>þ</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>Ξ</sup>^<sup>2</sup>

2τι, <sup>1</sup>

∑ N<sup>ξ</sup> l,j¼<sup>1</sup>

> ∑ N<sup>ξ</sup> l,j¼<sup>1</sup>

<sup>χ</sup>1ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>^<sup>ι</sup>

<sup>χ</sup>2ð Þ <sup>j</sup>; <sup>l</sup> <sup>Ξ</sup>^<sup>2</sup>

<sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>^ <sup>þ</sup>

(

ð Þ<sup>t</sup> <sup>O</sup>^ <sup>b</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ �

; Ξ^ ι 1,j ð Þt

ð Þ<sup>t</sup> ; <sup>O</sup>^ <sup>b</sup>ð Þ� <sup>t</sup> <sup>D</sup>^ �

D E h i

� �)

l¼1 ∑ N<sup>ξ</sup>

j¼1

ι,l ð Þt

j¼1 ∑ N<sup>ξ</sup>

Ob with the excitation numbers operators <sup>N</sup>^ <sup>α</sup> <sup>¼</sup> <sup>Ξ</sup>^<sup>α</sup>

l¼1

h i

1,l

ι,l

Tr W t ^ ð ÞO^ ð Þ <sup>0</sup> h i

d Oh i <sup>b</sup>ð Þt dt <sup>¼</sup> <sup>1</sup>

Using this system of Eq. (20), we can numerically study the cooperative nonlinear exchanges through the vacuum field between the Ξ and D radiators situated at relative distance x. One can observe that the spontaneous generation of photon pair by the D atom is drastically modified by the time increase of the cooperative correlation between the radiators. Indeed considering that the decay rate of the D atom 1=τ<sup>d</sup> is smaller than similar rates of the cascade transition in the Ξ atom (τd=τξ,i ≃6; τd=ð Þ¼ 4τ12<sup>d</sup> 2), we can numerically represent this dependence as a function of the relative time, t=τd, and the relative distance between the radiators, x ¼ 2πr=λ0. As shown in Figure 4A, the decay rate of D atom is drastically modified at small distances between the radiators which is in accordance with the analytic expressions (19). Considering that both atoms Ξ and D are prepared in the excited state, we observe the significant enhancement of the two-photon emission rate of the D radiator under the influence of the Ξ decay process.

Let us simplify the system of Eq. (20) in order to solve it exactly. Indeed, when dipole-active Ξ atom is situated at small distance relative to the D radiator (x ≪ 1), the system of Eq. (20) is drastically simplified:

$$\begin{aligned} \frac{d}{dt}\langle\hat{N}\_2(t)\rangle &= -\frac{\langle\hat{N}\_2(t)\rangle}{\tau\_{t,2}} - \frac{\langle\hat{F}\_b(t)\rangle}{4\tau\_{12d}^b}, \\\\ \frac{d}{dt}\langle\hat{N}\_d(t)\rangle &= -\frac{\langle\hat{N}\_d(t)\rangle}{\tau\_d} + \frac{1}{4\tau\_{12d}^b} \langle\hat{F}\_b(t)\rangle, \\\\ \frac{d}{dt}\langle\hat{F}\_b(t)\rangle &= -\frac{\langle\hat{F}\_b(t)\rangle}{2} \left(\frac{1}{\tau\_d} + \frac{1}{\tau\_{t,2}}\right) \\\\ &\quad - \frac{1}{2\tau\_{12d}^b} [4\langle\hat{N}\_2(t)\hat{N}\_d(t)\rangle + \langle\hat{N}\_d(t)\rangle] \\\\ &\quad - \langle\hat{N}\_2(t)\rangle - 5\langle\hat{N}\_d\hat{N}\_i\rangle], \\\\ \frac{d}{dt}\langle\hat{N}\_2(t)\hat{N}\_d(t)\rangle &= -\langle\hat{N}\_2(t)\hat{N}\_d(t)\rangle \left[\frac{1}{\tau\_{t,2}} + \frac{1}{\tau\_d}\right], \\\\ \frac{d}{dt}\langle\hat{N}\_i(t)\hat{N}\_d(t)\rangle &= \frac{\langle\hat{N}\_2(t)\hat{N}\_d(t)\rangle}{\tau\_{t,2}} - \langle\hat{N}\_i(t)\hat{N}\_d(t)\rangle \left[\frac{1}{\tau\_{t,1}} + \frac{1}{\tau\_d}\right]. \end{aligned} \tag{21}$$

The exact solution of this linear system of equation can be represented through solution of characteristic equation <sup>Y</sup><sup>α</sup> <sup>¼</sup> <sup>∑</sup><sup>5</sup> <sup>j</sup>¼<sup>1</sup>C<sup>j</sup> <sup>α</sup> exp <sup>Θ</sup>jt � �, where <sup>α</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>; 5 and f g <sup>Y</sup><sup>α</sup> are the atomic functions, <sup>Y</sup>1ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �, <sup>Y</sup>2ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �, <sup>Y</sup>3ðÞ¼ <sup>t</sup> <sup>F</sup>^bð Þ<sup>t</sup> � �, <sup>Y</sup>4ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> �� �, and <sup>Y</sup>5ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> �� �; the solution of characteristic equation is

$$\begin{aligned} \Theta\_1 &= -\left(\frac{1}{\tau\_2} + \frac{1}{\tau\_d}\right); \quad \Theta\_2 = -\left(\frac{1}{\tau\_1} + \frac{1}{\tau\_d}\right); \quad \Theta\_3 = -\frac{1}{2}\left(\frac{1}{\tau\_d} + \frac{1}{\tau\_{\ast,2}}\right); \\\\ \Theta\_{4,5} &= -\frac{1}{2}\left\{\frac{1}{\tau\_{\ast,2}} + \frac{1}{\tau\_d} \pm \sqrt{\left(\frac{1}{\tau\_d} - \frac{1}{\tau\_{\ast,2}}\right)^2 - \frac{1}{\tau\_{12b}^2}}\right\}. \end{aligned} \tag{22}$$

in Eq. (22), the oscillatory decay of the atomic inversion is possible, when

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

DOI: http://dx.doi.org/10.5772/intechopen.83013

conditions:h i N<sup>1</sup> ¼ h i N<sup>ι</sup> ¼ 0, h i N<sup>2</sup> ¼ 1:, and h i Nd ¼ 0.

Figure 4.

23

In this process the rate of energy transfer from Ξ to D atoms represented in Figure 4B has the oscillator behavior. In the case of the excitation of D, the coupling between the radiators becomes more effective, when the virtual level of

1=τ<sup>d</sup> ¼ 1=τι, 2. In this case the solutions Θ4, <sup>5</sup> become complex. We observe such an oscillation of the atoms inversion of Ξ radiator prepared initially in the excited state.

(A) The decay rate �d Nh i <sup>d</sup>=dt of the dipole-forbidden transitions of the D radiator under the influence of Ξ three-level radiator. This solution of Eq. (21) is plotted as function of t=τ<sup>d</sup> and relative distance between the radiators x ¼ ω1r=c, for the following parameters of the system: h i¼ N<sup>1</sup> h i¼ N<sup>ι</sup> 0:, h i¼ N<sup>2</sup> 1:, h i¼ Nd 1, τd=τ<sup>1</sup> ¼ τd=τ<sup>2</sup> ¼ 6, and τd=ð Þ¼ 4τ12<sup>d</sup> 2. (B) The decay process of excited state ∣2i of three-level system (thick line) and the transfer of the excitation from the Ξ radiator to D atom (dashed line) in the process of cascade emission of Ξ atom situated at relative distance x < <1 for the same parameters of the system and excitation

The coefficients C<sup>j</sup> α � � are determined from the initial conditions. As follows from the numerical estimation plotted in Figure 4B and solutions of characteristic Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

#### Figure 4.

Using this system of Eq. (20), we can numerically study the cooperative nonlinear exchanges through the vacuum field between the Ξ and D radiators situated at relative distance x. One can observe that the spontaneous generation of photon pair by the D atom is drastically modified by the time increase of the cooperative correlation between the radiators. Indeed considering that the decay rate of the D atom 1=τ<sup>d</sup> is smaller than similar rates of the cascade transition in the Ξ atom (τd=τξ,i ≃6; τd=ð Þ¼ 4τ12<sup>d</sup> 2), we can numerically represent this dependence as a function of the relative time, t=τd, and the relative distance between the radiators, x ¼ 2πr=λ0. As shown in Figure 4A, the decay rate of D atom is drastically modified at small distances between the radiators which is in accordance with the analytic expressions (19). Considering that both atoms Ξ and D are prepared in the excited state, we observe the significant enhancement of the two-photon emission rate of

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Let us simplify the system of Eq. (20) in order to solve it exactly. Indeed, when dipole-active Ξ atom is situated at small distance relative to the D radiator (x ≪ 1),

> � <sup>F</sup>^bð Þ<sup>t</sup> � � 4τ<sup>b</sup> 12d ,

> > <sup>F</sup>^bð Þ<sup>t</sup> � �,

<sup>½</sup><sup>4</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>þ</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �

,

τι, <sup>1</sup> þ 1 τd � �

:

;

(22)

<sup>α</sup> exp <sup>Θ</sup>jt � �, where <sup>α</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>; 5 and

(21)

� <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup>

; <sup>Θ</sup><sup>3</sup> ¼ � <sup>1</sup>

9 = ;: 2

1 τd þ 1 τι,<sup>2</sup> � �

� ��,

τι,<sup>2</sup> þ 1 τd � �

The exact solution of this linear system of equation can be represented through

f g <sup>Y</sup><sup>α</sup> are the atomic functions, <sup>Y</sup>1ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �, <sup>Y</sup>2ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �, <sup>Y</sup>3ðÞ¼ <sup>t</sup> <sup>F</sup>^bð Þ<sup>t</sup> � �, <sup>Y</sup>4ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> �� �, and <sup>Y</sup>5ðÞ¼ <sup>t</sup> <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> �� �; the solution of characteristic

> τ1 þ 1 τd � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

from the numerical estimation plotted in Figure 4B and solutions of characteristic

� 1 τ2 12b

� � are determined from the initial conditions. As follows

<sup>j</sup>¼<sup>1</sup>C<sup>j</sup>

the D radiator under the influence of the Ξ decay process.

the system of Eq. (20) is drastically simplified:

dt <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � �

dt <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � ¼ � <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �

dt <sup>F</sup>^bð Þ<sup>t</sup> � � ¼ � <sup>F</sup>^bð Þ<sup>t</sup> � �

τι, <sup>2</sup>

τd

2

τι, <sup>2</sup>

; <sup>Θ</sup><sup>2</sup> ¼ � <sup>1</sup>

1 τd � 1 τι, <sup>2</sup> � �<sup>2</sup>

� <sup>1</sup> 2τ<sup>b</sup> 12d

dt <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � ¼ � <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>1</sup>

dt <sup>N</sup>^ <sup>ι</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � � <sup>¼</sup> <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> � �

solution of characteristic equation <sup>Y</sup><sup>α</sup> <sup>¼</sup> <sup>∑</sup><sup>5</sup>

þ 1 4τ<sup>b</sup> 12d

1 τd þ 1 τι, <sup>2</sup> � �

� <sup>N</sup>^ <sup>2</sup>ð Þ<sup>t</sup> � � � <sup>5</sup> <sup>N</sup>^ dN^ <sup>i</sup>

d

d

d

d

d

<sup>Θ</sup><sup>1</sup> ¼ � <sup>1</sup>

<sup>Θ</sup>4, <sup>5</sup> ¼ � <sup>1</sup>

The coefficients C<sup>j</sup>

τ2 þ 1 τd � �

> 1 τι, <sup>2</sup> þ 1 τd �

<sup>8</sup> <sup>s</sup> <sup>&</sup>lt;

α

2

:

equation is

22

(A) The decay rate �d Nh i <sup>d</sup>=dt of the dipole-forbidden transitions of the D radiator under the influence of Ξ three-level radiator. This solution of Eq. (21) is plotted as function of t=τ<sup>d</sup> and relative distance between the radiators x ¼ ω1r=c, for the following parameters of the system: h i¼ N<sup>1</sup> h i¼ N<sup>ι</sup> 0:, h i¼ N<sup>2</sup> 1:, h i¼ Nd 1, τd=τ<sup>1</sup> ¼ τd=τ<sup>2</sup> ¼ 6, and τd=ð Þ¼ 4τ12<sup>d</sup> 2. (B) The decay process of excited state ∣2i of three-level system (thick line) and the transfer of the excitation from the Ξ radiator to D atom (dashed line) in the process of cascade emission of Ξ atom situated at relative distance x < <1 for the same parameters of the system and excitation conditions:h i N<sup>1</sup> ¼ h i N<sup>ι</sup> ¼ 0, h i N<sup>2</sup> ¼ 1:, and h i Nd ¼ 0.

in Eq. (22), the oscillatory decay of the atomic inversion is possible, when 1=τ<sup>d</sup> ¼ 1=τι, 2. In this case the solutions Θ4, <sup>5</sup> become complex. We observe such an oscillation of the atoms inversion of Ξ radiator prepared initially in the excited state. In this process the rate of energy transfer from Ξ to D atoms represented in Figure 4B has the oscillator behavior. In the case of the excitation of D, the coupling between the radiators becomes more effective, when the virtual level of

the D atom is situated between the excited and ground states (see Figure 4B). As the virtual states of the D radiator is off from the resonance with the dipole-active transitions of the Ξ radiators, the excitation of D atom takes place only with the absorption of both emitted photons by the Ξ atom. The cooperative effects between the Ξ and D radiators are described by second-order correlation function <sup>G</sup><sup>2</sup> <sup>¼</sup> <sup>E</sup>^� ð Þ<sup>t</sup> <sup>E</sup>^� ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup> ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup> ð Þt D E <sup>¼</sup> <sup>G</sup><sup>0</sup> <sup>2</sup> <sup>þ</sup> <sup>α</sup> <sup>F</sup>^bð Þ<sup>t</sup> � �. Here <sup>G</sup><sup>0</sup> <sup>2</sup> was derived in Ref. [5]. The contribution to the second-order correlation function remains larger than the square value of the first-order correlation function G<sup>1</sup> ¼ E�ð Þt E<sup>þ</sup> h i ð Þt , so that we can conclude that new cooperative effects between single- and two-photon transitions of D and Ξ subsystems play an important role in the two-photon decay process. Let us now return to the V three-level system in scattering interaction with the D system as this is represented in Figure 1B. In accordance with master Eq. (10) and its analytic representation (15), we can obtain the following expression for arbitrary atomic operators <sup>O</sup>^ <sup>s</sup>ð Þ<sup>t</sup> .

<sup>d</sup> <sup>O</sup>^ <sup>s</sup>ð Þ<sup>t</sup> D E dt <sup>¼</sup> <sup>1</sup> 2τι, <sup>1</sup> ∑ Nv l,j¼<sup>1</sup> <sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>^ <sup>1</sup> ι,l ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 1,j ð Þt D E h i þ 1 2τι, <sup>2</sup> ∑ Nv l,j¼<sup>1</sup> <sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>^ <sup>2</sup> ι,l ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 2,j ð Þt D E h i þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup> <sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>^ <sup>þ</sup> <sup>l</sup> ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>j</sup> ð Þt D E h i � <sup>i</sup> 2τs sad ∑ N m¼1 ∑ Nv j¼1 ∑ Nv l¼1 Vsð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>V</sup>^ <sup>1</sup> ι,j ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þt h iV^ <sup>ι</sup> 2,l ð Þt D E þ i 2τs sad ∑ m,j,l¼<sup>1</sup> Usð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈½O^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ;V^ <sup>1</sup> ι,j ð Þ� <sup>t</sup> <sup>V</sup>^ <sup>ι</sup> 2l ð Þ<sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉 n <sup>þ</sup><sup>U</sup> <sup>∗</sup> <sup>s</sup> ð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈V^ <sup>1</sup> ι,j ð Þ<sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þ½ <sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup> ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 2,l ð Þ� t 〉 o þ H:c: (23)

The similar expression can be obtained for a Λ three-level system in interaction with D radiators, doing the substitution V^ <sup>β</sup> <sup>α</sup>,j ! <sup>Λ</sup><sup>α</sup> βj . For two atoms in each subsystem, an attractive peculiarity follows from this substitution. If Osð Þt is the inversion of the D atom, the direct modification of the D atomic excitation by Λ three-level atom is equal to zero <sup>Λ</sup>^ <sup>ι</sup> 1,l ð Þ<sup>t</sup> <sup>Λ</sup>^<sup>2</sup> ι,l ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þ<sup>t</sup> � � D E <sup>¼</sup> 0 due to the operator product Λ^<sup>ι</sup> 1,l ð Þ<sup>t</sup> <sup>Λ</sup>^ <sup>2</sup> ι,l ðÞ¼ t 0 for the same atom. In order to obtain the closed system of equation from master Eqs. (15)and (23), we consider the simple interaction of two atoms in the scattering process represented by the analytical scheme of Figure 1B. In this case we introduce the new indexes }s } and }a} instead of }1} and }2}, which correspond to the Stokes and anti-Stokes scattering frequencies ω<sup>s</sup> and ωa. Considering that the anti-Stokes frequency ω<sup>a</sup> is larger than Stokes ωs, one can approximate the exchange integrals (17) with expression

$$V\_s \simeq \frac{\sin\left(\chi\_a\right)}{\chi\_a} + i \frac{1 - \cos\left(\chi\_a\right)}{\chi\_a}.\tag{24}$$

respectively. The functions <sup>F</sup>^sð Þ <sup>t</sup>; xa

generalized equation (23).

<sup>2</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup>

relative distance xa ¼ 2πr=λ<sup>a</sup> in the three-dimensional representation.

<sup>E</sup>^sð Þ <sup>t</sup>; xa � � <sup>¼</sup> <sup>V</sup>^ <sup>1</sup>

25

Figure 5.

� � <sup>¼</sup> <sup>i</sup> <sup>V</sup>^ <sup>1</sup>

The decay process of the dipole-forbidden transitions of the D radiator under the influence of V three-level radiator for following parameter atom for following parameters of the system, h i Na ¼ 0:5, h i Ns ¼ 0:5, h i¼ Nd 1, τa=τ<sup>d</sup> ¼ 0:1, τa=τ<sup>s</sup> ¼ 6, and τa=τasd, (A) represents the decay rate �d Nh i <sup>d</sup>=dt and (B) represents the excitation of the D atom plotted as the numerical solution of the system of Eq. (25) as function of t=τaÞ and

h i D E , <sup>N</sup>^ dN^ <sup>s</sup>

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

DOI: http://dx.doi.org/10.5772/intechopen.83013

ð Þ<sup>t</sup> <sup>V</sup>^ <sup>2</sup> <sup>1</sup>ð Þt

polarization and population correlations between the atoms Ξ and D. For this two-atom system, we can obtain the following closed system of equations from

<sup>2</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup>

h i D E ,

� �, and N^ dN^ <sup>a</sup>

V^ 2 <sup>1</sup>ð Þt

� � describe the

Here xa <sup>¼</sup> <sup>ω</sup>ar=c. The mean values of the operators <sup>N</sup>^ <sup>s</sup> � � <sup>¼</sup> <sup>V</sup>^ <sup>2</sup> 2 D E, <sup>N</sup>^ <sup>a</sup> � � <sup>¼</sup> <sup>V</sup>^ <sup>1</sup> 1 D E, and N^ <sup>d</sup> � � are considered the populations of excited states of V and D radiators,

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

#### Figure 5.

the D atom is situated between the excited and ground states (see Figure 4B). As the virtual states of the D radiator is off from the resonance with the dipole-active transitions of the Ξ radiators, the excitation of D atom takes place only with the absorption of both emitted photons by the Ξ atom. The cooperative effects between the Ξ and D radiators are described by second-order correlation

<sup>¼</sup> <sup>G</sup><sup>0</sup>

Ref. [5]. The contribution to the second-order correlation function remains

<sup>2</sup> <sup>þ</sup> <sup>α</sup> <sup>F</sup>^bð Þ<sup>t</sup> � �. Here <sup>G</sup><sup>0</sup>

<sup>2</sup> was derived in

function <sup>G</sup><sup>2</sup> <sup>¼</sup> <sup>E</sup>^�

operators <sup>O</sup>^ <sup>s</sup>ð Þ<sup>t</sup> .

<sup>d</sup> <sup>O</sup>^ <sup>s</sup>ð Þ<sup>t</sup> D E

dt <sup>¼</sup> <sup>1</sup>

2τι, <sup>1</sup>

þ 1 2τι, <sup>2</sup>

þ 1 2τ<sup>d</sup> ∑ N l,j¼<sup>1</sup>

þ i 2τs sad

<sup>þ</sup><sup>U</sup> <sup>∗</sup>

with D radiators, doing the substitution V^ <sup>β</sup>

three-level atom is equal to zero <sup>Λ</sup>^ <sup>ι</sup>

1,l ð Þ<sup>t</sup> <sup>Λ</sup>^ <sup>2</sup> ι,l

ator product Λ^<sup>ι</sup>

and N^ <sup>d</sup>

24

� <sup>i</sup> 2τs sad ∑ N m¼1 ∑ Nv j¼1 ∑ Nv l¼1

∑ Nv l,j¼<sup>1</sup>

> ∑ Nv l,j¼<sup>1</sup>

∑ m,j,l¼<sup>1</sup>

<sup>s</sup> ð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈V^ <sup>1</sup>

Figure 1B. In this case we introduce the new indexes }s

approximate the exchange integrals (17) with expression

Vs <sup>≃</sup> sin ð Þ xa xa

Here xa <sup>¼</sup> <sup>ω</sup>ar=c. The mean values of the operators <sup>N</sup>^ <sup>s</sup>

ð Þ<sup>t</sup> <sup>E</sup>^�

ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup>

D E

ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup> ð Þt

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

larger than the square value of the first-order correlation function G<sup>1</sup> ¼ E�ð Þt E<sup>þ</sup> h i ð Þt , so that we can conclude that new cooperative effects between single- and two-photon transitions of D and Ξ subsystems play an important role in the two-photon decay process. Let us now return to the V three-level system in scattering interaction with the D system as this is represented in Figure 1B. In accordance with master Eq. (10) and its analytic representation (15), we can obtain the following expression for arbitrary atomic

<sup>χ</sup>að Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>^ <sup>1</sup>

<sup>χ</sup>sð Þ <sup>j</sup>; <sup>l</sup> <sup>V</sup>^ <sup>2</sup>

<sup>χ</sup>dð Þ <sup>j</sup>; <sup>l</sup> <sup>D</sup>^ <sup>þ</sup>

n

ι,j ð Þ<sup>t</sup> <sup>D</sup>^ �

ι,l ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup>

> ι,l ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup>

<sup>l</sup> ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup>

Vsð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> <sup>V</sup>^ <sup>1</sup>

Usð Þ <sup>j</sup>; <sup>m</sup>; <sup>l</sup> 〈½O^ ð Þ<sup>s</sup>

<sup>m</sup>ð Þ½ <sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup>

<sup>α</sup>,j ! <sup>Λ</sup><sup>α</sup> βj

The similar expression can be obtained for a Λ three-level system in interaction

subsystem, an attractive peculiarity follows from this substitution. If Osð Þt is the inversion of the D atom, the direct modification of the D atomic excitation by Λ

system of equation from master Eqs. (15)and (23), we consider the simple interaction of two atoms in the scattering process represented by the analytical scheme of

}2}, which correspond to the Stokes and anti-Stokes scattering frequencies ω<sup>s</sup> and ωa. Considering that the anti-Stokes frequency ω<sup>a</sup> is larger than Stokes ωs, one can

þ i

� � are considered the populations of excited states of V and D radiators,

1,l ð Þ<sup>t</sup> <sup>Λ</sup>^<sup>2</sup> ι,l ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 1,j ð Þt

> ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 2,j ð Þt

ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>j</sup> ð Þt

> ð Þ<sup>t</sup> ;V^ <sup>1</sup> ι,j ð Þ� <sup>t</sup> <sup>V</sup>^ <sup>ι</sup> 2l ð Þ<sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉

ð Þ<sup>t</sup> ;V^ <sup>ι</sup> 2,l ð Þ� t 〉 o

ð Þ<sup>t</sup> <sup>N</sup>^ <sup>d</sup>ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þ<sup>t</sup> � � D E

ðÞ¼ t 0 for the same atom. In order to obtain the closed

1 � cosð Þ xa xa

ð Þ<sup>t</sup> ; <sup>D</sup>^ � <sup>m</sup>ð Þt V^ ι 2,l ð Þt (23)

þ H:c:

¼ 0 due to the oper-

} and }a} instead of }1} and

: (24)

, N^ <sup>a</sup>

� � <sup>¼</sup> <sup>V</sup>^ <sup>1</sup>

1 D E ,

2 D E

� � <sup>¼</sup> <sup>V</sup>^ <sup>2</sup>

. For two atoms in each

h i

D E

D E h i

D E h i

D E h i

ι,j ð Þ<sup>t</sup> <sup>O</sup>^ ð Þ<sup>s</sup>

The decay process of the dipole-forbidden transitions of the D radiator under the influence of V three-level radiator for following parameter atom for following parameters of the system, h i Na ¼ 0:5, h i Ns ¼ 0:5, h i¼ Nd 1, τa=τ<sup>d</sup> ¼ 0:1, τa=τ<sup>s</sup> ¼ 6, and τa=τasd, (A) represents the decay rate �d Nh i <sup>d</sup>=dt and (B) represents the excitation of the D atom plotted as the numerical solution of the system of Eq. (25) as function of t=τaÞ and relative distance xa ¼ 2πr=λ<sup>a</sup> in the three-dimensional representation.

respectively. The functions <sup>F</sup>^sð Þ <sup>t</sup>; xa � � <sup>¼</sup> <sup>i</sup> <sup>V</sup>^ <sup>1</sup> <sup>2</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup> V^ 2 <sup>1</sup>ð Þt h i D E , <sup>E</sup>^sð Þ <sup>t</sup>; xa � � <sup>¼</sup> <sup>V</sup>^ <sup>1</sup> <sup>2</sup>ð Þ<sup>t</sup> <sup>D</sup>^ � ð Þt D E � <sup>D</sup>^ <sup>þ</sup> ð Þ<sup>t</sup> <sup>V</sup>^ <sup>2</sup> <sup>1</sup>ð Þt h i D E , <sup>N</sup>^ dN^ <sup>s</sup> � �, and N^ dN^ <sup>a</sup> � � describe the polarization and population correlations between the atoms Ξ and D. For this two-atom system, we can obtain the following closed system of equations from generalized equation (23).

We can conclude that it is possible to study all cooperations two-photon process between single atoms in each system represented in Figure 1A–C. For example, the system of Eqs. (20) and (25) can be solved simultaneously taking into consideration scattering and two-photon transitions. In this case the effective energy transfer of the excitation between the atoms Ξ, V, and D radiator prepared in the special initial states can open the new possibilities of non-resonance interaction between the

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

DOI: http://dx.doi.org/10.5772/intechopen.83013

This chapter proposed the cooperative effects between three-level system and dipole-forbidden two-level systems in nonlinear interaction through the vacuum field during the spontaneous emission time. The possibility of cooperative migration of energy from one excited dipole-active three-level atom to another takes place with phase retardation effects and depends on the position of atoms in the system. This excitation transfer from dipole-active to dipole-forbidden subsystems takes place with phase dependence amplitudes, so that the cooperative excitation of the system consisted from two species of atoms depends on the retardation of radiation along the sample and geometry of the system. This follows from the excited or ground state of one of the radiators represented in Figures 4 and 5. As in Ref. [20], the exchanges between the Ξ (or V) three-level atom and D take place with the absorption and emission of two quanta, but in this chapter, we take into consideration the real and imaginary parts of exchange integrals. In this case, two

modify the dynamics of possible excitation of D atoms by Ξ and V radiators. The scattering transfer of the energy between the excited state of V three-level radiator and dipole-forbidden transitions of D two-level atoms are effective when the dipole-forbidden atom enters in the two-photon resonance with the energy difference between the two dipole transitions (Figures 1A and 5A). When the atom D is in the excited state, the emitted Stokes photon by one atom of the V systems can be absorbed by another radiator from the D subsystem, so that two radiators pass into the ground state generating two anti-Stokes photons with energies E<sup>0</sup> ¼ 2ℏωa. The opposite situation can be observed when D atom is prepared in the ground state.

Quantum Optics and Kinetic Processes Lab, Institute of Applied Physics of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

and <sup>E</sup>^b sð Þð Þ <sup>t</sup>; xa

, which

correlation functions introduced functions <sup>F</sup>^b sð Þð Þ <sup>t</sup>; xa

atomic subsystems.

4. Conclusions

Author details

Nicolae A. Enaki

27

Moldova, Chisinau, Republic of Moldova

provided the original work is properly cited.

\*Address all correspondence to: enakinicolae@yahoo.com

d dt <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; xa <sup>¼</sup> <sup>1</sup> τι,a <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa <sup>þ</sup> 1 τι,s <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa <sup>þ</sup> 1 τs sad 1 � cosð Þ xa xa <sup>E</sup>^sð Þ <sup>t</sup>; xa , d dt <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> τι,a <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa þ 1 2τ<sup>s</sup> sad sin ð Þ xa xa <sup>F</sup>^sð Þ <sup>t</sup>; xa � <sup>1</sup> � cosð Þ xa xa 〈E^sðt; xaÞ〉 , d dt <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> τι,s <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa � <sup>1</sup> 2τ<sup>s</sup> sad sin ð Þ xa xa <sup>F</sup>^sð Þ <sup>t</sup>; xa <sup>þ</sup> 1 � cosð Þ xa xa 〈E^sðt; xaÞ〉 , d dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> τd <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa þ 1 2τ<sup>s</sup> sad sin ð Þ xa xa <sup>F</sup>^sð Þ <sup>t</sup>; xa � <sup>1</sup> � cosð Þ xa xa 〈E^sðt; xaÞ〉 , d dt <sup>F</sup>^sð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> 2 1 τd þ 1 τι,s þ 1 τι,a <sup>F</sup>^sð Þ <sup>t</sup>; xa þ 1 τs sad sin ð Þ xa xa <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa � <sup>1</sup> � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa N^ <sup>a</sup>ð Þ <sup>t</sup>; xa , d dt <sup>E</sup>^sð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> 2 1 τd þ 1 τι,s þ 1 τι,a <sup>E</sup>^sð Þ <sup>t</sup>; xa � <sup>1</sup> � cosð Þ xa xaτ<sup>s</sup> sad <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa <sup>þ</sup> <sup>1</sup> � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa N^ <sup>a</sup>ð Þ <sup>t</sup>; xa d dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> τd þ 1 τι,s <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa � <sup>1</sup> 2τs sad sin ð Þ xa xa <sup>F</sup>^sð Þ <sup>t</sup>; xa <sup>þ</sup> 1 � cosð Þ xa xa 〈E^sðt; xaÞ〉 , d dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa ¼ � <sup>1</sup> τd þ 1 τιa <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa : (25)

As follows from the system (25), and numerical simulation plotted in Figure 5 the first N^ <sup>d</sup> =τ<sup>d</sup> and second terms 1=τ<sup>s</sup> sad <sup>F</sup>^<sup>s</sup> describe the generation rate of entangled photon pairs and scattering rate with absorption of Stokes photon and generation of two anti-Stokes photons by the system formed from V and D atoms. When the time tends to infinity, all excited atomic energies E<sup>0</sup> ¼ ℏω<sup>a</sup> þ ℏω<sup>s</sup> þ ℏω<sup>d</sup> of three-level V and two-level D atoms are emitted by the system. Taking into account the conservation law in the scattering process ω<sup>a</sup> � ω<sup>s</sup> � ω<sup>d</sup> ¼ 0, we observe that this cooperation between the atoms becomes predominant, when the collective scattering rate 1=τ<sup>s</sup> sad increases. In other words, the probability of absorption of Stokes photon ℏω<sup>s</sup> which is accompanied with the generation of the new anti-Stokes photon ℏω<sup>a</sup> by D atom becomes possible. In this case two atoms represented in the Figure 1B can generate an entangled anti-Stokes photons with energy E<sup>0</sup> ¼ 2ℏωa. The possibility of the excitation transfer between the atoms Ξ and D represented in Figure 4B can be found in the special preparation of the system.

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

We can conclude that it is possible to study all cooperations two-photon process between single atoms in each system represented in Figure 1A–C. For example, the system of Eqs. (20) and (25) can be solved simultaneously taking into consideration scattering and two-photon transitions. In this case the effective energy transfer of the excitation between the atoms Ξ, V, and D radiator prepared in the special initial states can open the new possibilities of non-resonance interaction between the atomic subsystems.

#### 4. Conclusions

d

d

d

d

d

d

d

d

the first N^ <sup>d</sup>

scattering rate 1=τ<sup>s</sup>

26

dt <sup>N</sup>^ <sup>ι</sup>ð Þ <sup>t</sup>; xa <sup>¼</sup> <sup>1</sup>

dt <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa

dt <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa

dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa

dt <sup>F</sup>^sð Þ <sup>t</sup>; xa

dt <sup>E</sup>^sð Þ <sup>t</sup>; xa

¼ � <sup>1</sup>

¼ � <sup>1</sup>

¼ � <sup>1</sup>

¼ � <sup>1</sup>

¼ � <sup>1</sup>

dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa

dt <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa

τι,a

þ 1 2τ<sup>s</sup> sad

τι,a

τι,s

τd

2

2

1 τd þ 1 τι,s þ 1 τι,a

1 τd þ 1 τι,s þ 1 τι,a

� <sup>1</sup> � cosð Þ xa xaτ<sup>s</sup> sad

=τ<sup>d</sup> and second terms 1=τ<sup>s</sup>

sin ð Þ xa xa

� <sup>1</sup> 2τ<sup>s</sup> sad

þ 1 2τ<sup>s</sup> sad

þ 1 τs sad

¼ � <sup>1</sup>

¼ � <sup>1</sup>

<sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa <sup>þ</sup>

> <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa

> > sin ð Þ xa xa

<sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa 

> sin ð Þ xa xa

<sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa 

> sin ð Þ xa xa

τd þ 1 τι,s 

τd þ 1 τιa 

Figure 4B can be found in the special preparation of the system.

� <sup>1</sup> 2τs sad

1 τι,s

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

<sup>F</sup>^sð Þ <sup>t</sup>; xa

<sup>F</sup>^sð Þ <sup>t</sup>; xa <sup>þ</sup>

<sup>F</sup>^sð Þ <sup>t</sup>; xa

<sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa <sup>þ</sup>

� <sup>1</sup> � cosð Þ xa

� <sup>1</sup> � cosð Þ xa

<sup>F</sup>^sð Þ <sup>t</sup>; xa 

<sup>E</sup>^sð Þ <sup>t</sup>; xa 

<sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa

As follows from the system (25), and numerical simulation plotted in Figure 5

sad F^<sup>s</sup>

Stokes photon ℏω<sup>s</sup> which is accompanied with the generation of the new anti-Stokes photon ℏω<sup>a</sup> by D atom becomes possible. In this case two atoms represented in the Figure 1B can generate an entangled anti-Stokes photons with energy E<sup>0</sup> ¼ 2ℏωa. The possibility of the excitation transfer between the atoms Ξ and D represented in

entangled photon pairs and scattering rate with absorption of Stokes photon and generation of two anti-Stokes photons by the system formed from V and D atoms. When the time tends to infinity, all excited atomic energies E<sup>0</sup> ¼ ℏω<sup>a</sup> þ ℏω<sup>s</sup> þ ℏω<sup>d</sup> of three-level V and two-level D atoms are emitted by the system. Taking into account the conservation law in the scattering process ω<sup>a</sup> � ω<sup>s</sup> � ω<sup>d</sup> ¼ 0, we observe that this cooperation between the atoms becomes predominant, when the collective

sin ð Þ xa xa

<sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa

1 τs sad

xa

1 � cosð Þ xa xa

xa

� <sup>1</sup> � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa

<sup>þ</sup> <sup>1</sup> � <sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa

<sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>s</sup>ð Þ <sup>t</sup>; xa 

> <sup>F</sup>^sð Þ <sup>t</sup>; xa <sup>þ</sup>

<sup>N</sup>^ <sup>d</sup>ð Þ <sup>t</sup>; xa <sup>N</sup>^ <sup>a</sup>ð Þ <sup>t</sup>; xa :

sad increases. In other words, the probability of absorption of

,

1 � cosð Þ xa xa

describe the generation rate of

1 � cosð Þ xa xa

〈E^sðt; xaÞ〉

〈E^sðt; xaÞ〉

〈E^sðt; xaÞ〉

<sup>E</sup>^sð Þ <sup>t</sup>; xa ,

,

,

,

N^ <sup>a</sup>ð Þ <sup>t</sup>; xa

N^ <sup>a</sup>ð Þ <sup>t</sup>; xa

〈E^sðt; xaÞ〉

,

(25)

This chapter proposed the cooperative effects between three-level system and dipole-forbidden two-level systems in nonlinear interaction through the vacuum field during the spontaneous emission time. The possibility of cooperative migration of energy from one excited dipole-active three-level atom to another takes place with phase retardation effects and depends on the position of atoms in the system. This excitation transfer from dipole-active to dipole-forbidden subsystems takes place with phase dependence amplitudes, so that the cooperative excitation of the system consisted from two species of atoms depends on the retardation of radiation along the sample and geometry of the system. This follows from the excited or ground state of one of the radiators represented in Figures 4 and 5. As in Ref. [20], the exchanges between the Ξ (or V) three-level atom and D take place with the absorption and emission of two quanta, but in this chapter, we take into consideration the real and imaginary parts of exchange integrals. In this case, two correlation functions introduced functions <sup>F</sup>^b sð Þð Þ <sup>t</sup>; xa and <sup>E</sup>^b sð Þð Þ <sup>t</sup>; xa , which modify the dynamics of possible excitation of D atoms by Ξ and V radiators. The scattering transfer of the energy between the excited state of V three-level radiator and dipole-forbidden transitions of D two-level atoms are effective when the dipole-forbidden atom enters in the two-photon resonance with the energy difference between the two dipole transitions (Figures 1A and 5A). When the atom D is in the excited state, the emitted Stokes photon by one atom of the V systems can be absorbed by another radiator from the D subsystem, so that two radiators pass into the ground state generating two anti-Stokes photons with energies E<sup>0</sup> ¼ 2ℏωa. The opposite situation can be observed when D atom is prepared in the ground state.

#### Author details

Nicolae A. Enaki Quantum Optics and Kinetic Processes Lab, Institute of Applied Physics of Moldova, Chisinau, Republic of Moldova

\*Address all correspondence to: enakinicolae@yahoo.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Dicke RH. Coherence in spontaneous radiation processes. Physics Review. 1954;93:99-110

[2] Florian R, Schwan LO, Schmid D. Superradiance and high-gain mirrorless laser activity of O <sup>2</sup> -centers in KCl. Solid State Communications. 1982;42: 55-57

[3] Rautian SG, Chernobrod BM. Cooperative effect in Raman scattering of light. Soviet Physics—JETP. 1977;45: 705-708

[4] Inouye S, Pfau T, Gupta S, Chikkatur AP, Görlitz A, Pritchard DE, et al. Phase-coherent amplification of atomic matter waves. Nature. 1999;402: 641-644

[5] Enaki NA. Superradiation from twophoton spontaneous decay. Soviet Physics—JETP. 1988;67:2033-2038

[6] Yuan L, Hokr BH, Traverso AJ, et al. Theoretical analysis of the coherencebrightened laser in air. Physical Review A. 2013;023826:87

[7] Schwendimann P. Damping effects in two-colour superfluorescence. Optica Acta: International Journal of Optics. 1984;31:107-114

[8] Haake F, Reibold R. Two-color superfluorescence from three-level systems. Physics Letters A. 1982;92: 29-31

[9] Andreev AV, Enaki NA, Ilinskii YA. Superfluorescence in a three-level system. Theoretical and Mathematical Physics. 1985;64:960-965

[10] Pando Lambruschini CL. A semiclassical approach for two-color four-mode solid-state superfluorescence. Optics Communications. 1991;85: 291-298

[11] Traverso AJ, Sanchez-Gonzalez R, Yuan L, et al. Coherence brightened laser source for atmospheric remote sensing. Proceedings of the National Academy of Sciences of the United States of America. 2012;109:15185-15190 Žurnal ėksperimental'noj i teoretičeskoj

DOI: http://dx.doi.org/10.5772/intechopen.83013

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

[20] Enaki NA, Rosca T. Cooperative effects between three subsystems in two-photon and Raman resonances. In:

ROMOPTO 2012: Tenth conference on optics: Micro- to Nanophotonics III,

Proceedings of the SPIE 8882,

fiziki. 1990;98:783-796

88820L; 2013. 11 p

29

[12] Enaki NA. Mutual cooperative effects between single- and two-photon super-fluorescent processes through vacuum field. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics. 2012;66(98)

[13] Enaki NA. Non-Linear Cooperative Effects in Open Quantum Systems: Entanglement and Second Order Coherence. NY, USA: Nova Science Publishers; 2015. p. 325

[14] Bonifacio R, Lugiato LA. Cooperative radiation processes in twolevel systems: Superfluorescence. Physical Review A. 1974;11:1507-1521

[15] Lehmberg RH. Radiation from an Natom system. I. General formalism. Physical Review A. 1970;2:883

[16] Lugiato LA, Oldano C, Narducci CLM. Cooperative frequency locking and stationary spatial structures in lasers. Journal of the Optical Society of America B: Optical Physics. 1988;5:879

[17] Wang T, Yelin SF, Eyler1 RCEE, Farooqi SM, Gould PL, Kostrun M, et al. Superradiance in ultracold Rydberg gases. Physical Review A. 2007;75 (033802)

[18] Paradis E, Barrett B, Kumarakrishnan A, Zhang R, Raithel G. Observation of superfluorescent emissions from laser-cooled atoms. Physical Review A. 2008;77:043419

[19] Enaki NA. The photon quantum statistics in the processes of two quanta optical nutation. Soviet Physics.

Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects DOI: http://dx.doi.org/10.5772/intechopen.83013

Žurnal ėksperimental'noj i teoretičeskoj fiziki. 1990;98:783-796

References

1954;93:99-110

laser activity of O

55-57

705-708

641-644

A. 2013;023826:87

1984;31:107-114

29-31

291-298

28

[1] Dicke RH. Coherence in spontaneous radiation processes. Physics Review.

[2] Florian R, Schwan LO, Schmid D. Superradiance and high-gain mirrorless

Solid State Communications. 1982;42:

[4] Inouye S, Pfau T, Gupta S, Chikkatur AP, Görlitz A, Pritchard DE, et al. Phase-coherent amplification of atomic matter waves. Nature. 1999;402:

[5] Enaki NA. Superradiation from twophoton spontaneous decay. Soviet Physics—JETP. 1988;67:2033-2038

[6] Yuan L, Hokr BH, Traverso AJ, et al. Theoretical analysis of the coherencebrightened laser in air. Physical Review

[7] Schwendimann P. Damping effects in two-colour superfluorescence. Optica Acta: International Journal of Optics.

[8] Haake F, Reibold R. Two-color superfluorescence from three-level systems. Physics Letters A. 1982;92:

[9] Andreev AV, Enaki NA, Ilinskii YA. Superfluorescence in a three-level system. Theoretical and Mathematical

Physics. 1985;64:960-965

[10] Pando Lambruschini CL. A semiclassical approach for two-color four-mode solid-state superfluorescence. Optics Communications. 1991;85:

[3] Rautian SG, Chernobrod BM. Cooperative effect in Raman scattering of light. Soviet Physics—JETP. 1977;45:

<sup>2</sup> -centers in KCl.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

[11] Traverso AJ, Sanchez-Gonzalez R, Yuan L, et al. Coherence brightened laser source for atmospheric remote sensing. Proceedings of the National Academy of Sciences of the United States of America. 2012;109:15185-15190

[12] Enaki NA. Mutual cooperative effects between single- and two-photon super-fluorescent processes through vacuum field. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics. 2012;66(98)

[13] Enaki NA. Non-Linear Cooperative Effects in Open Quantum Systems: Entanglement and Second Order Coherence. NY, USA: Nova Science

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[15] Lehmberg RH. Radiation from an Natom system. I. General formalism. Physical Review A. 1970;2:883

[16] Lugiato LA, Oldano C, Narducci CLM. Cooperative frequency locking and stationary spatial structures in lasers. Journal of the Optical Society of America B: Optical Physics. 1988;5:879

[17] Wang T, Yelin SF, Eyler1 RCEE, Farooqi SM, Gould PL, Kostrun M, et al. Superradiance in ultracold Rydberg gases. Physical Review A. 2007;75

Kumarakrishnan A, Zhang R, Raithel G.

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[20] Enaki NA, Rosca T. Cooperative effects between three subsystems in two-photon and Raman resonances. In: Proceedings of the SPIE 8882, ROMOPTO 2012: Tenth conference on optics: Micro- to Nanophotonics III, 88820L; 2013. 11 p

Chapter 3

Abstract

1. Introduction

31

Viktor I. Gerasimenko

Processes of Creation and

Propagation of Correlations in

particular, correlations characterizing the condensed states of systems.

quantum kinetic equation, correlation of states, scaling limit 2000 Mathematics Subject Classification: 35Q40; 47D06

typical particle of large quantum particle system.

Keywords: von Neumann hierarchy, nonlinear quantum BBGKY hierarchy,

In this chapter, we consider mathematical problems concerning the description

As known, the marginal correlation operators give an equivalent approach to the description of the evolution of states of quantum systems of many particles in comparison with marginal density operators [1]. The physical interpretation of marginal correlation operators is that the macroscopic characteristics of fluctuations of mean values of observables are determined by them on the microscopic level [1, 2]. Traditionally marginal correlation operators are introduced by means of the cluster expansions of the marginal density operators [2–4]. In articles [5, 6] an approach based on the definition of the marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy was developed. As a result of which, it is established that the marginal correlation

of processes of a creation and a propagation of correlations in quantum manyparticle systems, namely, correlations in quantum systems both finitely and infinitely many particles and the description of correlations by means of the state of

Large Quantum Particle System

We review new approaches to the description of the evolution of states of large quantum particle systems by means of the marginal correlation operators. Using the definition of marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy, we establish that a sequence of such operators is governed by the nonlinear quantum BBGKY hierarchy. The constructed nonperturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in large quantum particle systems. Furthermore, we consider the problem of the rigorous description of collective behavior of quantum manyparticle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation with initial correlations, in

#### Chapter 3

## Processes of Creation and Propagation of Correlations in Large Quantum Particle System

Viktor I. Gerasimenko

#### Abstract

We review new approaches to the description of the evolution of states of large quantum particle systems by means of the marginal correlation operators. Using the definition of marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy, we establish that a sequence of such operators is governed by the nonlinear quantum BBGKY hierarchy. The constructed nonperturbative solution of the Cauchy problem to this hierarchy of nonlinear evolution equations describes the processes of the creation and the propagation of correlations in large quantum particle systems. Furthermore, we consider the problem of the rigorous description of collective behavior of quantum manyparticle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation with initial correlations, in particular, correlations characterizing the condensed states of systems.

Keywords: von Neumann hierarchy, nonlinear quantum BBGKY hierarchy, quantum kinetic equation, correlation of states, scaling limit 2000 Mathematics Subject Classification: 35Q40; 47D06

#### 1. Introduction

In this chapter, we consider mathematical problems concerning the description of processes of a creation and a propagation of correlations in quantum manyparticle systems, namely, correlations in quantum systems both finitely and infinitely many particles and the description of correlations by means of the state of typical particle of large quantum particle system.

As known, the marginal correlation operators give an equivalent approach to the description of the evolution of states of quantum systems of many particles in comparison with marginal density operators [1]. The physical interpretation of marginal correlation operators is that the macroscopic characteristics of fluctuations of mean values of observables are determined by them on the microscopic level [1, 2].

Traditionally marginal correlation operators are introduced by means of the cluster expansions of the marginal density operators [2–4]. In articles [5, 6] an approach based on the definition of the marginal correlation operators within the framework of dynamics of correlations governed by the von Neumann hierarchy was developed. As a result of which, it is established that the marginal correlation operators are governed by the hierarchy of nonlinear evolution equations, known as the quantum nonlinear BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy, and its solution is represented in the form of series, the generating operator of every term of which are the corresponding-order cumulant of groups of nonlinear operators of the von Neumann hierarchy for correlation operators [7].

obeying Maxwell-Boltzmann statistics. Further an inverse group to group (1) will

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

operators (1) is determined in the sense of the strong convergence of the space

� � ¼ �i Hn <sup>f</sup> <sup>n</sup> � <sup>f</sup> <sup>n</sup>Hn

� � <sup>f</sup> <sup>n</sup> <sup>≐</sup> � <sup>i</sup> <sup>Φ</sup> <sup>j</sup>

set 1ð Þ ; …; s into ∣P∣ nonempty mutually disjoint subsets Xj, and the set

� � � � <sup>∣</sup> <sup>¼</sup> <sup>∣</sup>P∣. On the space <sup>L</sup><sup>1</sup>

sequences <sup>f</sup> <sup>¼</sup> <sup>f</sup> <sup>0</sup>; <sup>f</sup> <sup>1</sup>; …; <sup>f</sup> <sup>n</sup>; … � � of trace class operators <sup>f</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup>

following nonlinear one-parameter mapping is defined:

P:ð Þ¼ 1;…; s ∪ jXj

P0

� � � � ≐ ∑

Gð Þ¼ t; 1j f A1ð Þ t; 1 f <sup>1</sup>ð Þ1 ,

A3ð Þ t; 1; 2; 3 f <sup>1</sup>ð Þ1 f <sup>1</sup>ð Þ2 f <sup>1</sup>ð Þ3 :

the following estimate is true:

where <sup>c</sup> � <sup>e</sup>3max 1; maxP:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>∪</sup> iXi

∥A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

� � � � fs

<sup>n</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup>

dual operator to the generator of the Heisenberg equation for observables) [2], the

Let the symbol <sup>∑</sup><sup>P</sup>:ð Þ¼ <sup>1</sup>;…; <sup>s</sup> <sup>∪</sup> jXj denote the sum over all possible partitions P of the

� � � � consists from elements which are subsets Xj <sup>⊂</sup> ð Þ <sup>1</sup>; …; <sup>s</sup> of the set

A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

where the generating operator A∣P∣ð Þt of this expansion is the ∣P∣th-order cumulant of the groups of operators (1) defined by the following expansion [2]:

:ð Þ f g <sup>X</sup><sup>1</sup> ;…;f g <sup>X</sup>∣P<sup>∣</sup> <sup>¼</sup> <sup>∪</sup> kZk

and θ is the declusterization mapping: θ f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

Below we adduce the examples of mapping expansions (3):

Gð Þ¼ t; 1; 2j f A1ð Þ t; f g 1; 2 f <sup>2</sup>ð Þþ 1; 2 A<sup>1</sup>þ<sup>1</sup>ð Þ t; 1; 2 f <sup>1</sup>ð Þ1 f <sup>1</sup>ð Þ2 ,

A<sup>1</sup>þ<sup>1</sup>ð Þ t; 2; f g 1; 3 f <sup>1</sup>ð Þ2 f <sup>2</sup>ð Þþ 1; 3 A<sup>1</sup>þ<sup>1</sup>ð Þ t; 3; f g 1; 2 f <sup>1</sup>ð Þ3 f <sup>2</sup>ð Þþ 1; 2

Gð Þ¼ t; 1; 2; 3jf A1ð Þ t; f g 1; 2; 3 f <sup>3</sup>ð Þþ 1; 2; 3 A<sup>1</sup>þ<sup>1</sup>ð Þ t; 1; f g 2; 3 f <sup>1</sup>ð Þ1 f <sup>2</sup>ð Þþ 2; 3

∥L<sup>1</sup>

<sup>∥</sup>Gð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup><sup>j</sup> <sup>f</sup> <sup>∥</sup>L1ð Þ <sup>H</sup><sup>s</sup> <sup>≤</sup><sup>s</sup> ! <sup>e</sup>

∥ f∣Xi<sup>∣</sup>

parameter mapping (3) is a bounded strong continuous group of nonlinear operators.

� � � � Y

ð Þ �<sup>1</sup> <sup>∣</sup>P<sup>0</sup>

ð Þ H<sup>s</sup> , s ≥1, the mapping Gð Þ t; 1; …; sj f is defined, and, according to the

ð Þ <sup>H</sup><sup>s</sup> ≤ ∣P∣! e

<sup>∥</sup>L<sup>1</sup> <sup>H</sup>∣Xi ð Þ<sup>∣</sup> � �. On the space <sup>L</sup><sup>1</sup>

∣P∣ ∥ fs ∥L<sup>1</sup> ð Þ <sup>H</sup><sup>s</sup> ,

2s c s

<sup>j</sup>¼<sup>1</sup><sup>N</sup> <sup>∗</sup>

int is defined by means of the operator of a two-body interaction poten-

1; j 2 � � <sup>f</sup> <sup>n</sup> � <sup>f</sup> <sup>n</sup><sup>Φ</sup> <sup>j</sup>

ð Þj is a free motion generator of the von Neumann equation (the

� �≐ N <sup>∗</sup>

ðÞþ<sup>j</sup> <sup>ϵ</sup>∑<sup>n</sup> j <sup>1</sup> , j <sup>n</sup> of the group of

<sup>n</sup> f <sup>n</sup>, (2)

<sup>2</sup> <sup>¼</sup> <sup>1</sup><sup>N</sup> <sup>∗</sup> int j 1; j 2 � �, where

1; j 2

<sup>n</sup>¼<sup>0</sup> <sup>L</sup><sup>1</sup>

ð Þ H<sup>n</sup> of

ð Þ H<sup>n</sup> and f <sup>0</sup> ∈ C, the

� �, s≥1, (3)

� � � � , and we

ð Þ¼ <sup>F</sup> <sup>H</sup> <sup>⊕</sup><sup>∞</sup>

f∣Xj<sup>∣</sup> Xj

Y Zk ⊂P<sup>0</sup>

G∗ ∣θð Þ Zk ∣

, (5)

ð Þ F <sup>H</sup> , one-

ð Þ t; θð Þ Zk ,

(4)

Xj ⊂P

<sup>∣</sup>�<sup>1</sup> <sup>j</sup>P<sup>0</sup> ð Þ j � <sup>1</sup> !

� � � � <sup>≐</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> .

be denoted by G<sup>∗</sup>

the operator N <sup>∗</sup>

f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

For fs ∈L<sup>1</sup>

inequality

33

operator N <sup>∗</sup>

ð Þ H<sup>n</sup> by the operator

tial Φ by the formula N <sup>∗</sup>

ð Þ 1; …; s , i.e., ∣ f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

Gð Þ t; 1; …; sjf ≐ ∑

L1

n � ��<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82836

limt!0 1 t G∗

denoted a scaling parameter by ϵ . 0.

that has the following structure: N <sup>∗</sup>

ðÞ¼ <sup>t</sup> <sup>G</sup><sup>∗</sup>

<sup>n</sup>ð Þ �t . On its domain of the definition, the infinitesimal generator N <sup>∗</sup>

<sup>n</sup>ð Þt f <sup>n</sup> � f <sup>n</sup>

int j 1; j 2

In the chapter, we also consider the problem of the rigorous description of the evolution of correlations in quantum many-particle systems by means of a oneparticle (marginal) density operator that is a solution of the generalized quantum kinetic equation with initial correlations [8]. We remark that initial states specified by correlations are typical for the condensed states of many-particle systems in contrast to their gaseous state [1].

We note that in modern researches, the conventional approach to the problem of the rigorous derivation of kinetic equations lies in the construction of various scaling limits of a solution of equations, describing the evolution of the state of many-particle systems [9], in particular, a mean field limit of a perturbative solution of the BBGKY hierarchy for a sequence of marginal density operators [10–17].

#### 2. Dynamics of quantum correlations

As known [1, 2], quantum systems of fixed number of particles are described in terms of observables and states. The functional of the mean value of observables defines a duality between observables and states, and as a consequence, there exist two approaches to the description of the evolution of quantum systems, namely, in terms of observables that are governed by the Heisenberg equation and in terms of states governed by the von Neumann equation for the density operator, respectively. An equivalent approach to the description of states of quantum systems is given by means of operators determined by the cluster expansions of the density operator which are interpreted as correlation operators. In this section we consider fundamental equations describing the evolution of correlations of quantum systems with a finite number of particles.

#### 2.1 Preliminaries

We denote by <sup>F</sup> <sup>H</sup> <sup>¼</sup> ⊕∞ <sup>n</sup>¼<sup>0</sup>H<sup>⊗</sup><sup>n</sup> the Fock space over the Hilbert space <sup>H</sup>, where <sup>H</sup><sup>⊗</sup><sup>n</sup> � <sup>H</sup><sup>n</sup> is the <sup>n</sup>-particle Hilbert space. Let <sup>L</sup><sup>1</sup> ð Þ H<sup>n</sup> be the space of trace class operators <sup>f</sup> <sup>n</sup> � <sup>f</sup> <sup>n</sup>ð Þ <sup>1</sup>; …; <sup>n</sup> <sup>∈</sup> <sup>L</sup><sup>1</sup> ð Þ H<sup>n</sup> that satisfy the symmetry condition f <sup>n</sup>ð Þ¼ 1; …; n f <sup>n</sup>ð Þ i1; …; in for arbitrary ð Þ i1; …; in ∈ð Þ 1; …; n and are equipped with the norm

$$\|f\_n\|\_{\mathfrak{L}^1(\mathcal{H}\_n)} = \mathrm{Tr}\_{\mathbf{1},\ldots,n} |f\_n(\mathbf{1},\ldots,n)|\_{\mathfrak{L}^1}$$

where Tr1,…,n are partial traces over 1, …, n particles. We denote by L<sup>1</sup> <sup>0</sup>ð Þ H<sup>n</sup> the everywhere dense set of finite sequences of degenerate operators with infinitely differentiable kernels with compact supports.

On the space of trace class operators L<sup>1</sup> ð Þ H<sup>n</sup> , it is defined as the one-parameter mapping G<sup>∗</sup> <sup>n</sup>ð Þt

$$\mathbb{R}^1 \ni t \mapsto \mathcal{G}\_n^\*(t) f\_n \doteq e^{-itH\_n} f\_n e^{itH\_n},\tag{1}$$

where the following units are used: m ¼ 1 is the mass of a particle, h ¼ 2πћ ¼ 1 is a Planck constant, and the self-adjoint operator Hn is the Hamiltonian of n particles, Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

obeying Maxwell-Boltzmann statistics. Further an inverse group to group (1) will be denoted by G<sup>∗</sup> n � ��<sup>1</sup> ðÞ¼ <sup>t</sup> <sup>G</sup><sup>∗</sup> <sup>n</sup>ð Þ �t .

On its domain of the definition, the infinitesimal generator N <sup>∗</sup> <sup>n</sup> of the group of operators (1) is determined in the sense of the strong convergence of the space L1 ð Þ H<sup>n</sup> by the operator

$$\lim\_{n \to 0} \frac{1}{t} \left( \mathcal{G}\_n^\*(t) f\_n - f\_n \right) = -i \left( H\_n f\_n - f\_n H\_n \right) \doteq \mathcal{N}\_n^\* f\_n. \tag{2}$$

that has the following structure: N <sup>∗</sup> <sup>n</sup> <sup>¼</sup> <sup>∑</sup><sup>n</sup> <sup>j</sup>¼<sup>1</sup><sup>N</sup> <sup>∗</sup> ðÞþ<sup>j</sup> <sup>ϵ</sup>∑<sup>n</sup> j <sup>1</sup> , j <sup>2</sup> <sup>¼</sup> <sup>1</sup><sup>N</sup> <sup>∗</sup> int j 1; j 2 � �, where the operator N <sup>∗</sup> ð Þj is a free motion generator of the von Neumann equation (the dual operator to the generator of the Heisenberg equation for observables) [2], the operator N <sup>∗</sup> int is defined by means of the operator of a two-body interaction potential Φ by the formula N <sup>∗</sup> int j 1; j 2 � � <sup>f</sup> <sup>n</sup> <sup>≐</sup> � <sup>i</sup> <sup>Φ</sup> <sup>j</sup> 1; j 2 � � <sup>f</sup> <sup>n</sup> � <sup>f</sup> <sup>n</sup><sup>Φ</sup> <sup>j</sup> 1; j 2 � � � � , and we denoted a scaling parameter by ϵ . 0.

Let the symbol <sup>∑</sup><sup>P</sup>:ð Þ¼ <sup>1</sup>;…; <sup>s</sup> <sup>∪</sup> jXj denote the sum over all possible partitions P of the set 1ð Þ ; …; s into ∣P∣ nonempty mutually disjoint subsets Xj, and the set f g X<sup>1</sup> ; …; X∣P<sup>∣</sup> � � � � consists from elements which are subsets Xj <sup>⊂</sup> ð Þ <sup>1</sup>; …; <sup>s</sup> of the set ð Þ 1; …; s , i.e., ∣ f g X<sup>1</sup> ; …; X∣P<sup>∣</sup> � � � � <sup>∣</sup> <sup>¼</sup> <sup>∣</sup>P∣. On the space <sup>L</sup><sup>1</sup> ð Þ¼ <sup>F</sup> <sup>H</sup> <sup>⊕</sup><sup>∞</sup> <sup>n</sup>¼<sup>0</sup> <sup>L</sup><sup>1</sup> ð Þ H<sup>n</sup> of sequences <sup>f</sup> <sup>¼</sup> <sup>f</sup> <sup>0</sup>; <sup>f</sup> <sup>1</sup>; …; <sup>f</sup> <sup>n</sup>; … � � of trace class operators <sup>f</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup> ð Þ H<sup>n</sup> and f <sup>0</sup> ∈ C, the following nonlinear one-parameter mapping is defined:

$$\mathcal{G}(t; \mathbf{1}, \ldots, s | \mathcal{f}) \doteq \sum\_{\mathbb{P}: (\mathbf{1}, \ldots, s) = \cup\_j X\_j} \mathfrak{A}\_{|\mathbb{P}|} \left( t, \{ \mathbf{X}\_1 \}, \ldots, \{ \mathbf{X}\_{|\mathbb{P}|} \} \right) \prod\_{X\_j \subset \mathbb{P}} f\_{|X\_j|} (X\_j), \quad s \ge \mathbf{1}, \tag{3}$$

where the generating operator A∣P∣ð Þt of this expansion is the ∣P∣th-order cumulant of the groups of operators (1) defined by the following expansion [2]:

$$\mathfrak{A}\_{|\mathcal{V}|}\left(t,\{X\_1\},\ldots,\{X\_{|\mathcal{V}|}\}\right) \doteq \sum\_{\substack{\mathcal{V}:\{\{X\_1\},\ldots,\{X\_{|\mathcal{V}|}\}\}=\cup\_k Z\_k}} (-1)^{|\mathcal{V}|-1} (|\mathcal{V}|-1)! \prod\_{Z\_k \subset \mathcal{V}} \mathcal{G}^\*\_{|\mathcal{V}(Z\_k)|}(t,\theta(Z\_k)),\tag{4}$$

and θ is the declusterization mapping: θ f g X<sup>1</sup> ; …; X∣P<sup>∣</sup> � � � � <sup>≐</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> . Below we adduce the examples of mapping expansions (3):

$$\begin{split} & \mathcal{G}(t; \mathbf{1}|f) = \mathfrak{A}\_{1}(t, \mathbf{1}) f\_{1}(\mathbf{1}), \\ & \mathcal{G}(t; \mathbf{1}, \mathbf{2}|f) = \mathfrak{A}\_{1}(t, \{1, \mathbf{2}\}) f\_{2}(\mathbf{1}, \mathbf{2}) + \mathfrak{A}\_{1+1}(t, \mathbf{1}, \mathbf{2}) f\_{1}(\mathbf{1}) f\_{1}(\mathbf{2}), \\ & \mathcal{G}(t; \mathbf{1}, \mathbf{2}, \mathbf{3}|f) = \mathfrak{A}\_{1}(t, \{1, \mathbf{2}, \mathbf{3}\}) f\_{3}(\mathbf{1}, \mathbf{2}, \mathbf{3}) + \mathfrak{A}\_{1+1}(t, \mathbf{1}, \{2, \mathbf{3}\}) f\_{1}(\mathbf{1}) f\_{2}(\mathbf{2}, \mathbf{3}) + \\ & \mathfrak{A}\_{1+1}(t, \mathbf{2}, \{1, \mathbf{3}\}) f\_{1}(\mathbf{2}) f\_{2}(\mathbf{1}, \mathbf{3}) + \mathfrak{A}\_{1+1}(t, \mathbf{3}, \{1, \mathbf{2}\}) f\_{1}(\mathbf{3}) f\_{2}(\mathbf{1}, \mathbf{2}) + \\ & \mathfrak{A}\_{3}(t, \mathbf{1}, \mathbf{2}, \mathbf{3}) f\_{1}(\mathbf{1}) f\_{1}(\mathbf{2}) f\_{1}(\mathbf{3}). \end{split}$$

For fs ∈L<sup>1</sup> ð Þ H<sup>s</sup> , s ≥1, the mapping Gð Þ t; 1; …; sj f is defined, and, according to the inequality

$$\|\mathfrak{A}\_{|\mathbb{P}|}\left(t,\{X\_1\},\ldots,\{X\_{|\mathbb{P}|}\}\right)f\_s\|\_{\mathfrak{L}^1(\mathcal{H}\_r)} \le |\mathbb{P}|!\ e^{|\mathbb{P}|}\|f\_s\|\_{\mathfrak{L}^1(\mathcal{H}\_r)^\nu}$$

the following estimate is true:

$$\|\mathcal{G}(t; \mathbf{1}, \dots, \mathbf{s} \vert f)\|\_{\mathcal{L}^1(\mathcal{H}\_t)} \le \mathfrak{s} \text{ } \mathfrak{e}^{2\mathfrak{s}} \mathcal{E} \text{ },\tag{5}$$

where <sup>c</sup> � <sup>e</sup>3max 1; maxP:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>∪</sup> iXi ∥ f∣Xi<sup>∣</sup> <sup>∥</sup>L<sup>1</sup> <sup>H</sup>∣Xi ð Þ<sup>∣</sup> � �. On the space <sup>L</sup><sup>1</sup> ð Þ F <sup>H</sup> , oneparameter mapping (3) is a bounded strong continuous group of nonlinear operators.

operators are governed by the hierarchy of nonlinear evolution equations, known as the quantum nonlinear BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy, and its solution is represented in the form of series, the generating operator of every term of which are the corresponding-order cumulant of groups of nonlinear

In the chapter, we also consider the problem of the rigorous description of the evolution of correlations in quantum many-particle systems by means of a oneparticle (marginal) density operator that is a solution of the generalized quantum kinetic equation with initial correlations [8]. We remark that initial states specified by correlations are typical for the condensed states of many-particle systems in

We note that in modern researches, the conventional approach to the problem of

As known [1, 2], quantum systems of fixed number of particles are described in terms of observables and states. The functional of the mean value of observables defines a duality between observables and states, and as a consequence, there exist two approaches to the description of the evolution of quantum systems, namely, in terms of observables that are governed by the Heisenberg equation and in terms of states governed by the von Neumann equation for the density operator, respectively. An equivalent approach to the description of states of quantum systems is given by means of operators determined by the cluster expansions of the density operator which are interpreted as correlation operators. In this section we consider fundamental equations describing the evolution of correlations of quantum systems

<sup>n</sup>¼<sup>0</sup>H<sup>⊗</sup><sup>n</sup> the Fock space over the Hilbert space <sup>H</sup>, where

ð Þ H<sup>n</sup> that satisfy the symmetry condition

�itHn f <sup>n</sup>e

f <sup>n</sup>ð Þ¼ 1; …; n f <sup>n</sup>ð Þ i1; …; in for arbitrary ð Þ i1; …; in ∈ð Þ 1; …; n and are equipped with the

where Tr1,…,n are partial traces over 1, …, n particles. We denote by L<sup>1</sup>

everywhere dense set of finite sequences of degenerate operators with infinitely

<sup>n</sup>ð Þt f <sup>n</sup> ≐ e

where the following units are used: m ¼ 1 is the mass of a particle, h ¼ 2πћ ¼ 1 is a Planck constant, and the self-adjoint operator Hn is the Hamiltonian of n particles,

ð Þ <sup>H</sup><sup>n</sup> <sup>¼</sup> Tr1,…,n<sup>∣</sup> <sup>f</sup> <sup>n</sup>ð Þ <sup>1</sup>; …; <sup>n</sup> <sup>∣</sup>,

ð Þ H<sup>n</sup> be the space of trace class

ð Þ H<sup>n</sup> , it is defined as the one-parameter

itHn , (1)

<sup>0</sup>ð Þ H<sup>n</sup> the

the rigorous derivation of kinetic equations lies in the construction of various scaling limits of a solution of equations, describing the evolution of the state of many-particle systems [9], in particular, a mean field limit of a perturbative solution of the BBGKY hierarchy for a sequence of marginal density operators [10–17].

operators of the von Neumann hierarchy for correlation operators [7].

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

contrast to their gaseous state [1].

with a finite number of particles.

We denote by <sup>F</sup> <sup>H</sup> <sup>¼</sup> ⊕∞

operators <sup>f</sup> <sup>n</sup> � <sup>f</sup> <sup>n</sup>ð Þ <sup>1</sup>; …; <sup>n</sup> <sup>∈</sup> <sup>L</sup><sup>1</sup>

<sup>H</sup><sup>⊗</sup><sup>n</sup> � <sup>H</sup><sup>n</sup> is the <sup>n</sup>-particle Hilbert space. Let <sup>L</sup><sup>1</sup>

differentiable kernels with compact supports. On the space of trace class operators L<sup>1</sup>

∥ f <sup>n</sup>∥L<sup>1</sup>

R<sup>1</sup> ∍ t ↦ G<sup>∗</sup>

2.1 Preliminaries

norm

mapping G<sup>∗</sup>

32

<sup>n</sup>ð Þt

2. Dynamics of quantum correlations

#### 2.2 The von Neumann hierarchy for correlation operators

The evolution of all possible states of a quantum system of non-fixed, i.e., arbitrary but finite, number of identical particles, obeying the Maxwell-Boltzmann statistics, can be described by means of the sequence g tðÞ¼ g0; g1ð Þt ; …; gs ð Þ<sup>t</sup> ; … � �∈L<sup>1</sup> ð Þ F <sup>H</sup> of the correlation operators gs ðÞ¼ t gs ð Þ t; 1; …; s , s ≥1, governed by the Cauchy problem of the von Neumann hierarchy [5]:

$$\frac{\partial}{\partial t}\mathbf{g}\_s(t, \mathbf{1}, \dots, s) = \mathcal{N}\_s^\* \mathbf{g}\_s(t, \mathbf{1}, \dots, s) + \tag{6}$$

$$\mathop{\rm c}\limits\_{\mathbf{P}: \langle \mathbf{1}, \dots, s \rangle = X\_1 \cup X\_2} \sum\_{i\_1 \in X\_1} \sum\_{i\_2 \in X\_2} \mathcal{N}\_{\text{int}}^\*(i\_1, i\_2) \mathbf{g}\_{|X\_1|}(t, X\_1) \mathbf{g}\_{|X\_2|}(t, X\_2), \tag{7}$$

$$\mathbf{g}\_s(t) \big|\_{t=0} = \mathbf{g}\_s^{0, \epsilon}, \quad s \ge 1, \tag{7}$$

(iteration) expansion as a result of the application of analogs of the Duhamel

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

The following statement is true [6]. In the case of bounded interaction potentials for t∈ R, a solution of the Cauchy problem of the von Neumann hierarchy (6) and (7) is determined by a sequence of correlation operators represented by formula

The stated above results can be extended to quantum systems of bosons and

An equivalent approach in describing the states of quantum systems of many particles consists in describing states by means of marginal density operators governed by the BBGKI hierarchy or by means of operators determined by their cluster expansions, which are interpreted as marginal correlation operators [1]. On the microscopic scale, the macroscopic characteristics of fluctuations of observables are directly determined by the marginal correlation operators. Such approach allows us to describe the evolution of correlations in quantum systems both with finite and

3.1 The hierarchy of evolution equations for marginal correlation operators

Traditionally marginal correlation operators are determined by means of the cluster expansions of the marginal density operators [2–4]. We introduce the marginal correlation operators in the framework of the solution of the Cauchy problem for the von Neumann hierarchy (6) and (7) by the following series expansions:

According to estimate (5), series (10) exists and the following estimate holds:

We remark that the macroscopic characteristics of fluctuations of observables are directly determined by marginal correlation operators (10), for example, the

where <sup>A</sup>ð Þ<sup>1</sup> D EðÞ¼ <sup>t</sup> Tr1 <sup>a</sup>1ð Þ<sup>1</sup> <sup>G</sup>1ð Þ <sup>t</sup>; <sup>1</sup> is a mean-value functional of the additive-

Then the evolution of all possible states of large quantum particle systems, obeying the Maxwell-Boltzmann statistics, can be described by means of the

ators governed by the Cauchy problem of the following hierarchy of nonlinear

evolution equations (the nonlinear quantum BBGKY hierarchy):

c<sup>n</sup>, where

� �, is represented by the formula [1]

ð Þ <sup>0</sup> <sup>∥</sup>L<sup>1</sup> <sup>H</sup>∣Xi ð Þ<sup>∣</sup>

3. The evolution of correlations in large quantum particle systems

ð Þ H<sup>n</sup> , it is a strong solution, and for arbitrary initial data

Tr<sup>s</sup>þ1,…,sþ<sup>n</sup> Gð Þ t; 1; …; s þ njgð Þ 0 , s ≥1: (10)

<sup>1</sup>ð Þ� <sup>1</sup> <sup>A</sup>ð Þ<sup>1</sup> D E<sup>2</sup>

þ Tr1, <sup>2</sup> a1ð Þ1 a1ð Þ2 G2ð Þ t; 1; 2 ,

ð Þt � �G1ð Þ <sup>t</sup>; <sup>1</sup>

ð Þ F <sup>H</sup> of marginal correlation oper-

equation to cumulants (4) of groups of operators (1).

<sup>0</sup>ð Þ <sup>H</sup><sup>n</sup> <sup>⊂</sup> <sup>L</sup><sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82836

ð Þ H<sup>n</sup> , it is a weak solution.

(8). If g0, <sup>ϵ</sup> <sup>n</sup> ∈ L<sup>1</sup>

fermions like in paper [6].

infinite number of particles.

Gsð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> ≐ ∑∞

ð Þ <sup>H</sup><sup>s</sup> <sup>≤</sup> <sup>s</sup>! <sup>2</sup>e<sup>2</sup> ð Þ<sup>s</sup>

<sup>c</sup> � <sup>e</sup>3max 1; maxP:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>∪</sup> iXi

<sup>A</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>0</sup>; <sup>a</sup>1ð Þ<sup>1</sup> ; …; <sup>∑</sup><sup>n</sup>

type observable [2].

35

∥Gsð Þt ∥L<sup>1</sup>

n¼0

cs ∑<sup>∞</sup>

1 n!

<sup>n</sup>¼<sup>0</sup> <sup>2</sup>e<sup>2</sup> ð Þ<sup>n</sup>

� �.

<sup>i</sup>1¼<sup>1</sup>a1ð Þ <sup>i</sup><sup>1</sup> ; …

sequence G tðÞ¼ ð Þ <sup>I</sup>; <sup>G</sup>1ð Þ<sup>t</sup> ; <sup>G</sup>2ð Þ<sup>t</sup> ; …; Gsð Þ<sup>t</sup> ; … <sup>∈</sup> <sup>L</sup><sup>1</sup>

<sup>A</sup>ð Þ<sup>1</sup> � <sup>A</sup>ð Þ<sup>1</sup> � � D E <sup>2</sup> � �ðÞ¼ <sup>t</sup> Tr1 <sup>a</sup><sup>2</sup>

∥g∣Xi<sup>∣</sup>

functional of the dispersion of the additive-type observables, i.e.,

g0, <sup>ϵ</sup> <sup>n</sup> ∈L<sup>1</sup>

where <sup>ϵ</sup> , 0 is a scaling parameter, the symbol <sup>∑</sup><sup>P</sup>:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>X</sup><sup>1</sup> <sup>∪</sup>X<sup>2</sup> means the sum over all possible partitions P of the set 1ð Þ ; …; s into two nonempty mutually disjoint subsets X<sup>1</sup> and X2, and the operator N <sup>∗</sup> <sup>s</sup> is defined on the subspace L<sup>1</sup> <sup>0</sup>ð Þ H<sup>s</sup> by formula (2).

We remark that correlation operators can be introduced by means of the cluster expansions [2] of the density operators (the kernel of a density operator is known as a density matrix) governed by a sequence of the von Neumann equations, and hence, they describe the evolution of states by an equivalent method in comparison with the density operators. For quantum systems of fixed number of particles, the state is described by finite sequence of correlation operators governed by a corresponding system of the von Neumann equations (6).

A solution (nonperturbative solution) of the Cauchy problem of the von Neumann hierarchy for correlation operators (6) and (7) is represented by group of nonlinear operators (3)

$$\mathbf{g}(t; \mathbf{1}, \ldots, \mathbf{s}) = \mathcal{G}(t; \mathbf{1}, \ldots, \mathbf{s} | \mathbf{g}(\mathbf{0})), \quad \mathbf{s} \ge \mathbf{1}, \tag{8}$$

where a sequence of initial correlation operators (7) is denoted by gð Þ¼ 0 g0; g 0, ϵ <sup>1</sup> ; …; <sup>g</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> ; … � � and <sup>g</sup><sup>0</sup> <sup>∈</sup> <sup>C</sup>.

We remark, if at initial time there are no correlations between particles, i.e., in the case of initial states, satisfying a chaos condition [2], a sequence of initial correlation operators takes the form gð Þ¼ 0 0; g 0, ϵ <sup>1</sup> ; <sup>0</sup>; …; <sup>0</sup>; … � �. Then solution (8) of the Cauchy problem of the von Neumann hierarchy (6) and (7) is represented by the following expansions:

$$\mathbf{g}\_s(t, \mathbf{1}, \ldots, s) = \mathfrak{A}\_s(t, \mathbf{1}, \ldots, s) \prod\_{i=1}^s \mathbf{g}\_1^{0, e}(i), \quad s \ge \mathbf{1}, s$$

where the operator Asð Þt is the sth-order cumulant of groups of operators (1) determined by the expansion

$$\mathfrak{A}\_{\mathbf{s}}(t,\mathbf{1},\ldots,\mathbf{s}) = \sum\_{\mathbf{P}:(\mathbf{1},\ldots,\mathbf{s})=\cup\_{i}X\_{i}} (-\mathbf{1})^{|\mathbf{P}|-1} (|\mathbf{P}|-\mathbf{1})! \prod\_{X\_{i}\subset\mathbf{P}} \mathcal{G}^{\*}\_{|\mathbf{X}\_{i}|}(t,X\_{i}),\tag{9}$$

and we used notations accepted in formula (3).

We remark also that nonperturbative solution (8) of the Cauchy problem of the von Neumann hierarchy (6) and (7) can be transformed to the perturbation Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

(iteration) expansion as a result of the application of analogs of the Duhamel equation to cumulants (4) of groups of operators (1).

The following statement is true [6]. In the case of bounded interaction potentials for t∈ R, a solution of the Cauchy problem of the von Neumann hierarchy (6) and (7) is determined by a sequence of correlation operators represented by formula (8). If g0, <sup>ϵ</sup> <sup>n</sup> ∈ L<sup>1</sup> <sup>0</sup>ð Þ <sup>H</sup><sup>n</sup> <sup>⊂</sup> <sup>L</sup><sup>1</sup> ð Þ H<sup>n</sup> , it is a strong solution, and for arbitrary initial data g0, <sup>ϵ</sup> <sup>n</sup> ∈L<sup>1</sup> ð Þ H<sup>n</sup> , it is a weak solution.

The stated above results can be extended to quantum systems of bosons and fermions like in paper [6].

#### 3. The evolution of correlations in large quantum particle systems

An equivalent approach in describing the states of quantum systems of many particles consists in describing states by means of marginal density operators governed by the BBGKI hierarchy or by means of operators determined by their cluster expansions, which are interpreted as marginal correlation operators [1]. On the microscopic scale, the macroscopic characteristics of fluctuations of observables are directly determined by the marginal correlation operators. Such approach allows us to describe the evolution of correlations in quantum systems both with finite and infinite number of particles.

#### 3.1 The hierarchy of evolution equations for marginal correlation operators

Traditionally marginal correlation operators are determined by means of the cluster expansions of the marginal density operators [2–4]. We introduce the marginal correlation operators in the framework of the solution of the Cauchy problem for the von Neumann hierarchy (6) and (7) by the following series expansions:

$$G\_{\varepsilon}(t, \mathbf{1}, \ldots, \mathbf{s}) \doteq \sum\_{n=0}^{\infty} \frac{\mathbf{1}}{n!} \operatorname{Tr}\_{t+1, \ldots, t+n} \mathcal{G}(t; \mathbf{1}, \ldots, \mathbf{s} + n \lfloor \mathbf{g}(\mathbf{0}) \rfloor), \quad \mathbf{s} \ge \mathbf{1}. \tag{10}$$

According to estimate (5), series (10) exists and the following estimate holds: ∥Gsð Þt ∥L<sup>1</sup> ð Þ <sup>H</sup><sup>s</sup> <sup>≤</sup> <sup>s</sup>! <sup>2</sup>e<sup>2</sup> ð Þ<sup>s</sup> cs ∑<sup>∞</sup> <sup>n</sup>¼<sup>0</sup> <sup>2</sup>e<sup>2</sup> ð Þ<sup>n</sup> c<sup>n</sup>, where

<sup>c</sup> � <sup>e</sup>3max 1; maxP:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>∪</sup> iXi ∥g∣Xi<sup>∣</sup> ð Þ <sup>0</sup> <sup>∥</sup>L<sup>1</sup> <sup>H</sup>∣Xi ð Þ<sup>∣</sup> � �.

We remark that the macroscopic characteristics of fluctuations of observables are directly determined by marginal correlation operators (10), for example, the functional of the dispersion of the additive-type observables, i.e.,

<sup>A</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>0</sup>; <sup>a</sup>1ð Þ<sup>1</sup> ; …; <sup>∑</sup><sup>n</sup> <sup>i</sup>1¼<sup>1</sup>a1ð Þ <sup>i</sup><sup>1</sup> ; … � �, is represented by the formula [1]

$$
\begin{split}
\left\langle \left( \mathbf{A}^{(1)} - \left\langle \mathbf{A}^{(1)} \right\rangle \right)^{2} \right\rangle (t) &= \operatorname{Tr}\_{1} \left( a\_{1}^{2}(\mathbf{1}) - \left\langle \mathbf{A}^{(1)} \right\rangle^{2} (t) \right) \mathbf{G}\_{1}(t, \mathbf{1}) \\ &+ \operatorname{Tr}\_{1,2} a\_{1}(\mathbf{1}) a\_{1}(\mathbf{2}) \mathbf{G}\_{2}(t, \mathbf{1}, \mathbf{2}),
\end{split}
$$

where <sup>A</sup>ð Þ<sup>1</sup> D EðÞ¼ <sup>t</sup> Tr1 <sup>a</sup>1ð Þ<sup>1</sup> <sup>G</sup>1ð Þ <sup>t</sup>; <sup>1</sup> is a mean-value functional of the additivetype observable [2].

Then the evolution of all possible states of large quantum particle systems, obeying the Maxwell-Boltzmann statistics, can be described by means of the sequence G tðÞ¼ ð Þ <sup>I</sup>; <sup>G</sup>1ð Þ<sup>t</sup> ; <sup>G</sup>2ð Þ<sup>t</sup> ; …; Gsð Þ<sup>t</sup> ; … <sup>∈</sup> <sup>L</sup><sup>1</sup> ð Þ F <sup>H</sup> of marginal correlation operators governed by the Cauchy problem of the following hierarchy of nonlinear evolution equations (the nonlinear quantum BBGKY hierarchy):

2.2 The von Neumann hierarchy for correlation operators

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

can be described by means of the sequence g tðÞ¼ g0; g1ð Þt ; …; gs

s gs

∑ i<sup>2</sup> ∈X<sup>2</sup>

∑ i<sup>1</sup> ∈X<sup>1</sup>

> gs ð Þt � �

corresponding system of the von Neumann equations (6).

ð Þþ t; 1; …; s

N <sup>∗</sup>

<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>g</sup>0, <sup>ε</sup>

where <sup>ϵ</sup> , 0 is a scaling parameter, the symbol <sup>∑</sup><sup>P</sup>:ð Þ¼ <sup>1</sup>;…;<sup>s</sup> <sup>X</sup><sup>1</sup> <sup>∪</sup>X<sup>2</sup> means the sum over all possible partitions P of the set 1ð Þ ; …; s into two nonempty mutually disjoint

We remark that correlation operators can be introduced by means of the cluster expansions [2] of the density operators (the kernel of a density operator is known as a density matrix) governed by a sequence of the von Neumann equations, and hence, they describe the evolution of states by an equivalent method in comparison with the density operators. For quantum systems of fixed number of particles, the state is described by finite sequence of correlation operators governed by a

A solution (nonperturbative solution) of the Cauchy problem of the von Neumann hierarchy for correlation operators (6) and (7) is represented by group of

We remark, if at initial time there are no correlations between particles, i.e., in

where a sequence of initial correlation operators (7) is denoted by

ð Þ¼ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>A</sup>sð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>Y</sup><sup>s</sup>

P:ð Þ¼ 1;…;s ∪ iXi

the case of initial states, satisfying a chaos condition [2], a sequence of initial

the Cauchy problem of the von Neumann hierarchy (6) and (7) is represented by

where the operator Asð Þt is the sth-order cumulant of groups of operators (1)

ð Þ �<sup>1</sup> <sup>∣</sup>P∣�<sup>1</sup>

We remark also that nonperturbative solution (8) of the Cauchy problem of the von Neumann hierarchy (6) and (7) can be transformed to the perturbation

intð Þ i1; i<sup>2</sup> g∣X1<sup>∣</sup>

ðÞ¼ t gs

ð Þ¼ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>N</sup> <sup>∗</sup>

the correlation operators gs

the von Neumann hierarchy [5]:

ϵ ∑ P:ð Þ¼ 1;…;s X<sup>1</sup> ∪ X<sup>2</sup>

subsets X<sup>1</sup> and X2, and the operator N <sup>∗</sup>

formula (2).

nonlinear operators (3)

0, ϵ

the following expansions:

determined by the expansion

<sup>1</sup> ; …; <sup>g</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> ; … � � and <sup>g</sup><sup>0</sup> <sup>∈</sup> <sup>C</sup>.

correlation operators takes the form gð Þ¼ 0 0; g

gs

Asð Þ¼ t; 1; …; s ∑

and we used notations accepted in formula (3).

gð Þ¼ 0 g0; g

34

∂ ∂t gs

The evolution of all possible states of a quantum system of non-fixed, i.e., arbitrary but finite, number of identical particles, obeying the Maxwell-Boltzmann statistics,

ð Þ<sup>t</sup> ; … � �∈L<sup>1</sup>

ð Þ t; 1; …; s , s ≥1, governed by the Cauchy problem of

ð Þ t;X<sup>1</sup> g∣X2<sup>∣</sup>

<sup>s</sup> is defined on the subspace L<sup>1</sup>

g tð Þ¼ ; 1; …; s Gð Þ t; 1; …; sjgð Þ 0 , s ≥1, (8)

0, ϵ

i¼1 g 0, ϵ

ð Þ jPj � 1 !

<sup>1</sup> ; <sup>0</sup>; …; <sup>0</sup>; … � �. Then solution (8) of

<sup>1</sup> ð Þi , s≥ 1,

Y Xi ⊂P

G∗ ∣Xi∣

ð Þ t;Xi , (9)

<sup>s</sup> , s ≥1, (7)

ð Þ t;X<sup>2</sup> ,

<sup>0</sup>ð Þ H<sup>s</sup> by

ð Þ F <sup>H</sup> of

(6)

$$\frac{\partial}{\partial t} G\_i(t, \mathbf{1}, \ldots, s) = \mathcal{N}\_i^\* G\_i(t, \mathbf{1}, \ldots, s) +$$

$$\begin{split} \mathop{\rm c}\limits\_{\mathbf{P} : (\mathbf{1}, \ldots, s) = X\_1 \cup X\_2} & \sum\_{i\_1 \in X\_1} \sum\_{i\_2 \in X\_2} \mathcal{N}\_{\text{int}}^\*(i\_1, i\_2) G\_{|\mathcal{X}\_1|}(t, X\_1) G\_{|\mathcal{X}\_2|}(t, X\_2) \\ \mathop{\rm c}\limits\_{i \in \mathcal{X}} \sum\_{i \in \mathcal{Y}} \mathcal{N}\_{\text{int}}^\*(i, s + 1) (G\_{i+1}(t, 1, \ldots, s + 1) + \\ & \sum\_{i \in \mathcal{Y}, s \ge 1} G\_{|\mathcal{X}\_1|}(t, X\_1) G\_{|\mathcal{X}\_2|}(t, X\_2)), \\ \mathop{\rm P}\limits\_{i} \mathbf{P} : (\mathbf{1}, \ldots, s + \mathbf{1}) = X\_1 \cup X\_2, \\ \mathop{\rm c}\limits\_{i} i \in X\_1; s + \mathbf{1} \in X\_2 \end{split} \tag{11}$$

$$G\_i(t)|\_{t = 0} = G\_i^{0, c}, \quad s \ge \mathbf{1}, \tag{12}$$

is the third-order cumulant (9) of groups of operators (1).

<sup>G</sup>ð Þ<sup>c</sup> <sup>¼</sup> <sup>0</sup>; <sup>G</sup>0, <sup>ϵ</sup>

marginal correlation operators (13) are represented by the following series

operators

expansions:

operators

<sup>F</sup>ð Þ<sup>c</sup> <sup>¼</sup> I, F<sup>0</sup>, <sup>ϵ</sup>

Fsð Þ¼ t; 1; …; s ∑

operators <sup>g</sup>ð Þ¼ <sup>0</sup> I, g<sup>0</sup>, <sup>ϵ</sup>

g<sup>0</sup>, <sup>ϵ</sup>

37

ginal correlation operators

<sup>s</sup> ð Þ <sup>1</sup>; …; <sup>s</sup> ≐ ∑∞

Gðt; 1; …; s þ njfÞ ¼ ∑

We remark that on the space L<sup>1</sup>

n¼0

ð Þ �<sup>1</sup> <sup>n</sup> <sup>1</sup> n!

P:ð Þ¼ 1;…;sþn ∪ kXk

sive relations (18) are represented by expansions (14).

of operators (1).

Gsð Þ¼ t; 1; …; s ∑

groups of operators (1).

<sup>1</sup> ð Þ<sup>1</sup> , …, � <sup>Q</sup><sup>n</sup>

∞ n¼0

1 n!

∞ n¼0

DOI: http://dx.doi.org/10.5772/intechopen.82836

1 n!

<sup>F</sup><sup>0</sup>, <sup>ϵ</sup> <sup>s</sup> ð Þ¼ <sup>1</sup>; …; <sup>s</sup> <sup>∑</sup>

<sup>i</sup>¼<sup>1</sup> <sup>F</sup><sup>0</sup>, <sup>ϵ</sup>

In the case of initial data specified by the sequence of marginal correlation

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

i.e., initial states satisfying a chaos property [9], according to definition (14),

Trsþ1,…,sþ<sup>n</sup> <sup>A</sup>sþnð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> <sup>Y</sup>sþ<sup>n</sup>

where the generating operator Asþnð Þt is the ð Þ s þ n th-order cumulant (9) of

We note that within the framework of the description of states by means of marginal density operators defined by cluster expansions over marginal correlation

> Y Xi ⊂P G<sup>0</sup>, <sup>ϵ</sup> ∣Xi∣

Tr<sup>s</sup>þ1,…,sþ<sup>n</sup> <sup>A</sup><sup>1</sup>þ<sup>n</sup>ð Þ <sup>t</sup>; f g <sup>1</sup>; …; <sup>s</sup> ; <sup>s</sup> <sup>þ</sup> <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> <sup>Y</sup><sup>s</sup>þ<sup>n</sup>

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þt is the 1ð Þ þ n th-order cumulant of groups

<sup>1</sup> ð Þ<sup>1</sup> , …, � <sup>g</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> ð Þ <sup>1</sup>; …; <sup>n</sup> , …<sup>Þ</sup> determined by means of the mar-

Tr<sup>s</sup>þ1,…,sþ<sup>n</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup> <sup>s</sup>þ<sup>n</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> , s≥1, (17)

<sup>A</sup>∣X1<sup>∣</sup> <sup>t</sup>;X1<sup>j</sup> … <sup>A</sup>∣X∣<sup>P</sup>∣∣ <sup>t</sup>;X∣P∣<sup>j</sup> <sup>f</sup> � �… � �, n <sup>≥</sup>0, (18)

ð Þ F <sup>H</sup> , the generating operator (14) of series

One of the possible methods to derive series expansion (13) for the marginal correlation operators lies in the substitution of the cluster expansions of groups of nonlinear operators (3) over cumulants (14) and the sequence of initial correlation

into the definition of marginal correlation operators (10). Indeed, developing

according to definition (17), we derive expressions (13). The solutions of recur-

expansion (13) can be represented as the 1ð Þ þ n th-order reduced cumulant of the

groups of nonlinear operators (3) of the von Neumann hierarchy [2]:

the generating operators of series (13) as the following cluster expansions:

P:ð Þ¼ 1;…;s ∪ iXi

density operators are represented by the following series expansions (a nonperturbative solution of the quantum BBGKY hierarchy [2]):

initial states described like to sequence (15) is specified by the sequence

<sup>1</sup> ; <sup>0</sup>; …; <sup>0</sup>; … � �, (15)

G0, <sup>ϵ</sup>

ð Þ Xi , s≥1,

i¼1

F<sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þi , s≥ 1,

<sup>1</sup> ð Þi , s ≥1, (16)

i¼1

<sup>1</sup> ð Þi , …Þ, and in the case of sequence (16), the marginal

where ϵ . 0 is a scaling parameter and we use accepted in hierarchy (6) notations.

If <sup>G</sup>ð Þ¼ <sup>0</sup> <sup>I</sup>; <sup>G</sup>0, <sup>ϵ</sup> <sup>1</sup> ð Þ<sup>1</sup> ; …; <sup>G</sup>0, <sup>ϵ</sup> <sup>s</sup> ð Þ <sup>1</sup>; …; <sup>s</sup> ; … is a sequence of initial marginal correlation operators (12), then a nonperturbative solution of the Cauchy problem (11) and (12) is represented by the following sequence of self-adjoint operators:

$$\mathcal{G}\_{\boldsymbol{s}}(t, \mathbf{1}, \ldots, \mathbf{s}) = \sum\_{n=0}^{\infty} \frac{\mathbf{1}}{n!} \operatorname{Tr}\_{\boldsymbol{\tau}+\mathbf{1}, \ldots, \boldsymbol{s}+n} \mathfrak{A}\_{1+n}(t; \{1, \ldots, \boldsymbol{s}\}, \boldsymbol{s}+\mathbf{1}, \ldots, \boldsymbol{s}+n | \mathcal{G}(\mathbf{0})), \quad \boldsymbol{s} \ge \mathbf{1}, \tag{13}$$

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þ t; f g 1; …; s ; s þ 1; …; s þ njGð Þ 0 of this series is the 1ð Þ þ n th-order cumulant of groups of nonlinear operators (3):

$$\begin{aligned} \mathfrak{A}\_{1+n}(t; \{1, \ldots, s\}, s+1, \ldots, s+n | G(\mathbf{0})) & \doteq \\ \sum\_{\mathbf{P} : (\{1, \ldots, s\}, s+1, \ldots, s+n) = \cup\_k X\_k} (-\mathbf{1})^{|\mathbf{P}|-1} (|\mathbf{P}|-\mathbf{1}) \mathcal{G} \{t; \theta(X\_1) | \ldots \mathcal{G} \{t; \theta(X\_{|\mathbf{P}|}) | G(\mathbf{0}) \} \ldots \}, \quad n \geq 0, \end{aligned} \tag{14}$$

and composition of mappings (3) of the corresponding noninteracting groups of particles we denote by G t; θð Þj X<sup>1</sup> …G t; θ X∣P<sup>∣</sup> <sup>j</sup>Gð Þ <sup>0</sup> … , for example,

$$\begin{aligned} \mathcal{G}(t; \mathbf{1}|\mathcal{G}(t; 2|f)) &= \mathfrak{A}\_1(t, \mathbf{1}) \mathfrak{A}\_1(t, \mathbf{2}) f\_2(\mathbf{1}, \mathbf{2}), \\ \mathcal{G}(t; \mathbf{1}, \mathbf{2}|\mathcal{G}(t; \mathbf{3}|f)) &= \mathfrak{A}\_1(t, \{\mathbf{1}, \mathbf{2}\}) \mathfrak{A}\_1(t, \mathbf{3}) f\_3(\mathbf{1}, \mathbf{2}, \mathbf{3}) + \mathfrak{B}\_2(t, \mathbf{1}, \mathbf{2}) \mathfrak{A}\_2(t, \mathbf{3}) f\_4(\mathbf{1}, \mathbf{2}) f\_5(\mathbf{1}, \mathbf{3}), \\ \mathfrak{A}\_2(t, \mathbf{1}, \mathbf{2}) \mathfrak{A}\_1(t, \mathbf{3}) \left( f\_1(\mathbf{1}) f\_2(\mathbf{2}, \mathbf{3}) + f\_1(\mathbf{2}) f\_2(\mathbf{1}, \mathbf{3}) \right). \end{aligned}$$

Below we adduce the examples of expansions (14). The first-order cumulant of the groups of nonlinear operators (3) is the same group of nonlinear operators, i.e.,

$$
\mathfrak{A}\_1(t; \{\mathbf{1}, \ldots, s\} | G(\mathbf{0})) = \mathcal{G}(t; \mathbf{1}, \ldots, s | G(\mathbf{0})) .
$$

In the case of s ¼ 2, the second-order cumulant of nonlinear operators (3) has the structure

$$\begin{split} \mathfrak{A}\_{1+1}(\mathfrak{r},\{1,2\},\mathfrak{J}(\mathbf{G}(\mathbf{0}))) &= \mathfrak{G}(\mathfrak{r};\mathbf{1},2,\mathfrak{J}(\mathbf{G}(\mathbf{0})) - \mathfrak{G}(\mathfrak{r};\mathbf{1},2|\mathfrak{G}(\mathbf{t};\mathbf{3})\mathbf{G}(\mathbf{0}))) = \\ \mathfrak{A}\_{1+1}(\mathfrak{r},\{1,2\},\mathfrak{J}(\mathbf{2},\mathbf{3}) + (\mathfrak{A}\_{1+1}(\mathfrak{r},\{1,2\},\mathfrak{J}) - \mathfrak{A}\_{1+1}(\mathfrak{r},\mathbf{2},\mathfrak{J})\mathfrak{A}\_{1}(\mathfrak{r},\mathbf{1})) \mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{1})\mathbf{G}\_{2}^{0,\varepsilon}(\mathbf{2},\mathfrak{J}) + \\ (\mathfrak{A}\_{1+1}(\mathfrak{r},\{1,2\},\mathfrak{J}) - \mathfrak{A}\_{1+1}(\mathfrak{r},\mathbf{1},\mathfrak{J})\mathfrak{A}\_{1}(\mathfrak{r},\mathbf{2})) \mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{2})\mathbf{G}\_{2}^{0,\varepsilon}(\mathbf{1},\mathbf{3}) + \\ \mathfrak{A}\_{1+1}(\mathfrak{r},\{1,2\},\mathfrak{J})\mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{3})\mathbf{G}\_{2}^{0,\varepsilon}(\mathbf{1},\mathbf{2}) + \mathfrak{A}\_{3}(\mathfrak{r},\mathbf{1},\mathbf{2},\mathfrak{J})\mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{1})\mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{2})\mathbf{G}\_{1}^{0,\varepsilon}(\mathbf{3}), \end{split}$$

where the operator

$$
\mathfrak{A}\_3(t, \mathbf{1}, \mathbf{2}, \mathbf{3}) = \mathfrak{A}\_{1+1}(t, \{\mathbf{1}, \mathbf{2}\}, \mathbf{3}) - \mathfrak{A}\_{1+1}(t, \mathbf{2}, \mathbf{3})\mathfrak{A}\_1(t, \mathbf{1}) - \mathfrak{A}\_{1+1}(t, \mathbf{1}, \mathbf{3})\mathfrak{A}\_1(t, \mathbf{2}) - 
$$

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

is the third-order cumulant (9) of groups of operators (1).

In the case of initial data specified by the sequence of marginal correlation operators

$$\mathbf{G}^{(\epsilon)} = \left(\mathbf{0}, \mathbf{G}\_1^{0,\epsilon}, \mathbf{0}, \dots, \mathbf{0}, \dots\right), \tag{15}$$

i.e., initial states satisfying a chaos property [9], according to definition (14), marginal correlation operators (13) are represented by the following series expansions:

$$G\_{\boldsymbol{\varepsilon}}(t, \mathbf{1}, \ldots, \boldsymbol{\varepsilon}) = \sum\_{n=0}^{\infty} \frac{1}{n!} \operatorname{Tr}\_{\boldsymbol{\varepsilon}+\mathbf{1}, \ldots, \boldsymbol{\varepsilon}+n} \mathfrak{A}\_{\boldsymbol{\varepsilon}+n}(t; \mathbf{1}, \ldots, \boldsymbol{\varepsilon} + n) \prod\_{i=1}^{\varepsilon+n} G\_1^{0, \boldsymbol{\varepsilon}}(i), \quad \boldsymbol{\varepsilon} \ge \mathbf{1}, \tag{16}$$

where the generating operator Asþnð Þt is the ð Þ s þ n th-order cumulant (9) of groups of operators (1).

We note that within the framework of the description of states by means of marginal density operators defined by cluster expansions over marginal correlation operators

$$F\_s^{0,e}(\mathbf{1},\ldots,\mathbf{s}) = \sum\_{\mathbf{P}:(\mathbf{1},\ldots,\mathbf{s})=\cup\_i X\_i} \prod\_{X\_i \subset \mathbf{P}} G\_{|X\_i|}^{0,e}(X\_i), \quad s \ge \mathbf{1},$$

initial states described like to sequence (15) is specified by the sequence <sup>F</sup>ð Þ<sup>c</sup> <sup>¼</sup> I, F<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ð Þ<sup>1</sup> , …, � <sup>Q</sup><sup>n</sup> <sup>i</sup>¼<sup>1</sup> <sup>F</sup><sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ð Þi , …Þ, and in the case of sequence (16), the marginal density operators are represented by the following series expansions (a nonperturbative solution of the quantum BBGKY hierarchy [2]):

$$F\_{\boldsymbol{\varepsilon}}(t, \mathbf{1}, \ldots, \boldsymbol{\varepsilon}) = \sum\_{n=0}^{\infty} \frac{1}{n!} \operatorname{Tr}\_{\boldsymbol{\varepsilon} + \mathbf{1}, \ldots, \boldsymbol{\varepsilon} + n} \mathfrak{A}\_{1 + n}(t; \{1, \ldots, \boldsymbol{\varepsilon}\}, \boldsymbol{\varepsilon} + \mathbf{1}, \ldots, \boldsymbol{\varepsilon} + n) \prod\_{i=1}^{\varepsilon + n} F\_1^{0, \boldsymbol{\varepsilon}}(i), \quad \boldsymbol{\varepsilon} \ge \mathbf{1}, \ldots, \boldsymbol{\varepsilon} + n$$

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þt is the 1ð Þ þ n th-order cumulant of groups of operators (1).

One of the possible methods to derive series expansion (13) for the marginal correlation operators lies in the substitution of the cluster expansions of groups of nonlinear operators (3) over cumulants (14) and the sequence of initial correlation operators <sup>g</sup>ð Þ¼ <sup>0</sup> I, g<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ð Þ<sup>1</sup> , …, � <sup>g</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> ð Þ <sup>1</sup>; …; <sup>n</sup> , …<sup>Þ</sup> determined by means of the marginal correlation operators

$$\mathbf{g}\_s^{0,\epsilon}(\mathbf{1},...,\mathbf{s}) \doteq \sum\_{n=0}^{\infty} (-1)^n \frac{\mathbf{1}}{n!} \operatorname{Tr}\_{\mathbf{s}+\mathbf{1},...,\mathbf{s}+n} \operatorname{G}\_{s+n}^{0,\epsilon}(\mathbf{1},...,\mathbf{s}+n), \quad \varepsilon \ge \mathbf{1},\tag{17}$$

into the definition of marginal correlation operators (10). Indeed, developing the generating operators of series (13) as the following cluster expansions:

$$\mathcal{G}(\mathfrak{t}; \mathbf{1}, \ldots, \mathfrak{s} + n|\mathfrak{f}) = \sum\_{\mathbf{P} : (\mathbf{1}, \ldots, \mathfrak{s} + n) = \cup\_k X\_k} \mathfrak{A}\_{|\mathbf{X}\_1|} \left( \mathfrak{t}; X\_1 | \ldots \mathfrak{A}\_{|\mathbf{X}\_{|\mathbf{P}|}|} \left( \mathfrak{t}; X\_{|\mathbf{P}|} | f \right) \ldots \right), \quad n \ge \mathbf{0}, \tag{18}$$

according to definition (17), we derive expressions (13). The solutions of recursive relations (18) are represented by expansions (14).

We remark that on the space L<sup>1</sup> ð Þ F <sup>H</sup> , the generating operator (14) of series expansion (13) can be represented as the 1ð Þ þ n th-order reduced cumulant of the groups of nonlinear operators (3) of the von Neumann hierarchy [2]:

∂ ∂t

ϵ ∑ P:ð Þ¼ 1;…;s X<sup>1</sup> ∪X<sup>2</sup>

ϵTrsþ<sup>1</sup> ∑ i∈Y N <sup>∗</sup>

If <sup>G</sup>ð Þ¼ <sup>0</sup> <sup>I</sup>; <sup>G</sup>0, <sup>ϵ</sup>

Gsð Þ¼ t; 1; …; s ∑

∑ P:ðf g 1;…;s ;sþ1;…;sþnÞ ¼ ∪ kXk

the structure

36

<sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ <sup>t</sup>; f g <sup>1</sup>; <sup>2</sup> ; <sup>3</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ <sup>t</sup>; f g <sup>1</sup>; <sup>2</sup> ; <sup>3</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

where the operator

∞ n¼0

A<sup>1</sup>þ<sup>n</sup>ð Þ t; f g 1; …; s ; s þ 1; …; s þ njGð Þ 0 ≐

particles we denote by G t; θð Þj X<sup>1</sup> …G t; θ X∣P<sup>∣</sup>

ð Þ <sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ� <sup>t</sup>; f g <sup>1</sup>; <sup>2</sup> ; <sup>3</sup> <sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ <sup>t</sup>; <sup>1</sup>; <sup>3</sup> <sup>A</sup>1ð Þ <sup>t</sup>; <sup>2</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þ<sup>3</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

ð Þ �<sup>1</sup> <sup>∣</sup>P∣�<sup>1</sup>

1 n!

Gsð Þ¼ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>N</sup> <sup>∗</sup>

∑ P : ð Þ¼ 1; …; s þ 1 X<sup>1</sup> ∪ X2, i∈X1; s þ 1∈X<sup>2</sup>

∑ i<sup>1</sup> ∈X<sup>1</sup>

<sup>s</sup> Gsð Þþ t; 1; …; s

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

N <sup>∗</sup>

intð Þð i; s þ 1 Gsþ1ð Þþ t; 1; …; s þ 1

Gsð Þj <sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>G</sup>0, <sup>ϵ</sup>

where ϵ . 0 is a scaling parameter and we use accepted in hierarchy (6) notations.

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þ t; f g 1; …; s ; s þ 1; …; s þ njGð Þ 0 of this series

and composition of mappings (3) of the corresponding noninteracting groups of

Gðt; 1; 2jGð Þ t; 3j f Þ ¼ A1ð Þ t; f g 1; 2 A1ð Þ t; 3 f <sup>3</sup>ð Þþ 1; 2; 3 <sup>A</sup>2ð Þ <sup>t</sup>; <sup>1</sup>; <sup>2</sup> <sup>A</sup>1ð Þ <sup>t</sup>; <sup>3</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>1</sup> <sup>f</sup> <sup>2</sup>ð Þþ <sup>2</sup>; <sup>3</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>2</sup> <sup>f</sup> <sup>2</sup>ð Þ <sup>1</sup>; <sup>3</sup> :

Below we adduce the examples of expansions (14). The first-order cumulant of the groups of nonlinear operators (3) is the same group of nonlinear operators, i.e.,

A1ðt; f gj 1; …; s Gð Þ 0 Þ ¼ Gð Þ t; 1; …; sjGð Þ 0 :

In the case of s ¼ 2, the second-order cumulant of nonlinear operators (3) has

<sup>3</sup> ð Þþ <sup>1</sup>; <sup>2</sup>; <sup>3</sup> ð Þ <sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ� <sup>t</sup>; f g <sup>1</sup>; <sup>2</sup> ; <sup>3</sup> <sup>A</sup><sup>1</sup>þ<sup>1</sup>ð Þ <sup>t</sup>; <sup>2</sup>; <sup>3</sup> <sup>A</sup>1ð Þ <sup>t</sup>; <sup>1</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þ<sup>2</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

A3ð Þ¼ t; 1; 2; 3 A<sup>1</sup>þ<sup>1</sup>ð Þ� t; f g 1; 2 ; 3 A<sup>1</sup>þ<sup>1</sup>ð Þ t; 2; 3 A1ð Þ� t; 1 A<sup>1</sup>þ<sup>1</sup>ð Þ t; 1; 3 A1ð Þ t; 2

<sup>2</sup> ð Þþ 1; 3

<sup>1</sup> ð Þ<sup>1</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þ<sup>2</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þ3 ,

ð Þ jPj � 1 !G t; θð Þj X<sup>1</sup> …G t; θ X∣P<sup>∣</sup>

<sup>j</sup>Gð Þ <sup>0</sup> … , for example,

is the 1ð Þ þ n th-order cumulant of groups of nonlinear operators (3):

Gð Þ¼ t; 1jGð Þ t; 2j f A1ð Þ t; 1 A1ð Þ t; 2 f <sup>2</sup>ð Þ 1; 2 ,

A<sup>1</sup>þ<sup>1</sup>ðt; f g 1; 2 ; 3jGð Þ 0 Þ ¼ Gðt; 1; 2; 3jGð Þ 0 Þ � Gðt; 1; 2jGð Þ t; 3jGð Þ 0 Þ ¼

<sup>2</sup> ð Þþ <sup>1</sup>; <sup>2</sup> <sup>A</sup>3ð Þ <sup>t</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

tion operators (12), then a nonperturbative solution of the Cauchy problem (11) and (12) is represented by the following sequence of self-adjoint operators:

intð Þ i1; i<sup>2</sup> G∣X1<sup>∣</sup>ð Þ t;X<sup>1</sup> G∣X2<sup>∣</sup>ð ÞÞþ t;X<sup>2</sup>

<sup>s</sup> , s≥1, (12)

<sup>j</sup>Gð Þ <sup>0</sup> … , n <sup>≥</sup>0,

(11)

(13)

(14)

<sup>1</sup> ð Þ<sup>1</sup> <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

<sup>2</sup> ð Þþ 2; 3

G∣X1<sup>∣</sup>ð Þ t;X<sup>1</sup> G∣X2<sup>∣</sup>ð ÞÞ t;X<sup>2</sup> ,

<sup>1</sup> ð Þ<sup>1</sup> ; …; <sup>G</sup>0, <sup>ϵ</sup> <sup>s</sup> ð Þ <sup>1</sup>; …; <sup>s</sup> ; … is a sequence of initial marginal correla-

Tr<sup>s</sup>þ1,…,sþ<sup>n</sup> A<sup>1</sup>þ<sup>n</sup>ð Þ t; f g 1; …; s ; s þ 1; …; s þ njGð Þ 0 , s≥1,

∑ i<sup>2</sup> ∈X<sup>2</sup>

$$\begin{split} & U\_{1+n}(t; \{1, \ldots, s\}, s+1, \ldots, s+n|G(0)) \doteq \\ & \sum\_{k=0}^{n} (-1)^{k} \frac{n!}{k!(n-k)!} \sum\_{\substack{\mathbf{P}, \mathbf{(\boldsymbol{\theta}(\{1, \ldots, s\}), s+1, \ldots, s+n-k) = \cup, X\_{i}}} \mathfrak{A}\_{|\mathbf{P}|}(t, \{X\_{1}\}, \ldots, \{X\_{|\mathbf{P}|}\}) \\ & \sum\_{k=0}^{k} \frac{k!}{k\_{1}!(k-k\_{1})!} \dots \sum\_{k\_{|\mathbf{P}|-1}=0}^{k\_{|\mathbf{P}|-2}} \frac{k\_{|\mathbf{P}|-2}!}{k\_{|\mathbf{P}|-1}!(k\_{|\mathbf{P}|-2}-k\_{|\mathbf{P}|-1})!} G\_{|\mathbf{X}\_{1}| + k - k\_{1}}^{0, \epsilon}(X\_{1}, \\ & s+n-k+\mathbf{1}, \ldots, s+n-k\_{1}) \dots G\_{|\mathbf{X}\_{|\mathbf{P}|}|+k\_{|\mathbf{P}|-1}}^{0, \epsilon}(X\_{|\mathbf{P}|}, s+n-k\_{|\mathbf{P}|-1}+\mathbf{1}, \ldots, s+n), \quad n \ge 0, \end{split} \tag{19}$$

We assume the existence of a mean field limit for initial marginal correlation

Then, taking into account equality (20), and since the nth term of series expansion (16) for s-particle marginal correlation operator is determined by the ð Þ s þ n thorder cumulant of asymptotically perturbed groups of operators (1), we establish

If for the initial marginal correlation operator equality (21) holds, then in the

where for arbitrary finite time interval, the limit one-particle marginal correla-

<sup>1</sup> ð Þ <sup>t</sup> � <sup>t</sup>1; <sup>1</sup> <sup>N</sup> <sup>∗</sup>

int j 1; j 2

As a result of differentiation in the sense of the norm convergence of the space

ð Þ H by the time variable of the operator represented by series expansion (23), we conclude that limit one-particle marginal correlation operator (23) is governed by

Then for pure states we derive the Hartree equation [2], indeed, in terms of the kernel g<sup>1</sup> t; q; q<sup>0</sup> ð Þ¼ ψð Þ t; q ψ t; q<sup>0</sup> ð Þ of operator (23), describing a pure-state, quantum

> ð dq<sup>0</sup>

We note that in the case of pure states, kinetic equation (24) can be reduced to the nonlinear Schrödinger equation [12] or to the Gross-Pitaevskii kinetic

∥ϵG1ð Þ�t g1ð Þt ∥L<sup>1</sup>

tion operator <sup>g</sup>1ð Þ <sup>t</sup>; <sup>1</sup> is given by the norm convergent series on the space <sup>L</sup><sup>1</sup>

Ynþ1 j <sup>n</sup>¼1 G∗ <sup>1</sup> tn; j n � �Y<sup>n</sup>þ<sup>1</sup>

.

ð Þ<sup>1</sup> <sup>g</sup>1ð Þþ <sup>t</sup>; <sup>1</sup> Tr2 <sup>N</sup> <sup>∗</sup>

g1ð Þt � � <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>0</sup>

Δqψð Þþ t; q

where the function Φ is a two-body interaction potential.

<sup>1</sup> ∥L1ð Þ <sup>H</sup> ¼ 0: (21)

ð Þ <sup>H</sup><sup>s</sup> ¼ 0, s ≥2: (22)

ð Þ H

(23)

ð Þ <sup>H</sup> ¼ 0,

intð Þ 1; 2

i¼1 g0 <sup>1</sup> ð Þi :

Y 2

<sup>1</sup> t<sup>1</sup> � t2; j

� � is defined according to formula

intð Þ 1; 2 g1ð Þ t; 1 g1ð Þ t; 2 , (24)

<sup>1</sup> : (25)

ψð Þ t; q ,

ð Þ H under the condition

1 � �…

j <sup>1</sup>¼1 G∗

<sup>1</sup> ð Þt is defined by (1). For bounded interaction

<sup>Φ</sup> <sup>q</sup> � <sup>q</sup><sup>0</sup> ð Þ <sup>ψ</sup> <sup>t</sup>; <sup>q</sup><sup>0</sup> j j ð Þ <sup>2</sup>

operator (or a one-particle density operator) in the following sense:

∥ϵG0, <sup>ϵ</sup> <sup>1</sup> � <sup>g</sup><sup>0</sup>

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

Gsð Þt ∥L<sup>1</sup>

case of s ¼ 1 for series expansion (16), the following equality is true:

limϵ!0

the property of the propagation of initial chaos (15):

DOI: http://dx.doi.org/10.5772/intechopen.82836

limϵ!0 ∥ϵs

limϵ!0

dtnTr2,…,nþ<sup>1</sup>G<sup>∗</sup>

intð Þ kn; n þ 1

potential, series (23) is norm convergent on the space L<sup>1</sup>

<sup>1</sup> ∥L1ð Þ <sup>H</sup>

the Cauchy problem of the quantum Vlasov kinetic equation:

kinetic equation (24) is reduced to the Hartree equation

1 2

g1ð Þ¼ t; 1 ∑

Yn in¼1 G∗

L1

∞ n¼0

ðt

dt1… tnð�1

0

In series expansion (23), the operator N <sup>∗</sup>

� ��<sup>1</sup>

<sup>g</sup>1ð Þ¼ <sup>t</sup>; <sup>1</sup> <sup>N</sup> <sup>∗</sup>

ψð Þ¼� t; q

n kn¼1 N <sup>∗</sup>

0

(2), and the group of operators G<sup>∗</sup>

<sup>1</sup> ð Þ tn � tn; in ∑

that <sup>t</sup> , <sup>t</sup><sup>0</sup> � <sup>2</sup>∥Φ∥<sup>L</sup>ð Þ <sup>H</sup><sup>2</sup> <sup>∥</sup>g<sup>0</sup>

∂ ∂t

i ∂ ∂t

equation [13].

39

as examples, we adduce the simplest examples of reduced cumulants (19):

$$\begin{split} & U\_{1}(t;\{1,\ldots,s\}|\mathcal{G}(\mathbf{0}))=\mathcal{G}(t;\mathbf{1},\ldots,s|\mathcal{G}(\mathbf{0}))=\\ & \sum\_{\mathbf{P}:(1,\ldots,s)=\iota,X\_{i}}\mathfrak{A}\_{|\mathbf{P}|}(t,\{X\_{1}\},\ldots,\{X\_{|\mathbf{P}|}\})\prod\_{X\_{i}\in\mathbf{P}}\mathfrak{G}^{0,\varepsilon}\_{|\mathbf{X}\_{i}|}(X\_{i}),\\ & U\_{1+1}(t;\{1,\ldots,s\},s+1|\mathcal{G}(\mathbf{0}))=\sum\_{\mathbf{P}:(1,\ldots,s+1)=\iota,X\_{i}}\mathfrak{A}\_{|\mathbf{P}|}(t,\{X\_{1}\},\ldots,\{X\_{|\mathbf{P}|}\})\prod\_{X\_{i}\in\mathbf{P}}\mathfrak{G}^{0,\varepsilon}\_{|\mathbf{X}\_{i}|}(X\_{i})-\\ & \sum\_{\mathbf{P}:(1,\ldots,s)=\iota,X\_{i}}\mathfrak{A}\_{|\mathbf{P}|}(t,\{X\_{1}\},\ldots,\{X\_{|\mathbf{P}|}\})\sum\_{j=1}^{|\mathbf{P}|}\mathfrak{G}^{0,\varepsilon}\_{|\mathbf{X}\_{j}|+1}(X\_{j},\boldsymbol{\varepsilon}+\mathbf{1})\prod\_{X\_{i}\in\mathbf{P}\_{i}}\mathfrak{G}^{0,\varepsilon}\_{|X\_{i}|}(X\_{i}).\\ & \mathbf{X}\_{i}\neq{\mathbf{P}\_{i}} \end{split}$$

We note also that a nonperturbative solution of the nonlinear quantum BBGKY hierarchy (13) or in the form of series expansions with generating operators (19) can be transformed to the perturbation (iteration) series as a result of the application of analogs of the Duhamel equation to cumulants (4) of groups of operators (1).

The following statement is true [7]. If max<sup>n</sup>≥<sup>1</sup>∥G<sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∥</sup>L<sup>1</sup> ð Þ <sup>H</sup><sup>n</sup> , <sup>2</sup>e<sup>3</sup> ð Þ�<sup>1</sup> , then in the case of bounded interaction potentials for t∈ R, a solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy (11) and (12) is determined by a sequence of marginal correlation operators represented by series expansions (13). If <sup>G</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup> <sup>0</sup>ð Þ <sup>H</sup><sup>n</sup> <sup>⊂</sup>L<sup>1</sup> ð Þ H<sup>n</sup> , it is a strong solution, and for arbitrary initial data <sup>G</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup> ð Þ H<sup>n</sup> , it is a weak solution.

#### 3.2 A mean field asymptotic behavior of marginal correlation operators

Now we deal with a scaling asymptotic behavior of the constructed marginal correlation operators in a mean field limit in the case of initial state satisfied condition (15).

Let us observe that if fs ∈ L<sup>1</sup> ð Þ H<sup>s</sup> , then for arbitrary finite time interval for an asymptotically perturbed first-order cumulant (9) of the groups of operators (1), i.e., for the strongly continuous group (1), the following equality is valid:

$$\lim\_{\epsilon \to 0} \|\mathcal{G}^\*\_{\boldsymbol{s}}(t, \mathbf{1}, \ldots, \boldsymbol{s}) f\_{\boldsymbol{s}} - \prod\_{j=1}^s \mathcal{G}^\*\_{\mathbf{1}}(t, j) f\_{\boldsymbol{s}}\|\_{\mathcal{L}^1(\mathcal{H}\_{\boldsymbol{s}})} = \mathbf{0}.$$

As a result of this for the ð Þ s þ n th-order cumulants of asymptotically perturbed groups of operators (1), the following equalities are true:

$$\lim\_{\varepsilon \to 0} \| \frac{1}{\varepsilon^n} \mathfrak{A}\_{\mathfrak{s}+n}(t, \mathbf{1}, \dots, \mathfrak{s} + n) f\_{\mathfrak{s}+n} \|\_{\mathfrak{L}^1(\mathcal{H}\_{\mathfrak{s}+n})} = \mathbf{0}, \quad \mathfrak{s} \ge \mathbf{2}. \tag{20}$$

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

We assume the existence of a mean field limit for initial marginal correlation operator (or a one-particle density operator) in the following sense:

$$\lim\_{e \to 0} \|\epsilon G\_1^{0,e} - \mathbf{g}\_1^0\|\_{\mathcal{L}^1(\mathcal{H})} = \mathbf{0}.\tag{21}$$

Then, taking into account equality (20), and since the nth term of series expansion (16) for s-particle marginal correlation operator is determined by the ð Þ s þ n thorder cumulant of asymptotically perturbed groups of operators (1), we establish the property of the propagation of initial chaos (15):

$$\lim\_{\varepsilon \to 0} \|e^{\varepsilon} \mathcal{G}\_{\varepsilon}(t)\|\_{\mathfrak{L}^{1}(\mathcal{H}\_{\varepsilon})} = \mathbf{0}, \quad \varepsilon \ge 2. \tag{22}$$

If for the initial marginal correlation operator equality (21) holds, then in the case of s ¼ 1 for series expansion (16), the following equality is true:

$$\lim\_{\epsilon \to 0} \|\epsilon G\_1(t) - g\_1(t)\|\_{\mathcal{L}^1(\mathcal{H})} = \mathbf{0},$$

where for arbitrary finite time interval, the limit one-particle marginal correlation operator <sup>g</sup>1ð Þ <sup>t</sup>; <sup>1</sup> is given by the norm convergent series on the space <sup>L</sup><sup>1</sup> ð Þ H

$$\begin{split} g\_{1}(t,\mathbf{1}) &= \sum\_{n=0}^{\infty} \left[ dt\_{1\dots n} \int\_{0}^{t\_{n-1}} dt\_{n} \operatorname{Tr}\_{2,\dots,n+1} \mathcal{G}\_{1}^{\*}(t-t\_{1},\mathbf{1}) \mathcal{N}\_{\operatorname{int}}^{\*}(\mathbf{1},2) \prod\_{j\_{1}=1}^{2} \mathcal{G}\_{1}^{\*}(t\_{1}-t\_{2},j\_{1}) \dots \right. \\ &\left. \prod\_{i\_{n}=1}^{n} \mathcal{G}\_{1}^{\*}(t\_{n}-t\_{n},i\_{n}) \sum\_{k\_{n}=1}^{n} \mathcal{N}\_{\operatorname{int}}^{\*}(k\_{n},n+1) \prod\_{j\_{n}=1}^{n+1} \mathcal{G}\_{1}^{\*}(t\_{n},j\_{n}) \prod\_{i=1}^{n+1} \mathcal{g}\_{1}^{0}(i) . \end{split} \tag{23}$$

In series expansion (23), the operator N <sup>∗</sup> int j 1; j 2 � � is defined according to formula (2), and the group of operators G<sup>∗</sup> <sup>1</sup> ð Þt is defined by (1). For bounded interaction potential, series (23) is norm convergent on the space L<sup>1</sup> ð Þ H under the condition that <sup>t</sup> , <sup>t</sup><sup>0</sup> � <sup>2</sup>∥Φ∥<sup>L</sup>ð Þ <sup>H</sup><sup>2</sup> <sup>∥</sup>g<sup>0</sup> <sup>1</sup> ∥L1ð Þ <sup>H</sup> � ��<sup>1</sup> .

As a result of differentiation in the sense of the norm convergence of the space L1 ð Þ H by the time variable of the operator represented by series expansion (23), we conclude that limit one-particle marginal correlation operator (23) is governed by the Cauchy problem of the quantum Vlasov kinetic equation:

$$\frac{\partial}{\partial t}\mathbf{g}\_1(t,\mathbf{1}) = \mathcal{N}^\*(\mathbf{1})\mathbf{g}\_1(t,\mathbf{1}) + \text{Tr}\_2\mathcal{N}^\*\_{\text{int}}(\mathbf{1},\mathbf{2})\mathbf{g}\_1(t,\mathbf{1})\mathbf{g}\_1(t,\mathbf{2}),\tag{24}$$

$$\left. \mathbf{g\_1(t)} \right|\_{t=0} = \mathbf{g\_1^0} \tag{25}$$

Then for pure states we derive the Hartree equation [2], indeed, in terms of the kernel g<sup>1</sup> t; q; q<sup>0</sup> ð Þ¼ ψð Þ t; q ψ t; q<sup>0</sup> ð Þ of operator (23), describing a pure-state, quantum kinetic equation (24) is reduced to the Hartree equation

$$i\frac{\partial}{\partial t}\psi(t,q) = -\frac{1}{2}\Delta\_q\psi(t,q) + \int dq'\Phi(q-q')|\psi(t,q')|^2\psi(t,q),$$

where the function Φ is a two-body interaction potential.

We note that in the case of pure states, kinetic equation (24) can be reduced to the nonlinear Schrödinger equation [12] or to the Gross-Pitaevskii kinetic equation [13].

U1þnð Þ t; f g 1; …; s ; s þ 1; …; s þ njGð Þ 0 ≐

… ∑ k∣P∣�<sup>2</sup> k∣P∣�1¼0

<sup>s</sup> <sup>þ</sup> <sup>n</sup> � <sup>k</sup> <sup>þ</sup> <sup>1</sup>, …, s <sup>þ</sup> <sup>n</sup> � <sup>k</sup>1Þ…G0, <sup>ϵ</sup>

<sup>k</sup>!ð Þ <sup>n</sup> � <sup>k</sup> ! <sup>∑</sup>

U1ðt; f gj 1; …; s Gð Þ 0 Þ ¼ Gðt; 1; …; sjGð Þ 0 Þ ¼

A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup> � � � � ∑

ð Þ H<sup>n</sup> , it is a weak solution.

U<sup>1</sup>þ<sup>1</sup>ðt; f g 1; …; s ; s þ 1jGð Þ 0 Þ ¼ ∑

� � � � Y

P:ðθð Þ f g 1;…;s ;sþ1;…;sþn�kÞ¼ ∪ iXi

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

k∣P∣�2! k∣P∣�1! k∣P∣�<sup>2</sup> � k∣P∣�<sup>1</sup> � �!

as examples, we adduce the simplest examples of reduced cumulants (19):

G0, <sup>ε</sup> ∣Xi∣ ð Þ Xi ,

We note also that a nonperturbative solution of the nonlinear quantum BBGKY hierarchy (13) or in the form of series expansions with generating operators (19)

case of bounded interaction potentials for t∈ R, a solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy (11) and (12) is determined by a sequence

Now we deal with a scaling asymptotic behavior of the constructed marginal correlation operators in a mean field limit in the case of initial state satisfied

asymptotically perturbed first-order cumulant (9) of the groups of operators (1),

j¼1 G∗ <sup>1</sup> ð Þ t; j fs

As a result of this for the ð Þ s þ n th-order cumulants of asymptotically perturbed

i.e., for the strongly continuous group (1), the following equality is valid:

<sup>s</sup> ð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> fs � <sup>Y</sup><sup>s</sup>

groups of operators (1), the following equalities are true:

ð Þ H<sup>n</sup> , it is a strong solution, and for arbitrary initial data

ð Þ H<sup>s</sup> , then for arbitrary finite time interval for an

∥L<sup>1</sup>

<sup>ϵ</sup><sup>n</sup> <sup>A</sup><sup>s</sup>þ<sup>n</sup>ð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> fsþ<sup>n</sup>∥L1ð Þ <sup>H</sup>sþ<sup>n</sup> <sup>¼</sup> <sup>0</sup>, s<sup>≥</sup> <sup>2</sup>: (20)

ð Þ <sup>H</sup><sup>s</sup> ¼ 0:

Xi ⊂P

P:ð Þ¼ 1;…;sþ1 ∪ iXi

can be transformed to the perturbation (iteration) series as a result of the application of analogs of the Duhamel equation to cumulants (4) of groups of

of marginal correlation operators represented by series expansions (13). If

3.2 A mean field asymptotic behavior of marginal correlation operators

The following statement is true [7]. If max<sup>n</sup>≥<sup>1</sup>∥G<sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∥</sup>L<sup>1</sup>

∣P∣ j¼1 G<sup>0</sup>, <sup>ε</sup> A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup> � � � �

ðX1,

(19)

<sup>∣</sup>X∣<sup>P</sup>∣∣þk∣P∣�<sup>1</sup> <sup>X</sup>∣P∣; <sup>s</sup> <sup>þ</sup> <sup>n</sup> � <sup>k</sup>∣P∣�<sup>1</sup> <sup>þ</sup> <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> � �, n≥0,

A∣P<sup>∣</sup> t; f g X<sup>1</sup> ; …; X∣P<sup>∣</sup>

<sup>∣</sup>Xj∣þ<sup>1</sup> Xj; <sup>s</sup> <sup>þ</sup> <sup>1</sup> � � <sup>Y</sup>

� � � � Y

Xi ⊂P, Xi 6¼ Xj Xi ⊂P

G<sup>0</sup>, <sup>ε</sup> ∣Xi∣ ð Þ Xi :

ð Þ <sup>H</sup><sup>n</sup> , <sup>2</sup>e<sup>3</sup> ð Þ�<sup>1</sup>

G<sup>0</sup>, <sup>ε</sup> ∣Xi∣ ð Þ� Xi

, then in the

G0, <sup>ϵ</sup> ∣X1∣þk�k<sup>1</sup>

∑ n k¼0

∑ k k1¼0

∑ P:ð Þ¼ 1;…;s ∪ iXi

∑ P:ð Þ¼ 1;…;s ∪ iXi

operators (1).

<sup>G</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup>

<sup>G</sup><sup>0</sup>, <sup>ϵ</sup> <sup>n</sup> <sup>∈</sup>L<sup>1</sup>

condition (15).

38

<sup>0</sup>ð Þ <sup>H</sup><sup>n</sup> <sup>⊂</sup>L<sup>1</sup>

Let us observe that if fs ∈ L<sup>1</sup>

limϵ!0 ∥G<sup>∗</sup>

limϵ!0 ∥ 1

ð Þ �<sup>1</sup> <sup>k</sup> <sup>n</sup>!

k! k1!ð Þ k � k<sup>1</sup> !

#### 4. The description of processes of a creation and a propagation of correlations by means of kinetic equations

In this section we consider mathematical problems concerning the description of processes of creation and propagation of correlations within framework of the state of typical particle of quantum systems of many particles; in other words, an approach to the description of evolution of correlations by means of quantum kinetic equations is developing.

Gsþnðt; θð Þ f g 1; …; s ; s þ 1; …; s þ nÞ ¼

DOI: http://dx.doi.org/10.5772/intechopen.82836

∑ D<sup>j</sup> : Zj ¼ ∪ lj

∣Dj∣ ≤ s þ n � n<sup>1</sup> � … � nj

… <sup>∑</sup> <sup>n</sup>�n1�…�nk�<sup>1</sup> nk¼1

> Xlj ,

Zj � s þ n � n<sup>1</sup> � … � nj þ 1; …; s þ n � n<sup>1</sup> � … � nj�<sup>1</sup>

1 ð Þ n � n<sup>1</sup> � … � nk !

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

ðt; θð Þ f g 1; …; s ; s þ 1; …; s þ n � n<sup>1</sup> � … � nkÞ�

sþn�n1�…�nj

i16¼…6¼i∣Dj∣¼1

In formula (29) the sum of all possible dissections [18] of the linearly ordered set

1 <sup>∣</sup>Dj∣! <sup>∑</sup>

� � on no more than

notations accepted above were used. We adduce simplest examples of generating

ðt; θð Þ f g 1; …; s Þ ¼ Asð ÞÞ t; 1; …; s g<sup>ϵ</sup>

<sup>A</sup>2ð Þ <sup>t</sup>; <sup>j</sup>; <sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>g</sup><sup>ϵ</sup>

A method of the construction of marginal correlation functionals (28) is based on the application of kinetic cluster expansions [2] to the generating operators of series (13). If <sup>∥</sup>G1ð Þ<sup>t</sup> <sup>∥</sup>L1ð Þ <sup>H</sup> , <sup>e</sup>�ð Þ <sup>3</sup>sþ<sup>2</sup> , then for arbitrary <sup>t</sup><sup>∈</sup> <sup>R</sup> series expansion (28)

ð Þ H<sup>s</sup> . We emphasize that marginal correlation functionals (28) describe all the possible correlations generated by dynamics of large quantum particle system with initial

Now we establish the evolution equation for one-particle (marginal) density operator (27). As a result of the differentiation over time variable of the operator represented by series expansion (27) in the sense of the norm convergence of the

generating operators of obtained series expansion, for one-particle density operator

ð Þ H , then due to the application of the kinetic cluster expansions [19] to the

4.2 The generalized quantum kinetic equation with initial correlations

<sup>s</sup> <sup>þ</sup> <sup>n</sup> � <sup>n</sup><sup>1</sup> � … � nj linearly ordered subsets is denoted by <sup>∑</sup><sup>D</sup>j:Zj<sup>¼</sup> <sup>∪</sup> <sup>l</sup>

ð Þ s þ n th-order scattering cumulant is defined by the formula

<sup>A</sup>�<sup>s</sup>þ<sup>n</sup>ð Þ <sup>t</sup>; <sup>θ</sup>ð Þ f g <sup>1</sup>; …; <sup>s</sup> ; <sup>s</sup> <sup>þ</sup> <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> <sup>≐</sup> <sup>A</sup><sup>s</sup>þ<sup>n</sup>ð Þ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> <sup>g</sup><sup>ϵ</sup>

G<sup>s</sup>þ<sup>1</sup>ðt; θð Þ f g 1; …; s ; s þ 1Þ ¼ A<sup>s</sup>þ<sup>1</sup>ð Þ t; 1; …; s þ 1 g<sup>ϵ</sup>

correlations by means of a one-particle density operator.

Ys i¼1 A�<sup>1</sup> <sup>1</sup> ð Þ t; i ∑ s j¼1

<sup>s</sup>ð Þ 1; …; s

converges in the norm of the space L<sup>1</sup>

(27), we derive the following identity:

�

<sup>s</sup>þ<sup>n</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> is specified initial correlations (26) and

Y Xl <sup>j</sup> ⊂ D<sup>j</sup>

1 ∣Xlj ∣! <sup>A</sup>�1þ∣Xl j <sup>∣</sup> t; ilj ;Xlj � �:

> j Xl j

<sup>s</sup>þ<sup>n</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> <sup>Y</sup><sup>s</sup>þ<sup>n</sup>

<sup>s</sup>ð Þ 1; …; s

<sup>s</sup>þ<sup>1</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup>ð Þ <sup>j</sup>; <sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>A</sup>�<sup>1</sup>

Ys i¼1 A�<sup>1</sup> <sup>1</sup> ð Þ t; i ,

Ysþ1 i¼1 A�<sup>1</sup> <sup>1</sup> ð Þ� t; i

<sup>1</sup> ð Þ <sup>t</sup>; <sup>j</sup> <sup>A</sup>�<sup>1</sup>

, and the

A�<sup>1</sup> <sup>1</sup> ð Þ t; i ,

<sup>1</sup> ð Þ t; s þ 1 :

i¼1

(29)

n! ∑ n k¼0

Y k

j¼1

ð Þ �<sup>1</sup> <sup>k</sup> <sup>∑</sup> n n1¼1

where the operator g<sup>ϵ</sup>

<sup>G</sup>sðt; <sup>θ</sup>ð Þ f g <sup>1</sup>; …; <sup>s</sup> Þ ¼ <sup>A</sup>�<sup>s</sup>

operators (29):

Asð Þ t; 1; …; s g<sup>ϵ</sup>

space L<sup>1</sup>

41

<sup>A</sup>�sþn�n1�…�nk

#### 4.1 Marginal correlation functionals of the state

Further we shall consider the case of initial states specified by a one-particle marginal density operator with correlations, namely, initial states specified by the following sequence of marginal correlation operators:

$$\mathbf{G}^{(\epsilon)} = \left( I, \mathbf{G}\_1^{0,\epsilon}(\mathbf{1}), \mathbf{g}\_2^{\epsilon}(\mathbf{1}, \mathbf{2}) \prod\_{i=1}^2 \mathbf{G}\_1^{0,\epsilon}(i), \dots, \mathbf{g}\_n^{\epsilon}(\mathbf{1}, \dots, n) \prod\_{i=1}^n \mathbf{G}\_1^{0,\epsilon}(i), \dots \right), \tag{26}$$

where the operators g<sup>ϵ</sup> <sup>n</sup>ð Þ� 1; …; n g<sup>ϵ</sup> <sup>n</sup> ∈L<sup>1</sup> <sup>0</sup>ð Þ H<sup>n</sup> , n ≥2 specified the initial correlations. We remark that such assumption about initial states is intrinsic for the kinetic description of many-particle systems. On the other hand, initial data (26) is typical for the condensed states of large quantum systems of particle, for example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition with the correlations which characterize the condensed state [1].

For initial states specified in terms of a one-particle density operator and correlation operators (26), the evolution of states given within the framework of the sequence G tðÞ¼ ð Þ I; G1ð Þt ; …; Gsð Þt ; … of marginal correlation operators (13) can be described by means of the sequence G tð Þ¼ jG1ð Þt ð Þ I; G1ð Þt ; G2ð Þ tjG1ð Þt ; …; Gsð Þ tjG1ð Þt , … of marginal correlation functionals: Gsð Þ t; 1; …; sjG1ð Þt , s ≥2, with respect to the one-particle correlation operator G1ð Þt governed by the kinetic equation [8].

In the case under consideration, the marginal correlation functionals Gsð Þ tjG1ð Þt , s≥ 2 are defined with respect to the one-particle (marginal) density operator

$$\begin{aligned} G\_1(t, 1) &= \\ \sum\_{n=0}^{\infty} \frac{1}{n!} \operatorname{Tr}\_{2, \dots, 1+n} \mathfrak{A}\_{1+n}(t, 1, \dots, n+1) & \sum\_{\substack{\mathbf{P} \colon \begin{array}{c} \\ \end{array} \\ \end{array}} & \prod\_{\substack{\mathbf{X}\_i \in \mathbb{P} \\ \end{array}} \mathfrak{g}\_{|\mathbf{X}\_i|}^{e}(\mathbf{X}\_i) \prod\_{i=1}^{n+1} \mathbf{G}\_1^{0,e}(i), \tag{27} \end{aligned} \tag{27}$$

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þt is the 1ð Þ� þ n th-order cumulant (4) of the groups of operators (1), and these functionals are represented by the series expansions:

$$\begin{aligned} \mathbf{G}\_{\boldsymbol{s}}(t, \mathbf{1}, \ldots, \mathbf{s} | \mathbf{G}\_{1}(t)) &= \\ \sum\_{n=0}^{\infty} \frac{1}{n!} \operatorname{Tr}\_{\boldsymbol{s}+1, \ldots, \boldsymbol{s}+n} \mathfrak{G}\_{\boldsymbol{s}+n}(t, \boldsymbol{\theta}(\{1, \ldots, \boldsymbol{s}\}), \boldsymbol{s}+1, \ldots, \boldsymbol{s}+n) \prod\_{i=1}^{\boldsymbol{s}+n} \mathbf{G}\_{1}(t, i), \quad \boldsymbol{s} \ge \mathbf{2}, \end{aligned} \tag{28}$$

where the ð Þ s þ n th-order generating operator G<sup>s</sup>þ<sup>n</sup>ð Þt , n ≥0 of this series is determined by the following expansion:

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

$$\begin{split} &\mathfrak{G}\_{s+n}(t,\theta\{\{1,\ldots,s\}\},s+1,\ldots,s+n) = \\ &n! \sum\_{k=0}^{n}(-1)^{k} \sum\_{n\_{1}=1}^{n} \cdots \sum\_{n\_{k}=1}^{n-n\_{1}-\ldots-n\_{k-1}} \frac{\mathbf{1}}{(n-n\_{1}-\ldots-n\_{k})!} \times \\ &\breve{\mathfrak{U}}\_{s+n-n\_{1}-\ldots-n\_{k}}(t,\theta(\{1,\ldots,s\}),s+1,\ldots,s+n-n\_{1}-\ldots-n\_{k}) \times \\ &\prod\_{j=1}^{k} \sum\_{\begin{subarray}{c} \sum\\ \sum l\_{j} \le Z\_{j} \end{subarray}} \frac{\mathbf{1}}{|\mathcal{D}\_{j}|!} \sum\_{i\_{1}\ne\ldots\ne i\_{l}=1}^{s+n-n\_{1}-\ldots-n\_{l}} \prod\_{\begin{subarray}{c} \mathbf{X}\_{l\_{j}} \subset \mathbf{D}\_{j} \end{subarray}} \frac{\mathbf{1}}{|\mathcal{X}\_{l\_{j}}|!} \breve{\mathfrak{A}}\_{1+|\mathcal{X}\_{l\_{j}}|} \left(t,i\_{l\_{j}},\mathbf{X}\_{l\_{j}}\right). \end{split} \tag{29}$$

In formula (29) the sum of all possible dissections [18] of the linearly ordered set Zj � s þ n � n<sup>1</sup> � … � nj þ 1; …; s þ n � n<sup>1</sup> � … � nj�<sup>1</sup> � � on no more than <sup>s</sup> <sup>þ</sup> <sup>n</sup> � <sup>n</sup><sup>1</sup> � … � nj linearly ordered subsets is denoted by <sup>∑</sup><sup>D</sup>j:Zj<sup>¼</sup> <sup>∪</sup> <sup>l</sup> j Xl j , and the ð Þ s þ n th-order scattering cumulant is defined by the formula

$$
\check{\mathfrak{A}}\_{s+n}(t, \theta(\{1, \ldots, s\}), \mathfrak{s}+1, \ldots, \mathfrak{s}+n) \doteq \mathfrak{A}\_{s+n}(t, \mathbf{1}, \ldots, \mathfrak{s}+n) \mathfrak{g}\_{s+n}^{e}(\mathbf{1}, \ldots, \mathfrak{s}+n) \prod\_{i=1}^{s+n} \mathfrak{A}\_{1}^{-1}(t, i),
$$

where the operator g<sup>ϵ</sup> <sup>s</sup>þ<sup>n</sup>ð Þ <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>n</sup> is specified initial correlations (26) and notations accepted above were used. We adduce simplest examples of generating operators (29):

$$\begin{split} \mathfrak{G}\_{\iota}(t, \theta(\{1, \ldots, s\})) &= \breve{\mathfrak{A}}\_{\iota}(t, \theta(\{1, \ldots, s\})) = \mathfrak{A}\_{\iota}(t, \mathtt{1}, \ldots, s) \big] \mathfrak{g}\_{\iota}^{\epsilon}(\mathtt{1}, \ldots, s) \prod\_{i=1}^{s} \mathfrak{A}\_{1}^{-1}(t, i), \\ \mathfrak{G}\_{\iota+1}(t, \theta(\{1, \ldots, s\}), s+1) &= \mathfrak{A}\_{\iota+1}(t, \mathtt{1}, \ldots, s+1) \mathfrak{g}\_{\iota+1}^{\epsilon}(\mathtt{1}, \ldots, s+1) \prod\_{i=1}^{s+1} \mathfrak{A}\_{1}^{-1}(t, i) - \\ \mathfrak{A}\_{\iota}(t, \mathtt{1}, \ldots, s) \mathfrak{g}\_{\iota}^{\epsilon}(\mathtt{1}, \ldots, s) \prod\_{i=1}^{s} \mathfrak{A}\_{1}^{-1}(t, i) \sum\_{j=1}^{s} \mathfrak{A}\_{2}(t, j, s+1) \mathfrak{g}\_{2}^{\epsilon}(j, s+1) \mathfrak{A}\_{1}^{-1}(t, j) \mathfrak{A}\_{1}^{-1}(t, s+1). \end{split}$$

A method of the construction of marginal correlation functionals (28) is based on the application of kinetic cluster expansions [2] to the generating operators of series (13). If <sup>∥</sup>G1ð Þ<sup>t</sup> <sup>∥</sup>L1ð Þ <sup>H</sup> , <sup>e</sup>�ð Þ <sup>3</sup>sþ<sup>2</sup> , then for arbitrary <sup>t</sup><sup>∈</sup> <sup>R</sup> series expansion (28) converges in the norm of the space L<sup>1</sup> ð Þ H<sup>s</sup> .

We emphasize that marginal correlation functionals (28) describe all the possible correlations generated by dynamics of large quantum particle system with initial correlations by means of a one-particle density operator.

#### 4.2 The generalized quantum kinetic equation with initial correlations

Now we establish the evolution equation for one-particle (marginal) density operator (27). As a result of the differentiation over time variable of the operator represented by series expansion (27) in the sense of the norm convergence of the space L<sup>1</sup> ð Þ H , then due to the application of the kinetic cluster expansions [19] to the generating operators of obtained series expansion, for one-particle density operator (27), we derive the following identity:

4. The description of processes of a creation and a propagation of

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

of typical particle of quantum systems of many particles; in other words, an approach to the description of evolution of correlations by means of quantum

In this section we consider mathematical problems concerning the description of processes of creation and propagation of correlations within framework of the state

Further we shall consider the case of initial states specified by a one-particle marginal density operator with correlations, namely, initial states specified by the

<sup>1</sup> ð Þ<sup>i</sup> ; …; <sup>g</sup><sup>ϵ</sup>

<sup>n</sup> ∈L<sup>1</sup>

tions. We remark that such assumption about initial states is intrinsic for the kinetic description of many-particle systems. On the other hand, initial data (26) is typical for the condensed states of large quantum systems of particle, for example, the equilibrium state of the Bose condensate satisfies the weakening of correlation condition with the correlations which characterize the condensed state [1].

For initial states specified in terms of a one-particle density operator and correlation operators (26), the evolution of states given within the framework of the sequence G tðÞ¼ ð Þ I; G1ð Þt ; …; Gsð Þt ; … of marginal correlation operators (13) can be described by means of the sequence G tð Þ¼ jG1ð Þt ð Þ I; G1ð Þt ; G2ð Þ tjG1ð Þt ; …; Gsð Þ tjG1ð Þt , … of marginal correlation functionals: Gsð Þ t; 1; …; sjG1ð Þt , s ≥2, with respect to the one-particle

P : ð Þ¼ 1; …; n þ 1 ∪ iXi

where the generating operator A<sup>1</sup>þ<sup>n</sup>ð Þt is the 1ð Þ� þ n th-order cumulant (4) of the groups of operators (1), and these functionals are represented by the series

where the ð Þ s þ n th-order generating operator G<sup>s</sup>þ<sup>n</sup>ð Þt , n ≥0 of this series is

!

<sup>n</sup>ð Þ <sup>1</sup>; …; <sup>n</sup> <sup>Y</sup><sup>n</sup>

i¼1

G<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ð Þi ; …

<sup>0</sup>ð Þ H<sup>n</sup> , n ≥2 specified the initial correla-

Y Xi ⊂P gϵ ∣Xi∣ ð Þ Xi Ynþ1 i¼1

Ysþn i¼1

, (26)

G<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ð Þi ,

<sup>G</sup>1ð Þ <sup>t</sup>; <sup>i</sup> , s≥2, (28)

(27)

correlations by means of kinetic equations

4.1 Marginal correlation functionals of the state

following sequence of marginal correlation operators:

<sup>2</sup>ð Þ <sup>1</sup>; <sup>2</sup> <sup>Y</sup> 2

i¼1

<sup>n</sup>ð Þ� 1; …; n g<sup>ϵ</sup>

correlation operator G1ð Þt governed by the kinetic equation [8].

Tr2,…, <sup>1</sup>þ<sup>n</sup> A<sup>1</sup>þ<sup>n</sup>ð Þ t; 1; …; n þ 1 ∑

Tr<sup>s</sup>þ1,…,sþ<sup>n</sup>G<sup>s</sup>þ<sup>n</sup>ð Þ t; θð Þ f g 1; …; s ; s þ 1; …; s þ n

In the case under consideration, the marginal correlation functionals Gsð Þ tjG1ð Þt , s≥ 2 are defined with respect to the one-particle (marginal) density

G<sup>0</sup>, <sup>ϵ</sup>

<sup>1</sup> ð Þ<sup>1</sup> ; <sup>g</sup><sup>ϵ</sup>

kinetic equations is developing.

<sup>G</sup>ð Þ<sup>c</sup> <sup>¼</sup> <sup>I</sup>; <sup>G</sup><sup>0</sup>, <sup>ϵ</sup>

operator

∑ ∞ n¼0

G1ð Þ¼ t; 1

1 n!

expansions:

∑ ∞ n¼0

40

1 n!

Gsðt; 1; …; sjG1ð Þt Þ ¼

determined by the following expansion:

where the operators g<sup>ϵ</sup>

$$\begin{split} \frac{\partial}{\partial t} \mathcal{G}\_{1}(t, \mathbf{1}) &= \mathcal{N}^{\*}(\mathbf{1}) \mathcal{G}\_{1}(t, \mathbf{1}) + \epsilon \operatorname{Tr}\_{2} \mathcal{N}^{\*}\_{\text{int}}(\mathbf{1}, \mathbf{2}) \mathcal{G}\_{1}(t, \mathbf{1}) \mathcal{G}\_{1}(t, \mathbf{2}) \\ &+ \epsilon \operatorname{Tr}\_{2} \mathcal{N}^{\*}\_{\text{int}}(\mathbf{1}, \mathbf{2}) \mathcal{G}\_{2}(t, \mathbf{1}, \mathbf{2}|\mathcal{G}\_{1}(t)), \end{split} \tag{30}$$

potentials series (32) is norm convergent on the space L<sup>1</sup>

<sup>1</sup> ∥L<sup>1</sup> ð Þ H

i1¼1 G∗ <sup>1</sup> ð Þ t; i<sup>1</sup> gs

lation operators in the case of arbitrary initial states, namely,

s i¼1 N <sup>∗</sup> ð Þi gs

where we used notations similar to accepted above.

∂ ∂t

of a one-particle density operator in a mean field approximation.

P : ð Þ¼ 1; …; s þ 1 X<sup>1</sup> ∪ X2, i∈ X1; s þ 1∈ X<sup>2</sup>

<sup>s</sup> , s≥1,

density operator represented by series expansion (32) is a solution of the Cauchy problem of the Vlasov-type quantum kinetic equation with initial correlations:

<sup>g</sup>1ð Þ¼ <sup>t</sup>; <sup>1</sup> <sup>N</sup> <sup>∗</sup>

g1ð Þt � � <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>0</sup>

<sup>1</sup> ð Þ <sup>t</sup>; <sup>i</sup><sup>1</sup> <sup>g</sup>2ð Þþ <sup>1</sup>; <sup>2</sup> <sup>I</sup> � �Y

It should be noted that limit marginal correlation functionals (33) describe the process of the evolution of correlations of large quantum particle systems by means

Similar to the derivation of kinetic equation (30), we establish that the one-particle

2

i2¼1

and consequently, for pure states we derive the Hartree-type equation with initial correlations. We point out that Eq. (34) is the non-Markovian quantum

Thus, we established that a mean field behavior of processes of the creation of correlations and the propagation of initial correlations in large quantum particle

ð Þ1 g1ð Þþ t; 1

G∗ 1 � ��<sup>1</sup>

tion functionals and of series expansion (27).

ð Þ¼ t; 1; …; s ∑

 gs ð Þt � � <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>0</sup>

marginal correlation functionals (28) in the following sense:

.

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

For marginal correlation functionals (28), the following statement is true [8]. Under conditions (21) and (31) on initial state (26), there exists a mean field limit of

Gsðt; <sup>1</sup>; …; <sup>s</sup>jG1ð Þ<sup>t</sup> Þ � gs <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup>jg1ð Þ<sup>t</sup> � �∥L1ð Þ <sup>H</sup><sup>s</sup> <sup>¼</sup> <sup>0</sup>, s<sup>≥</sup> <sup>2</sup>,

Ys i2¼1

G∗ 1 � ��<sup>1</sup>

ð Þ t; i<sup>2</sup>

s i¼1 N <sup>∗</sup>

ð Þ t;X<sup>1</sup> g∣X2<sup>∣</sup>

intð Þ i; s þ 1

ð ÞÞ t;X<sup>2</sup> ,

ð Þ t; i<sup>2</sup> g1ð Þ t; 1 g1ð Þ t; 2 , (34)

<sup>1</sup> , (35)

Ys j¼1

where the limit marginal correlation functionals gs <sup>t</sup>jg1ð Þ<sup>t</sup> � �, s<sup>≥</sup> <sup>2</sup>, are represented

ð Þ 1; …; s

The proof of these statements is based on the validity of equality (20) for cumulants of asymptotically perturbed groups of operators (1) and the explicit structure of the generating operators of series expansions (28) of marginal correla-

and, respectively, the limit one-particle density operator g1ð Þt is represented by

We remark that limit marginal correlation functionals (32) and (33) are a solution of the Cauchy problem of the quantum Vlasov hierarchy of nonlinear evolution equations [6], which describes a mean field asymptotic behavior of marginal corre-

ð Þþ t; 1; …; s Tr<sup>s</sup>þ<sup>1</sup>∑

g∣X1<sup>∣</sup>

� ��<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82836

that <sup>t</sup> , <sup>t</sup><sup>0</sup> � <sup>2</sup>∥Φ∥<sup>L</sup>ð Þ <sup>H</sup><sup>2</sup> <sup>∥</sup>g<sup>0</sup>

limϵ!0 ∥ϵs

gs <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup>jg1ð Þ<sup>t</sup> � � <sup>¼</sup> <sup>Y</sup><sup>s</sup>

by the expansions

series expansion (32).

∂ ∂t gs

Tr2 N <sup>∗</sup>

kinetic equation.

43

intð Þ <sup>1</sup>; <sup>2</sup> <sup>Y</sup> 2

i1¼1 G∗

systems are governed by kinetic equation (34).

<sup>ð</sup>gsþ<sup>1</sup>ð Þþ <sup>t</sup>; <sup>1</sup>; …; <sup>s</sup> <sup>þ</sup> <sup>1</sup> <sup>∑</sup>

ð Þ H under the condition

g1ð Þ t; j , (33)

where the second part of the collision integral in equality (30) is determined in terms of the marginal correlation functional represented by series expansions (28) in the case of s ¼ 2. This identity we treat as the quantum kinetic equation, and we refer to this evolution equation as the generalized quantum kinetic equation with initial correlations.

We emphasize that the coefficients in an expansion of the collision integral of the non-Markovian kinetic equation (30) are determined by the operators specified initial correlations (26).

On the space L<sup>1</sup> ð Þ H for the Cauchy problem of the established generalized quantum kinetic equation with initial correlations, the following statement is true [19]. If ∥G0, <sup>ϵ</sup> <sup>1</sup> ∥L<sup>1</sup> ð Þ <sup>H</sup> , <sup>e</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>9</sup> ð Þ ð Þ �<sup>1</sup> , a global in time solution of the Cauchy problem of kinetic equation (30) is determined by series expansion (27). For initial data G<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ∈L<sup>1</sup> <sup>0</sup>ð Þ H , it is a strong solution, and for an arbitrary initial data, it is a weak solution.

The proof of this existence statement is similar to the proof in the case of the generalized quantum kinetic equation given in [18].

#### 4.3 On a propagation of initial correlations in a mean field limit

Further we establish the mean field asymptotic behavior of constructed marginal correlation functionals (28) in the case of initial states specified by the one-particle density operator with correlations (26).

We assume the existence of a mean field limit of an initial one-particle density operator in sense (21) and for initial correlation operators as follows:

$$\lim\_{\epsilon \to 0} \|\mathbf{g}\_n^{\epsilon} - \mathbf{g}\_n\|\_{\mathcal{L}^1(\mathcal{H}\_n)} = \mathbf{0}, \quad n \ge 2. \tag{31}$$

Then in consequence of the validity of equalities (20) for one-particle density operator (27), the following statement is true [8]. If conditions (21) and (31) hold, then for series expansion (27) the equality holds:

$$\lim\_{\epsilon \to 0} \|\epsilon \mathbf{G}\_1(t) - \mathbf{g}\_1(t)\|\_{\mathfrak{L}^1(\mathcal{H})} = \mathbf{0},$$

where for finite time interval, the limit one-particle density operator g1ð Þt is represented by the following norm convergent series on the space L<sup>1</sup> ð Þ H :

$$\begin{split} \mathcal{G}\_{1}(t,\mathbf{1}) &= \sum\_{n=0}^{\infty} \Bigg[ \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!$$

In series expansion (32), the operator N <sup>∗</sup> int j 1; j 2 � � is defined according to formula (2), and the group of operators G<sup>∗</sup> <sup>1</sup> ð Þt is defined by (1). For bounded interaction

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

potentials series (32) is norm convergent on the space L<sup>1</sup> ð Þ H under the condition that <sup>t</sup> , <sup>t</sup><sup>0</sup> � <sup>2</sup>∥Φ∥<sup>L</sup>ð Þ <sup>H</sup><sup>2</sup> <sup>∥</sup>g<sup>0</sup> <sup>1</sup> ∥L<sup>1</sup> ð Þ H � ��<sup>1</sup> .

For marginal correlation functionals (28), the following statement is true [8]. Under conditions (21) and (31) on initial state (26), there exists a mean field limit of marginal correlation functionals (28) in the following sense:

$$\lim\_{\varepsilon \to 0} \|\boldsymbol{\epsilon}^{\boldsymbol{\varepsilon}} \mathcal{G}\_{\boldsymbol{\varepsilon}}(t, \mathbf{1}, \dots, \boldsymbol{s} | \mathcal{G}\_{\mathbf{1}}(t)) - \mathbf{g}\_{\boldsymbol{\varepsilon}}(t, \mathbf{1}, \dots, \boldsymbol{s} | \mathbf{g}\_{\mathbf{1}}(t))\|\_{\mathcal{L}^{1}(\mathcal{H}\_{\boldsymbol{\varepsilon}})} = \mathbf{0}, \quad \boldsymbol{s} \ge \mathbf{2},$$

where the limit marginal correlation functionals gs <sup>t</sup>jg1ð Þ<sup>t</sup> � �, s<sup>≥</sup> <sup>2</sup>, are represented by the expansions

$$\mathbf{g}\_s(t, \mathbf{1}, \dots, s | \mathbf{g}\_1(t)) = \prod\_{i\_1=1}^s \mathcal{G}\_1^\*(t, i\_1) \mathbf{g}\_s(\mathbf{1}, \dots, s) \prod\_{i\_2=1}^s \left(\mathcal{G}\_1^\*\right)^{-1}(t, i\_2) \prod\_{j=1}^s \mathbf{g}\_1(t, j), \tag{33}$$

and, respectively, the limit one-particle density operator g1ð Þt is represented by series expansion (32).

The proof of these statements is based on the validity of equality (20) for cumulants of asymptotically perturbed groups of operators (1) and the explicit structure of the generating operators of series expansions (28) of marginal correlation functionals and of series expansion (27).

We remark that limit marginal correlation functionals (32) and (33) are a solution of the Cauchy problem of the quantum Vlasov hierarchy of nonlinear evolution equations [6], which describes a mean field asymptotic behavior of marginal correlation operators in the case of arbitrary initial states, namely,

$$\begin{aligned} \frac{\partial}{\partial t} \mathbf{g}\_{\boldsymbol{s}}(t, \mathbf{1}, \ldots, \mathbf{s}) &= \sum\_{i=1}^{\ell} \mathcal{N}^{\ast}(i) \mathbf{g}\_{\boldsymbol{s}}(t, \mathbf{1}, \ldots, \mathbf{s}) + \operatorname{Tr}\_{\boldsymbol{s} + 1} \sum\_{i=1}^{\ell} \mathcal{N}^{\ast}\_{\text{int}}(i, \mathbf{s} + \mathbf{1}), \\ (\mathbf{g}\_{\boldsymbol{s} + 1}(t, \mathbf{1}, \ldots, \mathbf{s} + \mathbf{1}) + \sum\_{\substack{\mathbf{P} : \begin{array}{c} (\mathbf{I}, \ldots, \mathbf{s} + \mathbf{1}) \\ \mathbf{P} : (\mathbf{1}, \ldots, \mathbf{s} + \mathbf{1}) \end{array}} \mathbf{g}\_{|\mathcal{X}\_{1}|}(t, \mathbf{X}\_{1}) \mathbf{g}\_{|\mathcal{X}\_{2}|}(t, \mathbf{X}\_{2})), \\ \quad \quad \quad \quad \quad \quad \quad \boldsymbol{t} \in \mathcal{X}\_{1}, \mathbf{s} + \mathbf{1} \in \mathcal{X}\_{2} \\ \mathbf{g}\_{\boldsymbol{s}}(t) \big|\_{\boldsymbol{t} = \mathbf{0}} = \mathbf{g}\_{\boldsymbol{s}}^{0}, \quad \boldsymbol{s} \ge \mathbf{1}, \end{aligned}$$

where we used notations similar to accepted above.

It should be noted that limit marginal correlation functionals (33) describe the process of the evolution of correlations of large quantum particle systems by means of a one-particle density operator in a mean field approximation.

Similar to the derivation of kinetic equation (30), we establish that the one-particle density operator represented by series expansion (32) is a solution of the Cauchy problem of the Vlasov-type quantum kinetic equation with initial correlations:

$$\frac{\partial}{\partial t}\mathbf{g}\_1(t,\mathbf{1}) = \mathcal{N}^\*(\mathbf{1})\mathbf{g}\_1(t,\mathbf{1}) +$$

$$\left(\mathrm{Tr}\_2\mathcal{N}^\*\_{\mathrm{int}}(\mathbf{1},\mathbf{2})\prod\_{i\_1=1}^2\mathcal{G}^\*\_1(t,i\_1)\Big{(}\mathbf{g}\_2(\mathbf{1},\mathbf{2}) + I\Big{)}\prod\_{i\_2=1}^2\left(\mathcal{G}^\*\_1\right)^{-1}(t,i\_2)\mathbf{g}\_1(t,\mathbf{1})\mathbf{g}\_1(t,\mathbf{2}),\tag{34}$$

$$\boldsymbol{\varrho}\_\*(t)\Big{)}\Big{)}\Big{)} = \mathcal{g}^0\_\*.\tag{35}$$

$$\left. \mathbf{g}\_1(t) \right|\_{t=0} = \mathbf{g}\_1^0,\tag{35}$$
  $\text{and consequently, for pure states we derive the Hartree-type equation with}$ 

initial correlations. We point out that Eq. (34) is the non-Markovian quantum kinetic equation.

Thus, we established that a mean field behavior of processes of the creation of correlations and the propagation of initial correlations in large quantum particle systems are governed by kinetic equation (34).

∂ ∂t

initial correlations.

[19]. If ∥G0, <sup>ϵ</sup>

G<sup>0</sup>, <sup>ϵ</sup> <sup>1</sup> ∈L<sup>1</sup>

solution.

initial correlations (26). On the space L<sup>1</sup>

<sup>1</sup> ∥L<sup>1</sup>

g1ð Þ¼ t; 1 ∑

g∣Xi<sup>∣</sup> ð Þ Xi Ynþ1 i¼1 g0

Yn in¼1 G∗

Y Xi ⊂P

42

∞ n¼0 ðt

dt1… tnð�1

0

n kn¼1 N <sup>∗</sup>

In series expansion (32), the operator N <sup>∗</sup>

0

<sup>1</sup> ð Þ tn � tn; in ∑

(2), and the group of operators G<sup>∗</sup>

<sup>G</sup>1ð Þ¼ <sup>t</sup>; <sup>1</sup> <sup>N</sup> <sup>∗</sup>

<sup>þ</sup> <sup>ϵ</sup>Tr2 <sup>N</sup> <sup>∗</sup>

ð Þ <sup>H</sup> , <sup>e</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>9</sup> ð Þ ð Þ �<sup>1</sup>

generalized quantum kinetic equation given in [18].

limϵ!0 ∥g<sup>ϵ</sup>

limϵ!0

then for series expansion (27) the equality holds:

density operator with correlations (26).

4.3 On a propagation of initial correlations in a mean field limit

operator in sense (21) and for initial correlation operators as follows:

represented by the following norm convergent series on the space L<sup>1</sup>

dtnTr2,…,nþ<sup>1</sup>G<sup>∗</sup>

intð Þ kn; n þ 1

<sup>n</sup> � gn∥L<sup>1</sup>

ð Þ<sup>1</sup> <sup>G</sup>1ð Þþ <sup>t</sup>; <sup>1</sup> <sup>ϵ</sup>Tr2 <sup>N</sup> <sup>∗</sup>

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

intð Þ 1; 2 G2ð Þ t; 1; 2jG1ð Þt ,

where the second part of the collision integral in equality (30) is determined in terms of the marginal correlation functional represented by series expansions (28) in the case of s ¼ 2. This identity we treat as the quantum kinetic equation, and we refer to this evolution equation as the generalized quantum kinetic equation with

We emphasize that the coefficients in an expansion of the collision integral of the non-Markovian kinetic equation (30) are determined by the operators specified

quantum kinetic equation with initial correlations, the following statement is true

<sup>0</sup>ð Þ H , it is a strong solution, and for an arbitrary initial data, it is a weak

The proof of this existence statement is similar to the proof in the case of the

Further we establish the mean field asymptotic behavior of constructed marginal correlation functionals (28) in the case of initial states specified by the one-particle

We assume the existence of a mean field limit of an initial one-particle density

Then in consequence of the validity of equalities (20) for one-particle density operator (27), the following statement is true [8]. If conditions (21) and (31) hold,

∥ϵG1ð Þ�t g1ð Þt ∥L1ð Þ <sup>H</sup> ¼ 0,

<sup>1</sup> ð Þ <sup>t</sup> � <sup>t</sup>1; <sup>1</sup> <sup>N</sup> <sup>∗</sup>

int j 1; j 2 intð Þ 1; 2

� � ∑

<sup>1</sup> ð Þt is defined by (1). For bounded interaction

<sup>1</sup> ð Þi : (32)

Y 2

j <sup>1</sup>¼1 G∗

where for finite time interval, the limit one-particle density operator g1ð Þt is

Ynþ1 j <sup>n</sup>¼1 G∗ <sup>1</sup> tn; j n

of kinetic equation (30) is determined by series expansion (27). For initial data

ð Þ H for the Cauchy problem of the established generalized

intð Þ 1; 2 G1ð Þ t; 1 G1ð Þ t; 2

, a global in time solution of the Cauchy problem

ð Þ <sup>H</sup><sup>n</sup> ¼ 0, n≥2: (31)

ð Þ H :

1 � �…

<sup>1</sup> t<sup>1</sup> � t2; j

P : ð Þ¼ 1; …; n þ 1 ∪ iXi

� � is defined according to formula

(30)

Moreover, in the case under consideration, the processes of the creation of correlations generated by dynamics of many-particle systems and the propagation of initial correlations are described by the constructed marginal functionals of the state (28) governed by the non-Markovian generalized kinetic equation with initial correlations (26).

#### 5. Conclusion

In this chapter the process of a creation and a propagation of correlations in quantum many-particle systems has been described by means of the Cauchy problem of the quantum BBGKY hierarchy of nonlinear equations (11) and (12). A nonperturbative solution for a sequence of marginal correlation operators is represented in the form of series (13) the generating operator of every term of which are corresponding-order cumulant (14) of groups of nonlinear operators (3). In the case of initial state specified by a sequence of the marginal correlation operators that satisfy chaos property (15), the correlations generated by dynamics of large quantum particle system (16) are completely determined by the corresponding-order cumulants (4) of groups of operators (1). The obtained results can be extended to large quantum systems of bosons and fermions like in paper [6].

In the case of initial state satisfied condition (15), a mean field asymptotic behavior of the processes of a creation and a propagation of correlations was described. It was directly proven the property called the propagation of initial chaos (22), which underlies in mathematical derivation of effective evolution equations of systems of infinitely many particles [16].

The problem of the rigorous description of collective behavior of quantum many-particle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation [18] with initial correlations [19], for instance, the initial correlations, characterizing the condensed states [1], or initial correlations that influence on ultrafast relaxation processes in plasmas [4] has been also considered.

In particular, such an approach to the derivation of the Vlasov-type quantum kinetic equation with initial correlations (34) from underlying dynamics governed by the generalized quantum kinetic equation with initial correlations (30) enables to construct the higher-order corrections to the mean field evolution of large quantum systems of particle.

We note that in paper [20] other approach to the description of the propagation of initial correlations of large quantum particle systems in a mean field limit was developed, namely, the process of the propagation of initial correlations was described within the framework of the evolution of marginal observables governed by the dual BBGKY hierarchy [2, 21].

Author details

45

Viktor I. Gerasimenko

Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

DOI: http://dx.doi.org/10.5772/intechopen.82836

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: gerasym@imath.kiev.ua

provided the original work is properly cited.

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

#### Author details

Moreover, in the case under consideration, the processes of the creation of correlations generated by dynamics of many-particle systems and the propagation of initial correlations are described by the constructed marginal functionals of the state (28) governed by the non-Markovian generalized kinetic equation with initial

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

In this chapter the process of a creation and a propagation of correlations in quantum many-particle systems has been described by means of the Cauchy problem of the quantum BBGKY hierarchy of nonlinear equations (11) and (12). A nonperturbative solution for a sequence of marginal correlation operators is represented in the form of series (13) the generating operator of every term of which are corresponding-order cumulant (14) of groups of nonlinear operators (3). In the case of initial state specified by a sequence of the marginal correlation operators that satisfy chaos property (15), the correlations generated by dynamics

corresponding-order cumulants (4) of groups of operators (1). The obtained results can be extended to large quantum systems of bosons and fermions like in paper [6]. In the case of initial state satisfied condition (15), a mean field asymptotic behavior of the processes of a creation and a propagation of correlations was described. It was directly proven the property called the propagation of initial chaos (22), which underlies in mathematical derivation of effective evolution equations of

The problem of the rigorous description of collective behavior of quantum many-particle systems by means of a one-particle (marginal) correlation operator that is a solution of the generalized quantum kinetic equation [18] with initial correlations [19], for instance, the initial correlations, characterizing the condensed states [1], or initial correlations that influence on ultrafast relaxation processes in

In particular, such an approach to the derivation of the Vlasov-type quantum kinetic equation with initial correlations (34) from underlying dynamics governed by the generalized quantum kinetic equation with initial correlations (30) enables to construct the higher-order corrections to the mean field evolution of large quan-

We note that in paper [20] other approach to the description of the propagation of initial correlations of large quantum particle systems in a mean field limit was developed, namely, the process of the propagation of initial correlations was described within the framework of the evolution of marginal observables governed

of large quantum particle system (16) are completely determined by the

systems of infinitely many particles [16].

plasmas [4] has been also considered.

by the dual BBGKY hierarchy [2, 21].

tum systems of particle.

44

correlations (26).

5. Conclusion

Viktor I. Gerasimenko Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

\*Address all correspondence to: gerasym@imath.kiev.ua

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Bogolyubov MM. Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems. Kyiv: Rad. Shkola; 1949 (in Ukrainian)

[2] Gerasimenko VI. Hierarchies of quantum evolution equations and dynamics of many-particle correlations. In: Statistical Mechanics and Random Walks: Principles, Processes and Applications. N.Y.: Nova Science Publ., Inc.; 2012. p. 233

[3] Prigogine I. Non-Equilibrium Statistical Mechanics. New York: Wiley; 1962

[4] Bonitz M, Henning C, Block D. Complex plasmas: A laboratory for strong correlations. Reports on Progress in Physics. 2010;73:066501 (29 p)

[5] Gerasimenko VI, Shtyk VO. Evolution of correlations of quantum many-particle systems. Journal of Statistical Mechanics: Theory and Experiment. 2008;3:P03007 (24 p)

[6] Gerasimenko VI, Polishchuk DO. Dynamics of correlations of Bose and Fermi particles. Mathematical Models and Methods in Applied Sciences. 2011; 34(1):76

[7] Gerasimenko VI. Evolution of correlation operators of large quantum particle systems. Methods of Functional Analysis and Topology. 2017;23(2):123

[8] Gerasimenko VI. On the description of quantum correlations by means of a one-particle density operator. Proc IM NASU. 2017;14(1):116

[9] Spohn H. Kinetic equations from Hamiltonian dynamics. Reviews of Modern Physics. 1980;53:600-640

[10] Benedetto D, Castella F, Esposito R, Pulvirenti M. A short review on the derivation of the nonlinear quantum Boltzmann equations. Communications in Mathematical Sciences. 2007;5:55-71

[18] Gerasimenko VI, Tsvir ZA. A description of the evolution of quantum states by means of the kinetic equation. Journal of Physics A: Mathematical and Theoretical. 2010;43(48):485203

DOI: http://dx.doi.org/10.5772/intechopen.82836

Processes of Creation and Propagation of Correlations in Large Quantum Particle System

[19] Gerasimenko VI, Tsvir ZA. On quantum kinetic equations of manyparticle systems in condensed states. Physica A: Statistical Mechanics and its Applications. 2012;391(24):6362

[20] Gerasimenko VI. New approach to derivation of quantum kinetic equations with initial correlations. Carpathian Mathematical Publications. 2015;7:38-48

[21] Gerasimenko VI. Approaches to derivation of quantum kinetic equations. Ukrainian Journal de Physique. 2009;54(8–9):834

47

[11] Pezzotti F, Pulvirenti M. Mean-field limit and semiclassical expansion of quantum particle system. Annales Henri Poincaré. 2009;10:145-187

[12] Erdös L, Schlein B, Yau H-T. Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems. Inventiones Mathematicae. 2007; 167(3):515-614

[13] Erdös L, Schlein B, Yau H-T. Derivation of the Gross–Pitaevskii equation for the dynamics of Bose– Einstein condensate. Annals of Mathematics. 2010;172:291-370

[14] Boccato C, Cenatiempo S, Schlein B. Quantum many-body fluctuations around nonlinear Schrödinger dynamics. Annales Henri Poincaré. 2017;18(1):113-191

[15] Porta M, Rademacher S, Saffirio C, Schlein B. Mean field evolution of fermions with coulomb interaction. Journal of Statistical Physics. 2017; 166(6):1345-1364

[16] Golse F. On the dynamics of large particle systems in the mean field limit. Macroscopic and large scale phenomena: Coarse graining, mean field limits and ergodicity. Lecture Notes in Applied Mathematics and Mechanics. Springer. 2016;3:1-144

[17] Golse F, Mouhot C, Paul T. On the mean-field and classical limits of quantum mechanics. Communications in Mathematical Physics. 2016;343: 165-205

Processes of Creation and Propagation of Correlations in Large Quantum Particle System DOI: http://dx.doi.org/10.5772/intechopen.82836

[18] Gerasimenko VI, Tsvir ZA. A description of the evolution of quantum states by means of the kinetic equation. Journal of Physics A: Mathematical and Theoretical. 2010;43(48):485203

References

Ukrainian)

Inc.; 2012. p. 233

1962

34(1):76

46

[1] Bogolyubov MM. Lectures on Quantum Statistics. Problems of Statistical Mechanics of Quantum Systems. Kyiv: Rad. Shkola; 1949 (in

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

[10] Benedetto D, Castella F, Esposito R, Pulvirenti M. A short review on the derivation of the nonlinear quantum Boltzmann equations. Communications in Mathematical Sciences. 2007;5:55-71

[11] Pezzotti F, Pulvirenti M. Mean-field limit and semiclassical expansion of quantum particle system. Annales Henri

Poincaré. 2009;10:145-187

167(3):515-614

2017;18(1):113-191

166(6):1345-1364

2016;3:1-144

165-205

[12] Erdös L, Schlein B, Yau H-T. Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems. Inventiones Mathematicae. 2007;

[13] Erdös L, Schlein B, Yau H-T. Derivation of the Gross–Pitaevskii equation for the dynamics of Bose– Einstein condensate. Annals of Mathematics. 2010;172:291-370

[14] Boccato C, Cenatiempo S, Schlein B. Quantum many-body fluctuations around nonlinear Schrödinger dynamics. Annales Henri Poincaré.

[15] Porta M, Rademacher S, Saffirio C, Schlein B. Mean field evolution of fermions with coulomb interaction. Journal of Statistical Physics. 2017;

[16] Golse F. On the dynamics of large particle systems in the mean field limit. Macroscopic and large scale phenomena: Coarse graining, mean field limits and ergodicity. Lecture Notes in Applied Mathematics and Mechanics. Springer.

[17] Golse F, Mouhot C, Paul T. On the mean-field and classical limits of quantum mechanics. Communications in Mathematical Physics. 2016;343:

[2] Gerasimenko VI. Hierarchies of quantum evolution equations and dynamics of many-particle correlations. In: Statistical Mechanics and Random Walks: Principles, Processes and Applications. N.Y.: Nova Science Publ.,

[3] Prigogine I. Non-Equilibrium

[4] Bonitz M, Henning C, Block D. Complex plasmas: A laboratory for strong correlations. Reports on Progress in Physics. 2010;73:066501 (29 p)

[5] Gerasimenko VI, Shtyk VO. Evolution of correlations of quantum many-particle systems. Journal of Statistical Mechanics: Theory and Experiment. 2008;3:P03007 (24 p)

[6] Gerasimenko VI, Polishchuk DO. Dynamics of correlations of Bose and Fermi particles. Mathematical Models and Methods in Applied Sciences. 2011;

[7] Gerasimenko VI. Evolution of correlation operators of large quantum particle systems. Methods of Functional Analysis and Topology. 2017;23(2):123

NASU. 2017;14(1):116

[8] Gerasimenko VI. On the description of quantum correlations by means of a one-particle density operator. Proc IM

[9] Spohn H. Kinetic equations from Hamiltonian dynamics. Reviews of Modern Physics. 1980;53:600-640

Statistical Mechanics. New York: Wiley;

[19] Gerasimenko VI, Tsvir ZA. On quantum kinetic equations of manyparticle systems in condensed states. Physica A: Statistical Mechanics and its Applications. 2012;391(24):6362

[20] Gerasimenko VI. New approach to derivation of quantum kinetic equations with initial correlations. Carpathian Mathematical Publications. 2015;7:38-48

[21] Gerasimenko VI. Approaches to derivation of quantum kinetic equations. Ukrainian Journal de Physique. 2009;54(8–9):834

Chapter 4

Systems

Abstract

1. Introduction

49

Recent Progresses in Ab Initio

toward Understandings of

Functional Mechanisms of

Biological Macromolecular

Jiyoung Kang,Takuya Sumi and Masaru Tateno

characteristic to biological macromolecular systems.

protein catalyst, catalysis, enzyme, molecular evolution

Keywords: density functional theory (DFT), electron delocalization,

aminoacyl-tRNA synthetase (aaRS), transfer RNA (tRNA), hybrid ribozyme/

Why are the theoretical analyses employing ab initio quantum mechanics (QM) calculations required to understand biological systems? In an organism, so many catalytic reactions are present, for example, transcription, DNA replication and

In this chapter, we present recent advances of theoretical analyses toward understandings of functional mechanisms of biological macromolecular systems, employing ab initio electronic structure calculations. Two distinct types of triggers to invoke dramatic rearrangements of electronic structures in the reaction centers are revealed by full ab initio quantum mechanics (QM) calculations (first example) and hybrid ab initio QM/molecular mechanics (MM) molecular dynamics (MD) calculations (second example). First, we demonstrate dramatic rearrangements of molecular orbitals (MOs) induced by binding of a hydroxyl ion (OH) to the [4Fe-3S] cluster found in hydrogenases, which catalyzes both dissociation and production of dihydrogen (H2). This induces the significant delocalization of the LUMO, resulting in formation of electron transfer pathways required for the catalysis. Thus, in organisms, just a tiny species (e.g. OH ligand) can play a key role for the biological functions. Second, we indicate dynamical rearrangements of MOs occurring in the enzymatic reactions of RNA-protein complexes. As the catalysis proceeds, the reactive MOs, which do not belong to the frontier orbitals in the initial stages of the reaction, are dramatically reconstituted in the hybrid ab initio QM/MM MD simulations, resulting in the frontier orbitals, which is a feature

Electronic Structure Calculation

#### Chapter 4

Recent Progresses in Ab Initio Electronic Structure Calculation toward Understandings of Functional Mechanisms of Biological Macromolecular Systems

Jiyoung Kang,Takuya Sumi and Masaru Tateno

#### Abstract

In this chapter, we present recent advances of theoretical analyses toward understandings of functional mechanisms of biological macromolecular systems, employing ab initio electronic structure calculations. Two distinct types of triggers to invoke dramatic rearrangements of electronic structures in the reaction centers are revealed by full ab initio quantum mechanics (QM) calculations (first example) and hybrid ab initio QM/molecular mechanics (MM) molecular dynamics (MD) calculations (second example). First, we demonstrate dramatic rearrangements of molecular orbitals (MOs) induced by binding of a hydroxyl ion (OH) to the [4Fe-3S] cluster found in hydrogenases, which catalyzes both dissociation and production of dihydrogen (H2). This induces the significant delocalization of the LUMO, resulting in formation of electron transfer pathways required for the catalysis. Thus, in organisms, just a tiny species (e.g. OH ligand) can play a key role for the biological functions. Second, we indicate dynamical rearrangements of MOs occurring in the enzymatic reactions of RNA-protein complexes. As the catalysis proceeds, the reactive MOs, which do not belong to the frontier orbitals in the initial stages of the reaction, are dramatically reconstituted in the hybrid ab initio QM/MM MD simulations, resulting in the frontier orbitals, which is a feature characteristic to biological macromolecular systems.

Keywords: density functional theory (DFT), electron delocalization, aminoacyl-tRNA synthetase (aaRS), transfer RNA (tRNA), hybrid ribozyme/ protein catalyst, catalysis, enzyme, molecular evolution

#### 1. Introduction

Why are the theoretical analyses employing ab initio quantum mechanics (QM) calculations required to understand biological systems? In an organism, so many catalytic reactions are present, for example, transcription, DNA replication and

repair, protein biosynthesis, respiration, photosynthesis, and synthesis and degradation of biological compounds (metabolites) such as amino acids, nucleotide, and lipid. In order to understand the mechanisms of biological functions, analyses of the electronic structure changes for the catalytic reactions are essential.

must be explored by carefully providing multiple combinations of spin assignments and thereby should be determined by identifying the spin state with the optimal

To provide the total spin assignments, it is convenient to divide the systems into some fragmental moieties, such as a transition metal and its coordinated ligands as a subsystem, which leads to spin assignments for a part of the QM models. Although this could be helpful to get a reasonable solution, there is no warranty of the convergence and acquisition of the correct solution. In Section 2, we discuss more

Hybrid functional approximation was introduced by Becke [1] in 1993 and has become one of the most popular computational approaches. By incorporating a portion of the exact exchange energy originated from the Hartree-Fock theory coupled with the exchange-correlation energies, the hybrid functional approach improves molecular properties that poorly described with simple ab initio functionals, such as bond lengths, vibration frequencies, and atomization energies [2]. In general, a hybrid functional is described as a linear combination of the Hartree-Fock exact exchange energy functional and the exchange-correlation density functionals; their weights are determined by a fitting procedure such as to reproduce

1.3 Hybrid functional in density functional theory (DFT) calculation

experimental or highly advanced calculated thermochemical data.

<sup>X</sup> � ELSDA X <sup>þ</sup> aX EB<sup>88</sup>

exchange by Becke and Perdew-Wang, respectively.

<sup>X</sup> <sup>þ</sup> <sup>P</sup><sup>1</sup> <sup>P</sup>4ESlater

exchange and correlation, respectively, ELSDA

is the exact exchange energy, and EB<sup>88</sup>

EXC <sup>¼</sup> <sup>P</sup>2EHF

parameters,

51

Becke proposed the following exchange-correlation approximation [1],

a0 = 0.20, ax = 0.72, and ac = 0.81). Herein, subscripts X and C represent the

<sup>X</sup> <sup>þ</sup> <sup>P</sup>3ΔEnon�local X <sup>þ</sup> <sup>P</sup>6Elocal

Various combinations of nonlocal exchange functionals and correlation functionals can be employed here. In the B3LYP functional, the Becke's three-parameter functional and the LYP correlation functional were combined with the following

In an attempt to improve the B3LYP functional, the long-range correlations are incorporated into the cam-B3LYP [5] and LC-BLYP [6] functionals. However, to the best of our experiences on applying to biological macromolecular systems,

P<sup>1</sup> ¼ 1, P<sup>2</sup> ¼ 0:2, P<sup>3</sup> ¼ 0:72, P<sup>4</sup> ¼ 0:8, P<sup>5</sup> ¼ 0:81, P<sup>6</sup> ¼ 1: (3)

<sup>X</sup> � ELSDA X <sup>þ</sup> aC EPW<sup>91</sup>

where a0, ax, and ac are the parameters determined by the fitting procedure (i.e.,

energy within the framework of the local spin-density approximation (LSDA), Eexact

<sup>X</sup> and EPW<sup>91</sup>

The B3LYP functional, which has been one of the most widely used functionals in molecular quantum calculation fields, employs the nonlocal correlation provided by the LYP expression (Lee-Yang-Parr) and the Becke88 exchange functional [3], and VWN local-density approximation that was constructed by Volsko, Wilk, and Nusair (VWN) [4]. Thus, the general formula of the hybrid functionals can be written as follows (this is exploited in Gaussian software; http://gaussian.com/dft/),

<sup>X</sup> � ELSDA X , (1)

X

XC represents the exchange-correlation

<sup>X</sup> are the gradient corrections for the

<sup>C</sup> <sup>þ</sup> <sup>P</sup>5Enon�local

<sup>C</sup> : (2)

total energy value among all of those spin states.

DOI: http://dx.doi.org/10.5772/intechopen.83545

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

details of this issue.

EXC <sup>¼</sup> <sup>E</sup>LSDA

XC <sup>þ</sup> <sup>a</sup><sup>0</sup> Eexact

Up to date, QM calculations have been employed to understand many biochemical reactions, although the system sizes of such macromolecular systems are huge. In this section, we will briefly introduce several substantial issues in QM methods that have frequently been employed in analyses of biological systems. From biochemical and biophysical points of view, these descriptions are also relevant to the construction of the QM models, spin assignments, selection of QM/MM methods, the QM calculation methods, basis sets, and so on.

#### 1.1 Construction of QM model system

To obtain precise geometric and electronic structures employing the QM calculations, high-quality three-dimensional (3D) structures are indispensable. In most studies of biological macromolecular systems, the initial 3D structures for the theoretical analyses are retrieved from Protein Data Bank (PDB) website (https://www.rc sb.org/), which provides 3D structures of biological macromolecules analyzed employing X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, and electron microscope (EM) experiments. Currently, the PDB site contains more than 129,300 X-ray structures, 12,300 NMR structures, and 2400 EM structures.

Although state-of-the-art methodologies, such as the X-ray free electron laser (XFEL) and cryo-EM, provide high-quality 3D structures, the resolution of most experimental structures is still insufficient to observe hydrogen atoms. Indeed, only 0.5% of X-ray and EM structures are under 1.0 Å resolutions, and 80.3% are in the range of 1.4–2.8 Å resolutions. Thus, one needs to attach the hydrogen atoms in chemically appropriate manners. This is also an important issue, and so we need enough time to carefully identify the appropriate configurations for attachments of hydrogen atoms.

Since the computation costs of QM calculations are too large to include the entire biological macromolecular systems, QM models are usually extracted and thereby include the numbers of atoms in the ranges of 50–100 atoms. The truncated boundary carbons of the QM models are usually capped by the methyl group. Other crucial moieties in the systems, which can significantly affect geometric and electronic structures of the active centers, such as ligands of the transition metal binding sites and hydrogen-bonded waters, should also be included in the extracted QM models. Notably, to overcome the increase of computational costs by including large environmental moieties into the QM models such as the bulk water molecules as the solvent, hybrid quantum mechanics and molecular mechanics (MM) (i.e., classical mechanics) schemes have been developed up to date (the hybrid QM/MM calculation method is discussed in Section 1.3).

#### 1.2 Spin assignments of the system

As mentioned, we require 3D structures of the calculation models, basis sets, the (total) charges, spin multiplicities, and initial wave functions of the systems, to perform the QM calculations. Since the QM calculations may also suffer from the nonlinear, initial guess, and local minimum problems, starting from appropriate initial 3D structures and wave functions is very important to obtain the correct states. In particular, the total spin of the system including multiple transition metals Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

must be explored by carefully providing multiple combinations of spin assignments and thereby should be determined by identifying the spin state with the optimal total energy value among all of those spin states.

To provide the total spin assignments, it is convenient to divide the systems into some fragmental moieties, such as a transition metal and its coordinated ligands as a subsystem, which leads to spin assignments for a part of the QM models. Although this could be helpful to get a reasonable solution, there is no warranty of the convergence and acquisition of the correct solution. In Section 2, we discuss more details of this issue.

#### 1.3 Hybrid functional in density functional theory (DFT) calculation

Hybrid functional approximation was introduced by Becke [1] in 1993 and has become one of the most popular computational approaches. By incorporating a portion of the exact exchange energy originated from the Hartree-Fock theory coupled with the exchange-correlation energies, the hybrid functional approach improves molecular properties that poorly described with simple ab initio functionals, such as bond lengths, vibration frequencies, and atomization energies [2]. In general, a hybrid functional is described as a linear combination of the Hartree-Fock exact exchange energy functional and the exchange-correlation density functionals; their weights are determined by a fitting procedure such as to reproduce experimental or highly advanced calculated thermochemical data.

Becke proposed the following exchange-correlation approximation [1],

$$E\_{\rm XC} = E\_{\rm XC}^{\rm LSDA} + a\_0 \left( E\_X^{\rm exact} - E\_X^{\rm LSDA} \right) + a\_X \left( E\_X^{\rm B88} - E\_X^{\rm LSDA} \right) + a\_C \left( E\_X^{\rm PW91} - E\_X^{\rm LSDA} \right), \tag{1}$$

where a0, ax, and ac are the parameters determined by the fitting procedure (i.e., a0 = 0.20, ax = 0.72, and ac = 0.81). Herein, subscripts X and C represent the exchange and correlation, respectively, ELSDA XC represents the exchange-correlation energy within the framework of the local spin-density approximation (LSDA), Eexact X is the exact exchange energy, and EB<sup>88</sup> <sup>X</sup> and EPW<sup>91</sup> <sup>X</sup> are the gradient corrections for the exchange by Becke and Perdew-Wang, respectively.

The B3LYP functional, which has been one of the most widely used functionals in molecular quantum calculation fields, employs the nonlocal correlation provided by the LYP expression (Lee-Yang-Parr) and the Becke88 exchange functional [3], and VWN local-density approximation that was constructed by Volsko, Wilk, and Nusair (VWN) [4]. Thus, the general formula of the hybrid functionals can be written as follows (this is exploited in Gaussian software; http://gaussian.com/dft/),

$$E\_{\rm XC} = P\_2 E\_X^{\rm HF} + P\_1 \left( P\_4 E\_X^{\rm Slater} + P\_3 \Delta E\_X^{an-local} \right) + P\_6 E\_C^{local} + P\_5 E\_C^{an-local}.\tag{2}$$

Various combinations of nonlocal exchange functionals and correlation functionals can be employed here. In the B3LYP functional, the Becke's three-parameter functional and the LYP correlation functional were combined with the following parameters,

$$P\_1 = \mathbf{1}, P\_2 = \mathbf{0}.\mathbf{2}, P\_3 = \mathbf{0}.\mathbf{7}\mathbf{2}, P\_4 = \mathbf{0}.\mathbf{8}, P\_5 = \mathbf{0}.\mathbf{81}, P\_6 = \mathbf{1}.\tag{3}$$

In an attempt to improve the B3LYP functional, the long-range correlations are incorporated into the cam-B3LYP [5] and LC-BLYP [6] functionals. However, to the best of our experiences on applying to biological macromolecular systems,

repair, protein biosynthesis, respiration, photosynthesis, and synthesis and degradation of biological compounds (metabolites) such as amino acids, nucleotide, and lipid. In order to understand the mechanisms of biological functions, analyses of the

Up to date, QM calculations have been employed to understand many biochemical reactions, although the system sizes of such macromolecular systems are huge. In this section, we will briefly introduce several substantial issues in QM methods that have frequently been employed in analyses of biological systems. From biochemical and biophysical points of view, these descriptions are also relevant to the construction of the QM models, spin assignments, selection of QM/MM methods,

To obtain precise geometric and electronic structures employing the QM calcula-

Since the computation costs of QM calculations are too large to include the entire biological macromolecular systems, QM models are usually extracted and thereby include the numbers of atoms in the ranges of 50–100 atoms. The truncated

boundary carbons of the QM models are usually capped by the methyl group. Other crucial moieties in the systems, which can significantly affect geometric and electronic structures of the active centers, such as ligands of the transition metal binding sites and hydrogen-bonded waters, should also be included in the extracted QM models. Notably, to overcome the increase of computational costs by including large environmental moieties into the QM models such as the bulk water molecules as the solvent, hybrid quantum mechanics and molecular mechanics (MM) (i.e., classical mechanics) schemes have been developed up to date (the hybrid QM/MM

As mentioned, we require 3D structures of the calculation models, basis sets, the (total) charges, spin multiplicities, and initial wave functions of the systems, to perform the QM calculations. Since the QM calculations may also suffer from the nonlinear, initial guess, and local minimum problems, starting from appropriate initial 3D structures and wave functions is very important to obtain the correct states. In particular, the total spin of the system including multiple transition metals

tions, high-quality three-dimensional (3D) structures are indispensable. In most studies of biological macromolecular systems, the initial 3D structures for the theoretical analyses are retrieved from Protein Data Bank (PDB) website (https://www.rc sb.org/), which provides 3D structures of biological macromolecules analyzed employing X-ray crystallography, nuclear magnetic resonance (NMR) spectroscopy, and electron microscope (EM) experiments. Currently, the PDB site contains more than 129,300 X-ray structures, 12,300 NMR structures, and 2400 EM structures. Although state-of-the-art methodologies, such as the X-ray free electron laser (XFEL) and cryo-EM, provide high-quality 3D structures, the resolution of most experimental structures is still insufficient to observe hydrogen atoms. Indeed, only 0.5% of X-ray and EM structures are under 1.0 Å resolutions, and 80.3% are in the range of 1.4–2.8 Å resolutions. Thus, one needs to attach the hydrogen atoms in chemically appropriate manners. This is also an important issue, and so we need enough time to carefully identify the appropriate configurations for attachments of

electronic structure changes for the catalytic reactions are essential.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

the QM calculation methods, basis sets, and so on.

calculation method is discussed in Section 1.3).

1.2 Spin assignments of the system

1.1 Construction of QM model system

hydrogen atoms.

50

selection of functionals is not so simple, and thus, careful investigations based on computational trials are required to determine them. In the following two examples described in this chapter, such examinations were actually performed, and thereby, the B3LYP functional was adopted.

to the presence of the interactions between QM and MM regions, the total energy of

The inclusion of the energy term of QM-MM interactions, E(QM/MM), enables a more realistic description of the system, compared with isolated QM calculations. In terms of the treatment of the electrostatic interaction between the QM and MM regions, QM/MM methodologies are divided into two groups, subtractive and additive schemes. Herein, we discuss the advantages and disadvantages of the QM/MM

Subtractive schemes consist of the three steps as follows: (1) an MM calculation

The subscript indicates the type of calculation (QM or MM calculation), and the

on the entire system, (2) a QM calculation on the QM region, and (3) an MM calculation on the QM region. Then, QM/MM energy of the entire system can be

region on which the calculation is performed is described in parentheses. An advantage of the subtractive schemes is simplicity. Explicit descriptions of interactions between QM and MM regions are not required. In addition, artifacts that might be caused by using link atom schemes to cap the truncated bonds at the QM-MM boundary (described below) can be avoided. On the other hand, disadvantages of the subtractive schemes are the following. (1) Force fields are required for describing the QM region that often includes ligands and intermediate structures of enzymatic reactions; in general, reliable force fields of the molecules are not prepared, and additional QM calculations should be carried out for the parameterization every time a new system is studied. (2) The electrostatic interactions between the QM and MM regions are described at molecular mechanics level; that is, the interactions are calculated by the Coulomb interactions between fixed atomic charges in the QM and MM regions. Such descriptions cannot represent polarization

of the QM region induced by the environment surrounding the QM region.

effects. The energy expression for the additive schemes is given in Eq. (6):

<sup>E</sup>QM=MMð Þ¼ QM; MM <sup>E</sup>elec

charges can be used.

53

<sup>H</sup>^ elec

On the other hand, the additive schemes can take into account the polarization

A characteristic feature of this scheme is the presence of the energy term with respect to the interactions between QM and MM regions, described as follows:

QM=MMð Þþ QM; MM <sup>E</sup>vdW

qj r<sup>i</sup> � R<sup>j</sup> þ ∑ L k ∑ M j

To calculate electrostatic interactions, that is, the first term in the left side of Eq. (7), one-electron integrals in the QM Hamiltonian incorporating MM partial

> N i ∑ M j

QM=MMð Þ QM; MM

qj Qk R<sup>k</sup> � R<sup>j</sup>  (7)

(8)

QM=MMð Þ QM; MM

<sup>þ</sup> <sup>E</sup>bonded

QM=MMð Þ¼� QM; MM ∑

<sup>E</sup>addð Þ¼ ES <sup>E</sup>MMð Þþ MM <sup>E</sup>QMð Þþ QM <sup>E</sup>QM=MMð Þ QM; MM (6)

E ES ð Þ¼ E QM ð Þþ E MM ð Þþ E QM ð Þ =MM (4)

<sup>E</sup>subð Þ¼ ES <sup>E</sup>MMð Þþ ES <sup>E</sup>QMð Þ� QM <sup>E</sup>MMð Þ QM (5)

the entire system can be formally written as follows:

DOI: http://dx.doi.org/10.5772/intechopen.83545

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

methodologies in the comparison of these two schemes.

formulated as follows:

#### 1.4 Hybrid QM/MM calculation scheme

In 1976, Warshel and Levitt [7] developed a QM/MM method, in which QM calculation is combined with classical mechanics calculation, to obtain the electronic structures of the QM region with consideration of the environmental effects, such as protein, membrane, and solvent water molecules. In this strategy, the QM calculation is adopted to the active site (QM region), and for the remainder of the system, the MM calculation is adopted (MM regions) (Figure 1).

Great progresses have been achieved up to date for improvement of QM/MM calculation algorithms and their applications to biological systems [8–21]. Importance of the environments has been reported from many QM/MM studies. For example, polarization from MM region affects both electronic structure and geometric structure [22]. Recently, we reported that in huge biological macromolecular systems such as complexes of aminoacyl-tRNA synthetases (aaRSs) and their cognate tRNAs, dynamical, geometrical changes induced dramatical rearrangements of the electronic structures in the catalytic sites, which thus generated the productive states in the reactions [23–25] (see Section 3). By contrast, in Section 2, full (ab initio) QM calculations were solely employed for the analysis of the electronic structures of a transition metal cluster found in proteins, since in most cases, such a structure is buried in protein environments.

#### 1.5 Energy expression for QM/MM calculation

In the framework of QM/MM methodology, an entire system is divided into two regions: QM region, which is described by quantum mechanics principles, and MM region, which is described by molecular mechanics (i.e., classical mechanics). Due

#### Figure 1.

Hybrid QM/MM modeling of a biological macromolecular system (i.e., valyl-tRNA synthetase (ValRS) in complex with the cognate tRNA; see Section 3).

Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

to the presence of the interactions between QM and MM regions, the total energy of the entire system can be formally written as follows:

$$\mathbf{E(ES)} = \mathbf{E(QM)} + \mathbf{E(MM)} + \mathbf{E(QM/MM)}\tag{4}$$

The inclusion of the energy term of QM-MM interactions, E(QM/MM), enables a more realistic description of the system, compared with isolated QM calculations. In terms of the treatment of the electrostatic interaction between the QM and MM regions, QM/MM methodologies are divided into two groups, subtractive and additive schemes. Herein, we discuss the advantages and disadvantages of the QM/MM methodologies in the comparison of these two schemes.

Subtractive schemes consist of the three steps as follows: (1) an MM calculation on the entire system, (2) a QM calculation on the QM region, and (3) an MM calculation on the QM region. Then, QM/MM energy of the entire system can be formulated as follows:

$$E^{\rm sub}(\text{ES}) = E\_{\rm MM}(\text{ES}) + E\_{\rm QM}(\text{QM}) - E\_{\rm MM}(\text{QM}) \tag{5}$$

The subscript indicates the type of calculation (QM or MM calculation), and the region on which the calculation is performed is described in parentheses. An advantage of the subtractive schemes is simplicity. Explicit descriptions of interactions between QM and MM regions are not required. In addition, artifacts that might be caused by using link atom schemes to cap the truncated bonds at the QM-MM boundary (described below) can be avoided. On the other hand, disadvantages of the subtractive schemes are the following. (1) Force fields are required for describing the QM region that often includes ligands and intermediate structures of enzymatic reactions; in general, reliable force fields of the molecules are not prepared, and additional QM calculations should be carried out for the parameterization every time a new system is studied. (2) The electrostatic interactions between the QM and MM regions are described at molecular mechanics level; that is, the interactions are calculated by the Coulomb interactions between fixed atomic charges in the QM and MM regions. Such descriptions cannot represent polarization of the QM region induced by the environment surrounding the QM region.

On the other hand, the additive schemes can take into account the polarization effects. The energy expression for the additive schemes is given in Eq. (6):

$$E^{\rm add}(\rm ES) = E\_{\rm MM}(\rm MM) + E\_{\rm QM}(\rm QM) + E\_{\rm QM/MM}(\rm QM, MM) \tag{6}$$

A characteristic feature of this scheme is the presence of the energy term with respect to the interactions between QM and MM regions, described as follows:

$$\begin{aligned} E\_{\text{QM/MM}}(\text{QM, MM}) &= E\_{\text{QM/MM}}^{\text{elec}}(\text{QM, MM}) \\ &+ E\_{\text{QM/MM}}^{\text{bonded}}(\text{QM, MM}) + E\_{\text{QM/MM}}^{\text{evdW}}(\text{QM, MM}) \end{aligned} \tag{7}$$

To calculate electrostatic interactions, that is, the first term in the left side of Eq. (7), one-electron integrals in the QM Hamiltonian incorporating MM partial charges can be used.

$$\hat{H}\_{\text{QM/MM}}^{\text{elec}}(\text{QM}, \text{MM}) = -\sum\_{i}^{N} \sum\_{j}^{M} \frac{q\_{j}}{\left| \mathbf{r}\_{i} - \mathbf{R}\_{j} \right|} + \sum\_{k}^{L} \sum\_{j}^{M} \frac{q\_{j} Q\_{k}}{\left| \mathbf{R}\_{k} - \mathbf{R}\_{j} \right|} \tag{8}$$

selection of functionals is not so simple, and thus, careful investigations based on computational trials are required to determine them. In the following two examples described in this chapter, such examinations were actually performed, and thereby,

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

In 1976, Warshel and Levitt [7] developed a QM/MM method, in which QM calculation is combined with classical mechanics calculation, to obtain the electronic structures of the QM region with consideration of the environmental effects, such as protein, membrane, and solvent water molecules. In this strategy, the QM calculation is adopted to the active site (QM region), and for the remainder of the

Great progresses have been achieved up to date for improvement of QM/MM calculation algorithms and their applications to biological systems [8–21]. Importance of the environments has been reported from many QM/MM studies. For example, polarization from MM region affects both electronic structure and geometric structure [22]. Recently, we reported that in huge biological macromolecular systems such as complexes of aminoacyl-tRNA synthetases (aaRSs) and their cognate tRNAs, dynamical, geometrical changes induced dramatical rearrangements of the electronic structures in the catalytic sites, which thus generated the productive states in the reactions [23–25] (see Section 3). By contrast, in Section 2, full (ab initio) QM calculations were solely employed for the analysis of the electronic structures of a transition metal cluster found in proteins, since in most cases, such a

In the framework of QM/MM methodology, an entire system is divided into two regions: QM region, which is described by quantum mechanics principles, and MM region, which is described by molecular mechanics (i.e., classical mechanics). Due

Hybrid QM/MM modeling of a biological macromolecular system (i.e., valyl-tRNA synthetase (ValRS) in

system, the MM calculation is adopted (MM regions) (Figure 1).

the B3LYP functional was adopted.

1.4 Hybrid QM/MM calculation scheme

structure is buried in protein environments.

Figure 1.

52

complex with the cognate tRNA; see Section 3).

1.5 Energy expression for QM/MM calculation

The symbols q<sup>j</sup> are the MM partial charges located at Rj. Qk are the nuclear charges of the QM atoms at Rk, and ri represents electron positions. N, M, and L represent number of electrons, MM atoms to be incorporated into the one-electron integrals, and QM atoms, respectively. Using the additive scheme, the electronic structures of the QM region are affected by the charge distribution of the environment. An advanced approach is to consider polarization of the MM region by the QM region (i.e., to allow the partial charges to be changed according to changes in the electronic structure of the QM region). However, polarizable force fields with broader applications have not yet emerged, while many efforts to account for the polarization effects were made [26, 27].

#### 2. Ab initio QM analysis of electron transfer (ET) mechanisms in hydrogenase

#### 2.1 [NiFe] hydrogenase

Hydrogenase is an enzyme that can catalyze dihydrogen (H2) to water molecules and its reverse process [28]. Due to the reversible oxidation properties in the H2 catalysis, hydrogenase has been focused in biotechnological devices, such as generation of H2 from solar energy [29]. However, most [NiFe] hydrogenases, which are classified as standard hydrogenase, are sensitive to the explosion of the O2; that is, their activities decrease in an aerobic condition. By contrast, some hydrogenases preserve their activities even in the presence of O2, which are referred to as O2 tolerance.

Herein, we focus on membrane-bound [NiFe]-hydrogenases (MBHs), which are O2-tolerant hydrogenases. The 3D structures and active sites of MBHs are very similar to those of the standard hydrogenases except for the transition metal (iron) binding site that are located in the proximity of the catalytic center where Ni and Fe are bound, which are referred to as the proximal and [NiFe] active clusters, respectively (Figure 2).

Although the mechanisms of the O2-tolerance still remained to be resolved, the structural differences of the proximal clusters between MBHs and the standard hydrogenases were suggested to be responsible for the O2-tolerance mechanisms. More specifically, the proximal cluster of the standard hydrogenase contains [4Fe-4S]-4Cys cluster, while that of MBH contains [4Fe-3S]-6Cys cluster (Figure 2). In fact, two cysteine residues in MBH are replaced with glycine (Gly) residues in the standard [NiFe]-hydrogenases.

In addition, for the proximal cluster of MBH, three charge states have been reported; that is, the reduced, oxidized, and superoxidized states. Moreover, the 3D structure of the proximal cluster in MBH is also changed depending on those redox states. Moreover, combined crystallographic and spectroscopic analyses have recently suggested that a hydroxyl ion (OH) was attached to a Fe ion in the superoxidized states of the proximal cluster in Ralstonia eutropha MBH [30]. However, its functional role has still remained to be clarified.

structures of the optimum energy states in the presence and absence of the hydroxyl ion, we employed full (ab initio) QM calculations with the use of the B3LYP func-

Stereoview of the 3D structures of the entire structure (A) and proximal cluster (B) of Ralstonia eutropha

To build structural models, we employed the atomic coordinates of the proximal cluster in the superoxidized state of the crystal structure of Ralstonia eutropha MBH (PDB ID: 4IUD). Our computation models included the iron-sulfur cluster (i.e.,

tional as mentioned above (see Section 1.3).

MBH, and schematic representation of model 2 (C). © Kim et al. [31].

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

DOI: http://dx.doi.org/10.5772/intechopen.83545

2.2 Exploration of spin assignments

Figure 2.

55

In this section, we first introduce the way of how we investigated the electronic structures of the proximal cluster of Ralstonia eutropha MBH. Herein, systematic exploration of spin combinations of the proximal cluster was essential to obtain the reliable calculation data. For the calculation models, we constructed two structural models of the proximal cluster in the presence and absence of the hydroxyl ion and compared their detailed electronic structures, to reveal the functional role of the hydroxyl ion. To evaluate the total energy of various spin states and the electronic

Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

Figure 2.

The symbols q<sup>j</sup> are the MM partial charges located at Rj. Qk are the nuclear charges of the QM atoms at Rk, and ri represents electron positions. N, M, and L represent number of electrons, MM atoms to be incorporated into the one-electron integrals, and QM atoms, respectively. Using the additive scheme, the electronic structures of the QM region are affected by the charge distribution of the environment. An advanced approach is to consider polarization of the MM region by the QM region (i.e., to allow the partial charges to be changed according to changes in the electronic structure of the QM region). However, polarizable force fields with broader applications have not yet emerged, while many efforts to account for the

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

2. Ab initio QM analysis of electron transfer (ET) mechanisms in

Hydrogenase is an enzyme that can catalyze dihydrogen (H2) to water molecules and its reverse process [28]. Due to the reversible oxidation properties in the H2 catalysis, hydrogenase has been focused in biotechnological devices, such as generation of H2 from solar energy [29]. However, most [NiFe] hydrogenases, which are classified as standard hydrogenase, are sensitive to the explosion of the O2; that is, their activities decrease in an aerobic condition. By contrast, some hydrogenases preserve their activities even in the presence of O2, which are referred to as O2-

Herein, we focus on membrane-bound [NiFe]-hydrogenases (MBHs), which are

Although the mechanisms of the O2-tolerance still remained to be resolved, the structural differences of the proximal clusters between MBHs and the standard hydrogenases were suggested to be responsible for the O2-tolerance mechanisms. More specifically, the proximal cluster of the standard hydrogenase contains [4Fe-4S]-4Cys cluster, while that of MBH contains [4Fe-3S]-6Cys cluster

(Figure 2). In fact, two cysteine residues in MBH are replaced with glycine (Gly)

In addition, for the proximal cluster of MBH, three charge states have been reported; that is, the reduced, oxidized, and superoxidized states. Moreover, the 3D structure of the proximal cluster in MBH is also changed depending on those redox states. Moreover, combined crystallographic and spectroscopic analyses have recently suggested that a hydroxyl ion (OH) was attached to a Fe ion in the superoxidized states of the proximal cluster in Ralstonia eutropha MBH [30]. How-

In this section, we first introduce the way of how we investigated the electronic structures of the proximal cluster of Ralstonia eutropha MBH. Herein, systematic exploration of spin combinations of the proximal cluster was essential to obtain the reliable calculation data. For the calculation models, we constructed two structural models of the proximal cluster in the presence and absence of the hydroxyl ion and compared their detailed electronic structures, to reveal the functional role of the hydroxyl ion. To evaluate the total energy of various spin states and the electronic

residues in the standard [NiFe]-hydrogenases.

ever, its functional role has still remained to be clarified.

O2-tolerant hydrogenases. The 3D structures and active sites of MBHs are very similar to those of the standard hydrogenases except for the transition metal (iron) binding site that are located in the proximity of the catalytic center where Ni and Fe are bound, which are referred to as the proximal and [NiFe] active clusters, respec-

polarization effects were made [26, 27].

hydrogenase

tolerance.

54

tively (Figure 2).

2.1 [NiFe] hydrogenase

Stereoview of the 3D structures of the entire structure (A) and proximal cluster (B) of Ralstonia eutropha MBH, and schematic representation of model 2 (C). © Kim et al. [31].

structures of the optimum energy states in the presence and absence of the hydroxyl ion, we employed full (ab initio) QM calculations with the use of the B3LYP functional as mentioned above (see Section 1.3).

#### 2.2 Exploration of spin assignments

To build structural models, we employed the atomic coordinates of the proximal cluster in the superoxidized state of the crystal structure of Ralstonia eutropha MBH (PDB ID: 4IUD). Our computation models included the iron-sulfur cluster (i.e.,

[4Fe-3S]) and six Fe-coordinated cysteine (Cys) residues (i.e., Cys17, Cys19, Cys20, Cys120, Cys115, and Cys149). Moreover, we also included three amino acid residues (i.e., Ser21, Glu76, and His229), two crystal water molecules, and OH ion, all of which are coordinated to the iron-sulfur cluster.

distorted, compared with the simple Fe-S clusters that were analyzed in the previous studies. Moreover, the attachment of the hydroxyl ion (model 2) induced the distinct electronic structures when we compared with those of model 1 and the standard iron-sulfur clusters. Thus, due to the distorted geometrical structure and attachment of the hydroxyl ion, the equivalence of BSij and BSji cannot be assured in the present case. In fact, BS34 and BS43 of models 1 and 2 were definitely different in the total energy by 7.78 and 1.77 kcal/mol, respectively (here, the

In this manner, we determined the optimal spin states of the proximal cluster of

In the presence of O2, the inactive form is induced with respect to the [NiFe] catalytic site of MBHs (i.e., the Ni-B state). For the reactivation of the catalysis, the [NiFe] active site is required to be changed to another state (i.e., the Ni-SI state) [41]. Two recovery mechanisms have been suggested up to date. First, Volbeda et al. [42] suggested that formation of a dimer enhanced the reactivation of MBHs; that is, one that is inactivated can be recovered by the other that is activated in the dimer. In this mechanism, at least two electrons should be transferred from the activated MBH to the inactivated MBH, and then the received electrons are transferred through the distal, medial, proximal clusters, and the [NiFe] active site [42]. In the second mechanism suggested by Kurkin et al. [43], the reduction of the [NiFe] active site in the presence of H2 reactivates the inactive MBH, although the process requires a few seconds (note here that the O2-sensitive hydrogenases such as the standard [NiFe] hydrogenase require over 1 h to be reactivated). In this reactivation process, H2 cleavage reaction induces the Ni-SI state from the Ni-B state of the [NiFe] active site, and four electrons should be transferred from the [NiFe] active site to the proximal and medial clusters [43]. Notably, the directions of the electron transfers (ETs) are opposing between these two mechanisms that

To describe the differences of the electronic structures of the proximal cluster in the presence (model 2) and absence (model 1) of the hydroxyl ion [31], we focus on the lowest unoccupied molecular orbital (LUMO) here, since LUMO could be closely related to the ET, which is required to occur in both reactivation mechanisms as mentioned above. In fact, comparison of the LUMOs of models 1 and 2 led us to identify the effects of the hydroxyl ion on the LUMOs: In the absence of the hydroxyl ions (model 1), the LUMO was localized on Fe4 and SCys20, whereas in the presence of the hydroxyl ion (model 2), the LUMO was delocalized on SCys17, SCys19,

SCys20, NCys20, the hydroxyl ion, Fe1, Fe2, and Fe4 (Figure 3A and B).

To further approach the substantial meanings of differences found in the calculated electronic structures, we also investigated the ET pathways by adopting an empirical simulation methodology, which can identify possible ET pathways by minimizing the penalties of the steps mediated by covalent bonds, hydrogen bonds, and spaces [44, 45]. Based on the aforementioned two ET processes, which have been suggested so far, that is, (1) the medial to proximal clusters and (2) the [NiFe] active site to proximal cluster, we examined the validity of these

As a result of the analysis, we revealed that the optimal ET pathways were significantly overlapped with the delocalized LUMOs of model 2, which means that

the MBH in the presence and absence of the hydroxyl ion, as the lowest energy states, that is, BS12 and BS34 of models 1 and 2, respectively. In the subsequent

part, we describe the electronic structures of these spin states.

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

optimum total energy is set to 0 kcal/mol).

DOI: http://dx.doi.org/10.5772/intechopen.83545

2.3 Functional role of OH

have been suggested so far.

two ET pathways.

57

Six amino acid residues (Cys17, Cys115, Cys120, Cys149, Glu76, and His229) were truncated by Cα atoms with the attachment of methyl groups (▬CH3). As mentioned in the last section, we constructed another similar model that did not include the OH ion to reveal its effects, and thus 103 and 101 atoms were included in our structural models. These are referred to as the original (model 2) and Δ(OH) (model 1) models, respectively (Figure 2).

Spectroscopic experiments elucidated the formal charge and total spin of the [4Fe-3S] cluster in the superoxidized state as +5 and 1/2, respectively [32]. For each of the iron and sulfur ions in the [4Fe-3S] proximal cluster, we set the formal charge as Fe2+ or Fe3+, and S2, as found in the previous study [32]. Thus, the core consists of three Fe3+, one Fe2+, and 3S2 ions.

Then, we constructed simple small fragmental models that were extracted from the 3D structure of the proximal cluster core: each of three Fe ions labeled as Fe2, Fe3, and Fe4 form a tetrahedral structure in the [4Fe-3S] proximal cluster, while the other Fe ion labeled as Fe1 form bipyramidal structures together with SCys19, SCys17, S1, (O atom of the hydroxyl ion), and S2 atoms (Figure 2).

Thus, we built small models including only the core atoms (i.e., Fe1, SCys19, SCys17, S1, and S2) in the presence and absence of the OH and evaluated the total energy values of the models. The analysis revealed that the optimum spin states with the minimum total energies were composed of the high spin states of Fe ions, which is consistent with the previous experimental data [33].

Herein, to specify the spin assignments of the [4Fe-3S] cluster, we represent them employing the nomenclature, BSij; that is, BS is an abbreviation of the broken symmetry state, and i and j indicate the (serial) numbers of Fe ions, as follows. Due to the two constraints, that is, (1) Fe ions take the high spin states as found above, and (2) the total spin sum is 1/2 (experimental data), the possible spin combinations of Fe ions are deduced as 5/2, 4/2, and 5/2. We adopt the indices i and j that should be corresponding to the spin states of 4/2 and 5/2 (of Fe), respectively [34]. For example, BS12 represents that spin of Fe1 and Fe2 are assigned to the 4/2 and 5/2, respectively, and thus, 5/2 spin state is assigned to Fe3 and Fe4. Thus, BS12 represents (Fe1, Fe2, Fe3, Fe4) = (4/2, 5/2, +5/2, +5/2).

Based on these considerations, we found that 12 spin assignments are possible for each structural model, and thus, we performed 24 QM calculations, to identify the optimal spin states of the [4Fe-3S] proximal cluster in the presence and absence of the hydroxyl ion. We employed the Gaussian09 package for all QM calculations with the B3LYP functional [3, 35]. For the [4Fe-3S] core and atoms that are coordinated to the Fe ions, the triple-ζ valence polarized (TZVP) basis set [36, 37] was adopted, and for the other atoms, the 6-311G\*\* basis set [38] was employed. We performed geometry optimization with all hydrogen atoms being movable.

As a result of the analysis, we found that the total energy of BS12, BS21, BS13, and BS31 was smaller than the other states in model 1 and that the total energy of BS12, BS21, BS34, and BS43 was lower than the other states in model 2. Thus, we indicated that the favorable spin assignments were depending on the presence and absence of hydroxyl ion in the proximal cluster.

Note here that in previous studies employing DFT calculations and the simple iron-sulfur clusters, such as [2Fe2S], [3Fe4S], and [4Fe4S], BSij and BSji were shown to be identical [39, 40]. However, this equivalence of BSij and BSji was not satisfied in the present case, since the [4Fe-3S] proximal cluster in the MBH is

Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

distorted, compared with the simple Fe-S clusters that were analyzed in the previous studies. Moreover, the attachment of the hydroxyl ion (model 2) induced the distinct electronic structures when we compared with those of model 1 and the standard iron-sulfur clusters. Thus, due to the distorted geometrical structure and attachment of the hydroxyl ion, the equivalence of BSij and BSji cannot be assured in the present case. In fact, BS34 and BS43 of models 1 and 2 were definitely different in the total energy by 7.78 and 1.77 kcal/mol, respectively (here, the optimum total energy is set to 0 kcal/mol).

In this manner, we determined the optimal spin states of the proximal cluster of the MBH in the presence and absence of the hydroxyl ion, as the lowest energy states, that is, BS12 and BS34 of models 1 and 2, respectively. In the subsequent part, we describe the electronic structures of these spin states.

#### 2.3 Functional role of OH

[4Fe-3S]) and six Fe-coordinated cysteine (Cys) residues (i.e., Cys17, Cys19, Cys20, Cys120, Cys115, and Cys149). Moreover, we also included three amino acid residues (i.e., Ser21, Glu76, and His229), two crystal water molecules, and OH ion,

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

in our structural models. These are referred to as the original (model 2) and

SCys17, S1, (O atom of the hydroxyl ion), and S2 atoms (Figure 2).

which is consistent with the previous experimental data [33].

BS12 represents (Fe1, Fe2, Fe3, Fe4) = (4/2, 5/2, +5/2, +5/2).

absence of hydroxyl ion in the proximal cluster.

56

Six amino acid residues (Cys17, Cys115, Cys120, Cys149, Glu76, and His229) were truncated by Cα atoms with the attachment of methyl groups (▬CH3). As mentioned in the last section, we constructed another similar model that did not include the OH ion to reveal its effects, and thus 103 and 101 atoms were included

Spectroscopic experiments elucidated the formal charge and total spin of the [4Fe-3S] cluster in the superoxidized state as +5 and 1/2, respectively [32]. For each of the iron and sulfur ions in the [4Fe-3S] proximal cluster, we set the formal charge as Fe2+ or Fe3+, and S2, as found in the previous study [32]. Thus, the core consists

Then, we constructed simple small fragmental models that were extracted from the 3D structure of the proximal cluster core: each of three Fe ions labeled as Fe2, Fe3, and Fe4 form a tetrahedral structure in the [4Fe-3S] proximal cluster, while the other Fe ion labeled as Fe1 form bipyramidal structures together with SCys19,

Thus, we built small models including only the core atoms (i.e., Fe1, SCys19, SCys17, S1, and S2) in the presence and absence of the OH and evaluated the total energy values of the models. The analysis revealed that the optimum spin states with the minimum total energies were composed of the high spin states of Fe ions,

Herein, to specify the spin assignments of the [4Fe-3S] cluster, we represent them employing the nomenclature, BSij; that is, BS is an abbreviation of the broken symmetry state, and i and j indicate the (serial) numbers of Fe ions, as follows. Due to the two constraints, that is, (1) Fe ions take the high spin states as found above, and (2) the total spin sum is 1/2 (experimental data), the possible spin combinations of Fe ions are deduced as 5/2, 4/2, and 5/2. We adopt the indices i and j that should be corresponding to the spin states of 4/2 and 5/2 (of Fe), respectively [34]. For example, BS12 represents that spin of Fe1 and Fe2 are assigned to the 4/2 and 5/2, respectively, and thus, 5/2 spin state is assigned to Fe3 and Fe4. Thus,

Based on these considerations, we found that 12 spin assignments are possible for each structural model, and thus, we performed 24 QM calculations, to identify the optimal spin states of the [4Fe-3S] proximal cluster in the presence and absence of the hydroxyl ion. We employed the Gaussian09 package for all QM calculations with the B3LYP functional [3, 35]. For the [4Fe-3S] core and atoms that are coordinated to the Fe ions, the triple-ζ valence polarized (TZVP) basis set [36, 37] was adopted, and for the other atoms, the 6-311G\*\* basis set [38] was employed. We performed geometry optimization with all hydrogen atoms being movable.

As a result of the analysis, we found that the total energy of BS12, BS21, BS13, and BS31 was smaller than the other states in model 1 and that the total energy of BS12, BS21, BS34, and BS43 was lower than the other states in model 2. Thus, we indicated that the favorable spin assignments were depending on the presence and

Note here that in previous studies employing DFT calculations and the simple iron-sulfur clusters, such as [2Fe2S], [3Fe4S], and [4Fe4S], BSij and BSji were shown to be identical [39, 40]. However, this equivalence of BSij and BSji was not satisfied in the present case, since the [4Fe-3S] proximal cluster in the MBH is

all of which are coordinated to the iron-sulfur cluster.

Δ(OH) (model 1) models, respectively (Figure 2).

of three Fe3+, one Fe2+, and 3S2 ions.

In the presence of O2, the inactive form is induced with respect to the [NiFe] catalytic site of MBHs (i.e., the Ni-B state). For the reactivation of the catalysis, the [NiFe] active site is required to be changed to another state (i.e., the Ni-SI state) [41]. Two recovery mechanisms have been suggested up to date. First, Volbeda et al. [42] suggested that formation of a dimer enhanced the reactivation of MBHs; that is, one that is inactivated can be recovered by the other that is activated in the dimer. In this mechanism, at least two electrons should be transferred from the activated MBH to the inactivated MBH, and then the received electrons are transferred through the distal, medial, proximal clusters, and the [NiFe] active site [42].

In the second mechanism suggested by Kurkin et al. [43], the reduction of the [NiFe] active site in the presence of H2 reactivates the inactive MBH, although the process requires a few seconds (note here that the O2-sensitive hydrogenases such as the standard [NiFe] hydrogenase require over 1 h to be reactivated). In this reactivation process, H2 cleavage reaction induces the Ni-SI state from the Ni-B state of the [NiFe] active site, and four electrons should be transferred from the [NiFe] active site to the proximal and medial clusters [43]. Notably, the directions of the electron transfers (ETs) are opposing between these two mechanisms that have been suggested so far.

To describe the differences of the electronic structures of the proximal cluster in the presence (model 2) and absence (model 1) of the hydroxyl ion [31], we focus on the lowest unoccupied molecular orbital (LUMO) here, since LUMO could be closely related to the ET, which is required to occur in both reactivation mechanisms as mentioned above. In fact, comparison of the LUMOs of models 1 and 2 led us to identify the effects of the hydroxyl ion on the LUMOs: In the absence of the hydroxyl ions (model 1), the LUMO was localized on Fe4 and SCys20, whereas in the presence of the hydroxyl ion (model 2), the LUMO was delocalized on SCys17, SCys19, SCys20, NCys20, the hydroxyl ion, Fe1, Fe2, and Fe4 (Figure 3A and B).

To further approach the substantial meanings of differences found in the calculated electronic structures, we also investigated the ET pathways by adopting an empirical simulation methodology, which can identify possible ET pathways by minimizing the penalties of the steps mediated by covalent bonds, hydrogen bonds, and spaces [44, 45]. Based on the aforementioned two ET processes, which have been suggested so far, that is, (1) the medial to proximal clusters and (2) the [NiFe] active site to proximal cluster, we examined the validity of these two ET pathways.

As a result of the analysis, we revealed that the optimal ET pathways were significantly overlapped with the delocalized LUMOs of model 2, which means that

the SCys17-HO-SCys19-Fe4 segmental electronic field, which is a main component of

To the best of our knowledge, this is the first work to have revealed the mechanisms of creation of the ET pathways in biological macromolecular systems [31]. Note here that a tiny molecular species, that is, OH, is a trigger to generate the ET pathways. Thus, organisms regulate the functions employing such a subtle factor but thereby dramatically change their physiological status. The present achievements could further be a solid basis toward sophisticated rational design of novel catalysts, reactions, and functional materials, through regulations of the elaborately

The origin and formation mechanisms of the delocalized LUMO are a very interesting issue. However, the limitation of space here does not allow us to describe it, and so for the detailed descriptions on this issue, see text and Figure 6(a) in our

the delocalized LUMO in the presence of the OH.

DOI: http://dx.doi.org/10.5772/intechopen.83545

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

constituted orbital-based electronic structures.

brane moieties, and solvent [17, 22, 46, 47].

3.1 Aminoacyl-tRNA synthetases (aaRSs)

3. Hybrid ab initio QM/MM molecular dynamics simulation

In the last section, we presented the effects of the attachment of a hydroxyl ligand on the electronic structure of the proximal cluster in the MBH and also showed that the effects could be closely related to the ET and recovery from the inactive form of the [NiFe] active site (i.e., the O2-tolerance of the MBH). Thus, the tiny species dramatically changes the electronic structures of the enzymes, as

In this section, we demonstrate that electronic structures are dynamically changed as catalytic reactions proceed. In order to investigate such dynamical properties of electronic structures in biological macromolecular systems, we built a

superparallel computers. This computational system enabled us to evaluate dynamical transitions of electronic structures in catalytic sites involving the environmental structures such as protein moieties, nucleic acid (RNA and DNA) moieties, mem-

We introduce editing reaction of aminoacyl-tRNA synthetase (aaRS), which forms a protein family composed of twenty enzymes, divided into two classes, that is, classes I and II [48]. Here, we focus on two class I aaRSs, that is, leucyl-tRNA synthetase (LeuRS) and valyl-tRNA synthetase (ValRS). We performed hybrid ab initio QM/MM MD simulations and thereby revealed that as the reactions proceeded, dynamical rearrangements of molecular orbitals (MOs) occurred, which was critical for both covalent bond formation and cleavage. We emphasize here that such dramatic transitions of electronic structures would be characteristic in catalytic reactions

In the central dogma, which demonstrates the flow of the genetic information (e.g., from gene coded in genome DNA toward protein), the adaptation between a codon (i.e., combination of three nucleotide bases) and an amino acid (aa) is required in the protein biosynthesis (i.e., translation), while synthesis of messenger RNA (mRNA) (i.e., transcription) occurs based on the rules of base pairing (e.g. the Watson-Crick and wobble base pairs). Thereby, base sequences of genes are converted to amino acid sequences of proteins. Here, aaRSs play a critical role by correctly recognizing and attaching both specific amino acid and cognate tRNA,

hybrid ab initio QM/MM molecular dynamics (MD) calculation system on

of biological macromolecular systems, as also indicated in hydrogenase.

report [31].

discussed above.

59

#### Figure 3.

Stereoview of the LUMOs in terms of the optimal spin states of models 1 (BS12) (A) and 2 (BS34) (B). In model 1 (i.e., Δ(OH)), the LUMOs are localized on Fe4 and SCys20 atoms. By contrast, in model 2 (i.e., involving OH), the LUMO is distributed and thus principally composed of SCys17, SCys19, SCys20, NCys20, the hydroxyl ion, Fe1, Fe2, and Fe4. To investigate the relationships between the effects of the hydroxyl ion and the ET mechanisms, the plausible ET pathways obtained by an empirical method (i.e., pathway) to search for the ET pathways (C) are compared with the distribution of the LUMO of model 2, which is identical to that shown in panel (B) (note that the directions for rendering of the structures are identical in the panels (A) and (B), while those are different between the panels (A)/(B) and (C)). The blue line with an arrow shows the ET pathway from the [NiFe] active site to the proximal cluster, and the red line with an arrow shows the ET pathway from the medial to proximal clusters. © Kim et al., [31].

the attachment of the hydroxyl ion to Fe1 may promote the ETs (Figure 3C). This also means that the attachment of the hydroxyl ion creates the ET pathways in the proximal cluster by inducing the electron delocalization of the LUMO, thus forming Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

the SCys17-HO-SCys19-Fe4 segmental electronic field, which is a main component of the delocalized LUMO in the presence of the OH.

The origin and formation mechanisms of the delocalized LUMO are a very interesting issue. However, the limitation of space here does not allow us to describe it, and so for the detailed descriptions on this issue, see text and Figure 6(a) in our report [31].

To the best of our knowledge, this is the first work to have revealed the mechanisms of creation of the ET pathways in biological macromolecular systems [31]. Note here that a tiny molecular species, that is, OH, is a trigger to generate the ET pathways. Thus, organisms regulate the functions employing such a subtle factor but thereby dramatically change their physiological status. The present achievements could further be a solid basis toward sophisticated rational design of novel catalysts, reactions, and functional materials, through regulations of the elaborately constituted orbital-based electronic structures.

#### 3. Hybrid ab initio QM/MM molecular dynamics simulation

In the last section, we presented the effects of the attachment of a hydroxyl ligand on the electronic structure of the proximal cluster in the MBH and also showed that the effects could be closely related to the ET and recovery from the inactive form of the [NiFe] active site (i.e., the O2-tolerance of the MBH). Thus, the tiny species dramatically changes the electronic structures of the enzymes, as discussed above.

In this section, we demonstrate that electronic structures are dynamically changed as catalytic reactions proceed. In order to investigate such dynamical properties of electronic structures in biological macromolecular systems, we built a hybrid ab initio QM/MM molecular dynamics (MD) calculation system on superparallel computers. This computational system enabled us to evaluate dynamical transitions of electronic structures in catalytic sites involving the environmental structures such as protein moieties, nucleic acid (RNA and DNA) moieties, membrane moieties, and solvent [17, 22, 46, 47].

We introduce editing reaction of aminoacyl-tRNA synthetase (aaRS), which forms a protein family composed of twenty enzymes, divided into two classes, that is, classes I and II [48]. Here, we focus on two class I aaRSs, that is, leucyl-tRNA synthetase (LeuRS) and valyl-tRNA synthetase (ValRS). We performed hybrid ab initio QM/MM MD simulations and thereby revealed that as the reactions proceeded, dynamical rearrangements of molecular orbitals (MOs) occurred, which was critical for both covalent bond formation and cleavage. We emphasize here that such dramatic transitions of electronic structures would be characteristic in catalytic reactions of biological macromolecular systems, as also indicated in hydrogenase.

#### 3.1 Aminoacyl-tRNA synthetases (aaRSs)

In the central dogma, which demonstrates the flow of the genetic information (e.g., from gene coded in genome DNA toward protein), the adaptation between a codon (i.e., combination of three nucleotide bases) and an amino acid (aa) is required in the protein biosynthesis (i.e., translation), while synthesis of messenger RNA (mRNA) (i.e., transcription) occurs based on the rules of base pairing (e.g. the Watson-Crick and wobble base pairs). Thereby, base sequences of genes are converted to amino acid sequences of proteins. Here, aaRSs play a critical role by correctly recognizing and attaching both specific amino acid and cognate tRNA,

the attachment of the hydroxyl ion to Fe1 may promote the ETs (Figure 3C). This also means that the attachment of the hydroxyl ion creates the ET pathways in the proximal cluster by inducing the electron delocalization of the LUMO, thus forming

pathway from the medial to proximal clusters. © Kim et al., [31].

Stereoview of the LUMOs in terms of the optimal spin states of models 1 (BS12) (A) and 2 (BS34) (B). In model 1 (i.e., Δ(OH)), the LUMOs are localized on Fe4 and SCys20 atoms. By contrast, in model 2 (i.e., involving OH), the LUMO is distributed and thus principally composed of SCys17, SCys19, SCys20, NCys20, the hydroxyl ion, Fe1, Fe2, and Fe4. To investigate the relationships between the effects of the hydroxyl ion and the ET mechanisms, the plausible ET pathways obtained by an empirical method (i.e., pathway) to search for the ET pathways (C) are compared with the distribution of the LUMO of model 2, which is identical to that shown in panel (B) (note that the directions for rendering of the structures are identical in the panels (A) and (B), while those are different between the panels (A)/(B) and (C)). The blue line with an arrow shows the ET pathway from the [NiFe] active site to the proximal cluster, and the red line with an arrow shows the ET

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Figure 3.

58

thereby forming aminoacyl-tRNA (i.e., aa-tRNAaa). This crucial catalytic reaction is referred to as aminoacylation.

If aaRSs misattach noncognate amino acids (or tRNAs) to the specific tRNAs (or amino acids), then the noncognate amino acids are incorporated into the protein as an incorrect amino acid residue, which is different from the correct one coded in the gene, since the tRNA recognizes its cognate codon by making base pairs between the codon (in mRNA) and the anticodon moiety of the tRNA. It should be noted here that only aaRSs determine the relationship between amino acids and codons for biosynthesis of proteins.

Based on the 3D structures, aaRSs are classified into two classes, that is, classes I and II [49, 50]. More specifically, catalytic cores of class I aaRSs contain the classical nucleotide-binding fold (i.e., the Rossmann fold). By contrast, the active sites of class II aaRSs possess an antiparallel β sheet flanked by α helices, which is the architecture completely different from the Rossmann fold. Based on the structural similarity of the catalytic and noncatalytic domains, each class is further classified into three subclasses a, b, and c [51, 52].

#### 3.2 Editing reaction of aaRSs

As mentioned above, high translational fidelity is essential in the decoding of genetic information from mRNA (i.e., base sequences) to protein (i.e., amino acid sequences). Notably, the aminoacylation reaction of aaRSs consists of two steps; that is, (1) activation of the amino acid yielding aminoacyl adenylate and (2) transfer of amino acid moiety of the aminoacyl adenylate to the 3<sup>0</sup> -end of the tRNA. However, misactivated amino acids and misaminoacylated tRNAs are generated occasionally, since some amino acids are structurally similar (e.g., leucine (Leu), isoleucine (Ile), and valine (Val)).

To ensure the translational fidelity, some aaRSs possess editing functions to correct such errors [53–60]. Correspondingly, two types of editing reactions are known, that is, pre- and posttransfer editing reactions, which hydrolyze a misactivated amino acid and misaminoacylated tRNA, respectively. Herein, we focus on the posttransfer editing reaction, since the catalytic mechanisms were remained to be elucidated for the last some decades.

#### 3.3 Ab initio QM/MM MD simulation of editing reaction

For the Leu system, we constructed a structural model of LeuRS in complex with a misaminoacylated tRNALeu (i.e., valyl-tRNALeu), where the 3<sup>0</sup> -end nucleotide (adenine 76; A76) is bound to the active site for the editing reaction in the connective polypeptide (CP) 1 domain [22] (Figure 4). Then, we performed classical MD simulation and succeeded in identification of the nucleophilic water for the editing reaction [23]. Employing this structural model, we performed hybrid ab initio QM/ MM MD simulations with the use of our QM/MM interface program [47] that connects QM and MM calculation engines (i.e., GAMESS [61] and AMBER [62], respectively).

To determine the reaction path, we employed an adiabatic mapping approach, in which hybrid ab initio QM/MM MD simulations were performed to enhance the conformational sampling, together with hybrid ab initio QM/MM geometry optimization being employed to reach the potential energy surface. This scheme enabled us to conduct more effective explorations of both conformations and electronic structures than previous schemes that employ only geometry optimizations [63]. We assumed some possible reaction pathways and performed hybrid ab initio QM/MM MD simulation for each pathway, which provided the estimation of the energy barrier

Figure 4.

61

(A) (Left) Entire 3D structure of the ValRSthereonyl-tRNAVal complex is shown as an example of structures of class Ia aaRSs (i.e., involving the Leu, Val, and Ile systems). (Right) Catalytic site of the editing reaction is

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

DOI: http://dx.doi.org/10.5772/intechopen.83545

Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

Figure 4.

(A) (Left) Entire 3D structure of the ValRSthereonyl-tRNAVal complex is shown as an example of structures of class Ia aaRSs (i.e., involving the Leu, Val, and Ile systems). (Right) Catalytic site of the editing reaction is

thereby forming aminoacyl-tRNA (i.e., aa-tRNAaa). This crucial catalytic reaction is

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

If aaRSs misattach noncognate amino acids (or tRNAs) to the specific tRNAs (or amino acids), then the noncognate amino acids are incorporated into the protein as an incorrect amino acid residue, which is different from the correct one coded in the gene, since the tRNA recognizes its cognate codon by making base pairs between the codon (in mRNA) and the anticodon moiety of the tRNA. It should be noted here that only aaRSs determine the relationship between amino acids and codons

Based on the 3D structures, aaRSs are classified into two classes, that is, classes I and II [49, 50]. More specifically, catalytic cores of class I aaRSs contain the classical nucleotide-binding fold (i.e., the Rossmann fold). By contrast, the active sites of class II aaRSs possess an antiparallel β sheet flanked by α helices, which is the architecture completely different from the Rossmann fold. Based on the structural similarity of the catalytic and noncatalytic domains, each class is further classified

As mentioned above, high translational fidelity is essential in the decoding of genetic information from mRNA (i.e., base sequences) to protein (i.e., amino acid sequences). Notably, the aminoacylation reaction of aaRSs consists of two steps; that is, (1) activation of the amino acid yielding aminoacyl adenylate and (2) transfer of amino acid moiety of the aminoacyl adenylate to the 3<sup>0</sup>

tRNA. However, misactivated amino acids and misaminoacylated tRNAs are generated occasionally, since some amino acids are structurally similar (e.g., leucine

To ensure the translational fidelity, some aaRSs possess editing functions to correct such errors [53–60]. Correspondingly, two types of editing reactions are known, that is, pre- and posttransfer editing reactions, which hydrolyze a misactivated amino acid and misaminoacylated tRNA, respectively. Herein, we focus on the posttransfer editing reaction, since the catalytic mechanisms were

For the Leu system, we constructed a structural model of LeuRS in complex with

(adenine 76; A76) is bound to the active site for the editing reaction in the connective polypeptide (CP) 1 domain [22] (Figure 4). Then, we performed classical MD simulation and succeeded in identification of the nucleophilic water for the editing reaction [23]. Employing this structural model, we performed hybrid ab initio QM/ MM MD simulations with the use of our QM/MM interface program [47] that connects QM and MM calculation engines (i.e., GAMESS [61] and AMBER [62],

To determine the reaction path, we employed an adiabatic mapping approach, in which hybrid ab initio QM/MM MD simulations were performed to enhance the conformational sampling, together with hybrid ab initio QM/MM geometry optimization being employed to reach the potential energy surface. This scheme enabled us to conduct more effective explorations of both conformations and electronic structures than previous schemes that employ only geometry optimizations [63]. We assumed some possible reaction pathways and performed hybrid ab initio QM/MM MD simu-

lation for each pathway, which provided the estimation of the energy barrier



referred to as aminoacylation.

for biosynthesis of proteins.

into three subclasses a, b, and c [51, 52].

(Leu), isoleucine (Ile), and valine (Val)).

remained to be elucidated for the last some decades.

3.3 Ab initio QM/MM MD simulation of editing reaction

a misaminoacylated tRNALeu (i.e., valyl-tRNALeu), where the 3<sup>0</sup>

3.2 Editing reaction of aaRSs

respectively).

60

shown (stereoview). The crystal structure of the complex (1IVS) is colored yellow (for amino acid and RNA backbones), green (for amino acid side chains), and orange (for nucleic acids). The crystal structure of the isolated CP1 domain (1WK9) is colored light blue (for amino acid backbone) and magenta (for amino acid side chains). (B, C) Schematic representations of fundamental reaction schemes of hybrid ribozyme/protein catalyst (left) and their variant systems. The black circles (broken line) show the catalytic site, and the macromolecules involving the catalysts are colored red. The mechanisms of the editing reactions in both Leu and Val systems are revealed to be common by our recent studies: Interestingly, the editing reaction is ribozymal together with assists of protein moiety (left panels in (B-C)), which is referred to as hybrid ribozyme/protein catalysis [63]. The ribozymal factor (i.e., 3<sup>0</sup> -OH of A76 of tRNA) activates the nucleophilic water molecule (as represented by the red arrow), and the protein (LeuRS and ValRS) moiety promotes the catalysis (the blue arrow) [63]. Nevertheless, in the "defective" mutated systems (i.e., replacements of the aforementioned 3<sup>0</sup> -OH with 3<sup>0</sup> -H), reductions of the editing activities were experimentally revealed to be distinct. In the Leu system (B), the editing activity significantly decreased by the mutation, whereas in the Val and Ile systems, those were preserved. Our modeling studies elucidated that these were due to the absence and presence of compensation mechanisms in the Leu and Val/Ile protein moieties, respectively. Actually, in our previous study, we constructed the atomistic structural model of the Val system (i.e., the ValRS�threonyl-tRNAVal (misaminoacylated) complex) and showed that for the Val system with a "defective" ribozymal activator of tRNAVal, the protein (ValRS) moiety could activate the nucleophilic water molecule (the red arrow), which is referred to as the protein enzyme (note that the definition of protein enzyme described here is different from that employed by Cech [64]). As discussed in the text, this transition from the hybrid ribozyme/protein catalyst toward the protein enzyme may fill a gap found in the evolutionary transition from the RNA world to the current RNP world, which could possibly occur in primordial biological macromolecular systems. © Sakabe et al., [25].

depending on the reaction pathway. Thus, we determined the optimal reaction pathway and thereby elucidated the mechanism of the editing reaction (Figure 4).

As a result of the analysis, we discovered a novel catalytic mechanism, as follows: the editing reaction was revealed to be driven by the O3<sup>0</sup> of the ribose moiety of the 3<sup>0</sup> -end nucleotide A76, which acts as the general base to activate the nucleophilic water. Surprisingly, the editing of the LeuRS�valyl-tRNALeu complex was revealed to be ribozymal [63, 65, 66].

Furthermore, we found that this ribozyme reaction was enhanced by protein, through the formation of a hydrogen bond with the catalytic core of tRNALeu. Since the catalytic cores of the conventional protein-dependent ribozymes such as ribosome and group I intron [67] are purely composed of RNAs [64], this finding, that is, direct contributions of the protein moiety on the ribozymal reaction, is novel, and thus, we referred to the LeuRS�valyl-tRNALeu complex as a "hybrid ribozyme/ protein catalyst" (Figure 4).

replaced with the 3<sup>0</sup>

LeuRS is replaced with Ala).

© Sakabe et al., [25].

a Δ(3<sup>0</sup>

Δ(2<sup>0</sup>

Table 1.

nism, in which the 2<sup>0</sup>

ophilic water).

with 3<sup>0</sup>

63

ValRS can replace the functional role of 3<sup>0</sup>

contains "unreactive" ribozymal functional group (i.e., 3<sup>0</sup>

Class Species Attached

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

Ia E. coli O2<sup>0</sup> 3<sup>0</sup>

ThrRS IIa E. coli O3<sup>0</sup> Δ(His73)c,d 104

thermophirus

DOI: http://dx.doi.org/10.5772/intechopen.83545

LeuRS Ia E. coli O2<sup>0</sup> Δ(3<sup>0</sup>

ValRS Ia E. coli O2<sup>0</sup> Δ(3<sup>0</sup>

IleRS Ia E. coli O3<sup>0</sup> Δ(2<sup>0</sup>

ProRS IIa E. coli O3<sup>0</sup> Δ(2<sup>0</sup>

PheRS IIc E. coli O2<sup>0</sup> Δ(3<sup>0</sup>

Escherichia coli ThrRS acts as a protein enzyme (see Figure 4C).



abyssi

LeuRS Ia T.

ThrRS IIa Pyrococcus



E. coli ThrRS acts as a protein enzyme. <sup>d</sup>

LeuRS (D342A) site

O2<sup>0</sup> 3<sup>0</sup>

O3<sup>0</sup> 2<sup>0</sup>



binding and the activation of the nucleophilic water [71].

intermediated by hybrid ribozyme/protein catalysts.


Activator Reduction




D276

E327



rate





10-fold [69]

5-fold [69, 70]



Reference




the editing activity of the variant Val system is maintained by an amino acid residue of ValRS acting as the possible general base (Figure 4). More specifically, Asp276 of

Summary of the nucleophile activators and the reduction rate of hybrid ribozyme/protein catalysts (i.e., LeuRS, ValRS, IleRS, PheRS, ProRS, Pyrococcus abyssi ThrRS, and ribosome) and a variant aaRS (D342 of E. coli

In this manner, the hybrid ribozyme/protein catalysts are operated in both Leu and Val systems. Moreover, this suggested that in the "defective" Val system that

We further suggested that the ribozymal mechanism that we discovered is common in the editing reaction of various aaRS systems beyond the classes (Table 1) [63]. In fact,Thermus thermophilus IleRS (class Ia) [77], Pyrococcus abyssi ThrRS (class IIa) [72, 74, 75, 78], and Enterococcus faecalis ProRS (class IIa) [71] showed the similar binding mode of the nucleophilic water in the catalytic site. Furthermore, Kumar et al. also suggested that the editing reaction of the complex of prolyl-tRNA synthetase (ProRS) and alanyl-tRNAPro exhibited a similar mecha-

These data are summarized in Figure 4 and Table 1. In this section, we further discuss the dynamical aspects of the electronic structures in the editing reaction of Leu and Val systems, investigated by hybrid ab initio QM/MM MD simulations of the LeuRS�valyl-tRNALeu and ValRS�threonyl-tRNAVal complexes, respectively.


A very recent experimental study conducted by Dulic et al. experimentally showed that the defective mutation of the O3<sup>0</sup> atom (i.e., 3<sup>0</sup> -OH of A76 was replaced with 3<sup>0</sup> -H) significantly reduced the activity of the Leu system (�10<sup>4</sup> -fold rate reduction) (Table 1) [68], and thus, the hybrid ribozyme/protein catalyst mechanism has been reasserted by the biochemical experiments.

Can we generalize this novel, hybrid ribozyme/protein catalytic reaction mechanism of the LeuRS�valyl-tRNALeu complex? Considering structural similarity of class Ia aaRSs, which involves LeuRS, valyl-tRNA synthetase (ValRS), and isoleucyl-tRNA synthetase (IleRS), they may share a common editing mechanism with LeuRS. However, an experimental conflict has still been left to be resolved as follows. While the aforementioned modification of the O3<sup>0</sup> reduced the editing reaction in the Leu system [63, 68], the identical modification was not severe in the reduction of the editing activity with respect to the Val and Ile systems (Table 1) [69].

To resolve this discrepancy of the experiments, we constructed a structural model of the complex of ValRS and misaminoacylated tRNA [63] and thereby suggested that the hybrid ribozyme/protein catalyst mechanism was also shared in the editing reaction of the Val system [25]. Furthermore, to explain how the variant Val system can maintain its catalytic activity (Table 1), we constructed a structural model involving the variant tRNAVal in which the 3<sup>0</sup> -OH (reactive) of A76 was


Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

© Sakabe et al., [25].

a Δ(3<sup>0</sup> -OH) represents the replacement of 3<sup>0</sup> -OH group of A76 with 3<sup>0</sup> -H atom. <sup>b</sup>

Δ(2<sup>0</sup> -OH) represents the replacement of 2<sup>0</sup> -OH group of A76 with 2<sup>0</sup> -H atom. <sup>c</sup>

E. coli ThrRS acts as a protein enzyme. <sup>d</sup>

Escherichia coli ThrRS acts as a protein enzyme (see Figure 4C).

#### Table 1.

depending on the reaction pathway. Thus, we determined the optimal reaction pathway and thereby elucidated the mechanism of the editing reaction (Figure 4).

shown (stereoview). The crystal structure of the complex (1IVS) is colored yellow (for amino acid and RNA backbones), green (for amino acid side chains), and orange (for nucleic acids). The crystal structure of the isolated CP1 domain (1WK9) is colored light blue (for amino acid backbone) and magenta (for amino acid side chains). (B, C) Schematic representations of fundamental reaction schemes of hybrid ribozyme/protein catalyst (left) and their variant systems. The black circles (broken line) show the catalytic site, and the macromolecules involving the catalysts are colored red. The mechanisms of the editing reactions in both Leu and Val systems are revealed to be common by our recent studies: Interestingly, the editing reaction is ribozymal together with assists of protein moiety (left panels in (B-C)), which is referred to as hybrid ribozyme/protein

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

represented by the red arrow), and the protein (LeuRS and ValRS) moiety promotes the catalysis (the blue arrow) [63]. Nevertheless, in the "defective" mutated systems (i.e., replacements of the aforementioned 3<sup>0</sup>

(misaminoacylated) complex) and showed that for the Val system with a "defective" ribozymal activator of tRNAVal, the protein (ValRS) moiety could activate the nucleophilic water molecule (the red arrow), which is referred to as the protein enzyme (note that the definition of protein enzyme described here is different from that employed by Cech [64]). As discussed in the text, this transition from the hybrid ribozyme/protein catalyst toward the protein enzyme may fill a gap found in the evolutionary transition from the RNA world to the current RNP world, which could possibly occur in primordial biological macromolecular systems. © Sakabe



As a result of the analysis, we discovered a novel catalytic mechanism, as follows: the editing reaction was revealed to be driven by the O3<sup>0</sup> of the ribose moiety

Furthermore, we found that this ribozyme reaction was enhanced by protein, through the formation of a hydrogen bond with the catalytic core of tRNALeu. Since the catalytic cores of the conventional protein-dependent ribozymes such as ribosome and group I intron [67] are purely composed of RNAs [64], this finding, that is, direct contributions of the protein moiety on the ribozymal reaction, is novel, and thus, we referred to the LeuRS�valyl-tRNALeu complex as a "hybrid ribozyme/

A very recent experimental study conducted by Dulic et al. experimentally

reduction) (Table 1) [68], and thus, the hybrid ribozyme/protein catalyst mecha-

To resolve this discrepancy of the experiments, we constructed a structural model of the complex of ValRS and misaminoacylated tRNA [63] and thereby suggested that the hybrid ribozyme/protein catalyst mechanism was also shared in the editing reaction of the Val system [25]. Furthermore, to explain how the variant Val system can maintain its catalytic activity (Table 1), we constructed a structural

Can we generalize this novel, hybrid ribozyme/protein catalytic reaction mechanism of the LeuRS�valyl-tRNALeu complex? Considering structural similarity of class Ia aaRSs, which involves LeuRS, valyl-tRNA synthetase (ValRS), and isoleucyl-tRNA synthetase (IleRS), they may share a common editing mechanism with LeuRS. However, an experimental conflict has still been left to be resolved as follows. While the aforementioned modification of the O3<sup>0</sup> reduced the editing reaction in the Leu system [63, 68], the identical modification was not severe in the reduction of the editing activity with respect to the Val and Ile systems






showed that the defective mutation of the O3<sup>0</sup> atom (i.e., 3<sup>0</sup>

nism has been reasserted by the biochemical experiments.

model involving the variant tRNAVal in which the 3<sup>0</sup>

philic water. Surprisingly, the editing of the LeuRS�valyl-tRNALeu complex was


of the 3<sup>0</sup>

et al., [25].

with 3<sup>0</sup>

with 3<sup>0</sup>

(Table 1) [69].

62

revealed to be ribozymal [63, 65, 66].

catalysis [63]. The ribozymal factor (i.e., 3<sup>0</sup>

protein catalyst" (Figure 4).

Summary of the nucleophile activators and the reduction rate of hybrid ribozyme/protein catalysts (i.e., LeuRS, ValRS, IleRS, PheRS, ProRS, Pyrococcus abyssi ThrRS, and ribosome) and a variant aaRS (D342 of E. coli LeuRS is replaced with Ala).

replaced with the 3<sup>0</sup> -H (unreactive). Based on this analysis [25], we suggested that the editing activity of the variant Val system is maintained by an amino acid residue of ValRS acting as the possible general base (Figure 4). More specifically, Asp276 of ValRS can replace the functional role of 3<sup>0</sup> -OH of A76 (i.e., activation of the nucleophilic water).

In this manner, the hybrid ribozyme/protein catalysts are operated in both Leu and Val systems. Moreover, this suggested that in the "defective" Val system that contains "unreactive" ribozymal functional group (i.e., 3<sup>0</sup> -OH of A76 is replaced with 3<sup>0</sup> -H), the function of a hybrid ribozyme/protein catalyst can be transferred to a protein enzyme. This could also be related to the evolutional transition from RNA enzymes (the RNA world) to protein enzymes (assisted by RNA) (the RNP world), intermediated by hybrid ribozyme/protein catalysts.

We further suggested that the ribozymal mechanism that we discovered is common in the editing reaction of various aaRS systems beyond the classes (Table 1) [63]. In fact,Thermus thermophilus IleRS (class Ia) [77], Pyrococcus abyssi ThrRS (class IIa) [72, 74, 75, 78], and Enterococcus faecalis ProRS (class IIa) [71] showed the similar binding mode of the nucleophilic water in the catalytic site. Furthermore, Kumar et al. also suggested that the editing reaction of the complex of prolyl-tRNA synthetase (ProRS) and alanyl-tRNAPro exhibited a similar mechanism, in which the 2<sup>0</sup> -OH group of A76 of tRNAPro was involved in the substrate binding and the activation of the nucleophilic water [71].

These data are summarized in Figure 4 and Table 1. In this section, we further discuss the dynamical aspects of the electronic structures in the editing reaction of Leu and Val systems, investigated by hybrid ab initio QM/MM MD simulations of the LeuRS�valyl-tRNALeu and ValRS�threonyl-tRNAVal complexes, respectively.

#### 3.4 Dynamic rearrangement of MOs in the editing reactions

For both Leu and Val systems, we suggested that the editing reactions occur in a similar manner [25, 63]. Actually, in both systems, the reactions were shown to be initiated by opening of the "H-gate": The H-gate is defined by a dihedral angle, C4<sup>0</sup> –C3<sup>0</sup> –O3<sup>0</sup> –HO3<sup>0</sup> of A76, and its opening represents the rotation of the dihedral angle by �100°, which thus leads to the nucleophilic attack of the water molecule.

In this manner, as the reaction proceeds, the energy level of the reactive MO seems to go up (Figure 5). Thus, if the nucleophilic attack would be achieved, the energy level of the reactive MO in the nucleophilic water could be raised up to that of the HOMO, which would thus result in hybridization of the HOMO and LUMO. In fact, the similar dynamical rearrangements in the electronic structure were also observed in our hybrid ab initio QM/MM calculations of the editing reaction occur-

For the Leu system, we further investigated the overall mechanism of the editing

However, when H-gate was open (state 3), the nucleophilic water approached the C atom, and the energy level of an MO that most contained the 2p orbital of Ow was elevated to HOMO�9 from the HOMO�14 observed in state 1. This elevation decreased the energy difference between the LUMO, which contained the reactive moiety (i.e., atomic orbitals of the carbonyl group of the substrate and the O2<sup>0</sup> atom

Schematic representation of the reaction states of editing mechanism of LeuRS system is shown in (A). Energy diagram of the editing (B) and molecular orbitals (C) of states 1, 4, 5, 6, and 7 are shown. © Kang et al., [66].

reaction employing the hybrid ab initio QM/MM MD simulations. Based on the orbital analysis of the trajectory of the hybrid MD simulations, we found more detailed dynamical properties of the electronic structures (Figure 6): In the initial stage of the editing reaction, the HOMO did not contain the 2p orbital of the Ow atom of the nucleophilic water, even though it attacked the C atom, which seemed

ring in the LeuRS�valyl-tRNALeu complex [65, 66].

DOI: http://dx.doi.org/10.5772/intechopen.83545

Figure 6.

65

to be inconsistent with the frontier orbital theory (FOT) [79].

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

For the Val system, employing the hybrid ab initio QM/MM calculations, we evaluated the electronic structures for the two distinct H-gate states, that is, the opened and closed states [25, 63]. The resultant data showed that the LUMO was located in the C atom in both closed and opened H-gate states (Figure 5). By contrast, the energy levels of the MOs that include the 2p orbital of oxygen atom of the nucleophilic water (Ow) (i.e., the "reactive" MO) were different depending on the two distinct states of the H-gate: In the opened H-gate state, we observed the MO as HOMO�6, while in the closed H-gate state, the MO was observed as HOMO�11, for which the energy level was much lower compared with that of the former state.

#### Figure 5.

Schematic picture of the H-gate closed and H-gate open states (A). Stereoview of LUMO and molecular orbitals (MOs) of the catalytic site including the nucleophilic water molecule for H-gate closed and H-gate opened conformations. HOMO-11 for H-gate closed and HOMO-6 for H-gate opened conformations. The ribose and threonine moieties of the substrate, Val215, Asp276, Asp279, and the three water molecules are included as the QM region. (B) Energy diagrams concerning the LUMOs and HOMO-11 (HOMO-6) in H-gate closed state (H-gate opened state). © Sakabe et al. [25].

Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

In this manner, as the reaction proceeds, the energy level of the reactive MO seems to go up (Figure 5). Thus, if the nucleophilic attack would be achieved, the energy level of the reactive MO in the nucleophilic water could be raised up to that of the HOMO, which would thus result in hybridization of the HOMO and LUMO. In fact, the similar dynamical rearrangements in the electronic structure were also observed in our hybrid ab initio QM/MM calculations of the editing reaction occurring in the LeuRS�valyl-tRNALeu complex [65, 66].

For the Leu system, we further investigated the overall mechanism of the editing reaction employing the hybrid ab initio QM/MM MD simulations. Based on the orbital analysis of the trajectory of the hybrid MD simulations, we found more detailed dynamical properties of the electronic structures (Figure 6): In the initial stage of the editing reaction, the HOMO did not contain the 2p orbital of the Ow atom of the nucleophilic water, even though it attacked the C atom, which seemed to be inconsistent with the frontier orbital theory (FOT) [79].

However, when H-gate was open (state 3), the nucleophilic water approached the C atom, and the energy level of an MO that most contained the 2p orbital of Ow was elevated to HOMO�9 from the HOMO�14 observed in state 1. This elevation decreased the energy difference between the LUMO, which contained the reactive moiety (i.e., atomic orbitals of the carbonyl group of the substrate and the O2<sup>0</sup> atom

Figure 6.

Schematic representation of the reaction states of editing mechanism of LeuRS system is shown in (A). Energy diagram of the editing (B) and molecular orbitals (C) of states 1, 4, 5, 6, and 7 are shown. © Kang et al., [66].

3.4 Dynamic rearrangement of MOs in the editing reactions

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

C4<sup>0</sup> –C3<sup>0</sup> –O3<sup>0</sup>

former state.

Figure 5.

64

(H-gate opened state). © Sakabe et al. [25].

For both Leu and Val systems, we suggested that the editing reactions occur in a similar manner [25, 63]. Actually, in both systems, the reactions were shown to be initiated by opening of the "H-gate": The H-gate is defined by a dihedral angle,

angle by �100°, which thus leads to the nucleophilic attack of the water molecule. For the Val system, employing the hybrid ab initio QM/MM calculations, we evaluated the electronic structures for the two distinct H-gate states, that is, the opened and closed states [25, 63]. The resultant data showed that the LUMO was located in the C atom in both closed and opened H-gate states (Figure 5). By contrast, the energy levels of the MOs that include the 2p orbital of oxygen atom of the nucleophilic water (Ow) (i.e., the "reactive" MO) were different depending on the two distinct states of the H-gate: In the opened H-gate state, we observed the MO as HOMO�6, while in the closed H-gate state, the MO was observed as HOMO�11, for which the energy level was much lower compared with that of the

Schematic picture of the H-gate closed and H-gate open states (A). Stereoview of LUMO and molecular orbitals (MOs) of the catalytic site including the nucleophilic water molecule for H-gate closed and H-gate opened conformations. HOMO-11 for H-gate closed and HOMO-6 for H-gate opened conformations. The ribose and threonine moieties of the substrate, Val215, Asp276, Asp279, and the three water molecules are included as the QM region. (B) Energy diagrams concerning the LUMOs and HOMO-11 (HOMO-6) in H-gate closed state

–HO3<sup>0</sup> of A76, and its opening represents the rotation of the dihedral

of A76), and the MO involving the 2p orbital of Ow, from 7.0 eV to 6.2 eV. In state 4, this energy gap further decreased to 5.4 eV, and the MO involving 2p orbital of Ow became the HOMO, while the LUMO remained to be localized on the C atom.

this reaction stage is followed by the subsequent phase of the covalent bond formation and cleavage. The obtained picture could be found in functional mechanisms of other various biological macromolecular systems, and thus, the generality of the presented novel picture is further amenable to future theoretical and experimental works. Thereby, this picture could be considered to be a characteristic feature in

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

This work was partly supported by Grants-in-Aid from the Ministry of Educa-

tion, Culture, Sports, Science and Technology (MEXT), under contract Nos. 21340108 and 25287099. Computations were performed using computer facilities of the Computer Center for Agriculture, Forestry, and Fisheries Research, MAFF, Japan, and the Supercomputer Center, Institute for Solid State Physics, The University of Tokyo. JK was supported by the Korea Research Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry

biological macromolecular systems.

DOI: http://dx.doi.org/10.5772/intechopen.83545

of Science and ICT (NRF-2017H1D3A1A01053094).

The authors declare to have no conflicting interests.

, Takuya Sumi<sup>2</sup> and Masaru Tateno<sup>2</sup>

Yonsei University, Seoul, Republic of Korea

provided the original work is properly cited.

\*Address all correspondence to: tateno1611@gmail.com

1 Center for Systems and Translational Brain Sciences, Institute of Human

Complexity and Systems Science, System Science Center for Brain and Cognition,

2 Graduate School of Life Science, University of Hyogo, Kamigori-cho, Hyogo,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*

Acknowledgements

Conflict of interest

Author details

Jiyoung Kang<sup>1</sup>

Japan

67

In this stage, both HOMO and LUMO were the reactive MOs responsible for the catalytic reaction, and thus, the electronic structure was fully consistent with the FOT. In fact, the OwC covalent bond was formed by the interaction between the 2p orbital of Ow (HOMO) and the 2p orbital of C (LUMO). Here, we referred to such dynamical rearrangements of the electronic structure as the dynamical induction of the reactive MOs for the HOMO and LUMO, that is, DIRH and DIRL, respectively [65, 66].

In our preliminary studies of other biological macromolecular systems, we have also observed the DIRH and DIRL mechanisms (unpublished data). Future theoretical and experimental analyses are amenable to examine the generality of this picture on dynamical rearrangements of electronic structures occurring in the reaction cycles of biological macromolecular systems. In addition, aaRSs are closely relevant to the molecular evolution and the origin of life, which should thus be considered from ab initio QM calculations. So, the present achievements are also related to those evolutional issues, although we do not describe them herein due to space limitation (for more details, see the literature [25]).

#### 4. Conclusions

In this chapter, we introduced investigations employing ab initio QM calculations and hybrid ab initio QM/MM MD calculations. For the latter, a catalytic reaction site is considered at ab initio QM level, and the other parts, such as the remainder in protein structures and solvent water molecules, are considered at MM (i.e., classical) level, and thus, we can consider the entire system with reasonable computational costs, to evaluate the electronic structure of the catalytic active site. In both analyses of biological macromolecular systems, we revealed the significant reconstitutions of the electronic structures in the reaction cycles.

In the first example, we demonstrated the detailed electronic structures of a crucial functional site, the proximal cluster, in the MBH, which contains multiple transition metals as [4Fe-3S] and is closely related to the ET. We analyzed the effects of the OH ion that was experimentally identified in the proximal cluster, to the ET mechanisms. Thereby, we revealed that the OH ion created the ET pathways by inducing the delocalization of the LUMO of the proximal cluster.

This means that tiny molecular species (e.g., OH) can induce dramatic rearrangements of the electronic structure in the biological macromolecular systems, which thus generates the ET pathways. This is the first work to point out the mechanisms to create the ET pathways in biological macromolecular systems. In this manner, organisms regulate the biological functions employing such a subtle factor but thereby dramatically change their physiological status.

In the second example, we investigated dynamical changes of the electronic structures in the catalytic reaction cycles of the LeuRSvalyl-tRNALeu and ValRS threonyl-tRNAVal complexes, employing our hybrid ab initio QM/MM MD calculation system, which is a state-of-the-art theoretical technique to elucidate the functional mechanisms of biological macromolecular systems. As a consequence, we revealed that the dynamical geometrical changes induced the dramatic rearrangemets of the electronic structures.

Thereby, the reactive MOs, which are positioned energetically far from the Fermi levels in the initial stages of the reaction cycles, are dynamically rearranged, but those MOs become the HOMO and LUMO, as the reaction cycles proceed. Thus, Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

this reaction stage is followed by the subsequent phase of the covalent bond formation and cleavage. The obtained picture could be found in functional mechanisms of other various biological macromolecular systems, and thus, the generality of the presented novel picture is further amenable to future theoretical and experimental works. Thereby, this picture could be considered to be a characteristic feature in biological macromolecular systems.

#### Acknowledgements

of A76), and the MO involving the 2p orbital of Ow, from 7.0 eV to 6.2 eV. In state 4, this energy gap further decreased to 5.4 eV, and the MO involving 2p orbital of Ow became the HOMO, while the LUMO remained to be localized on the C atom.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

respectively [65, 66].

4. Conclusions

In this stage, both HOMO and LUMO were the reactive MOs responsible for the catalytic reaction, and thus, the electronic structure was fully consistent with the FOT. In fact, the OwC covalent bond was formed by the interaction between the 2p orbital of Ow (HOMO) and the 2p orbital of C (LUMO). Here, we referred to such dynamical rearrangements of the electronic structure as the dynamical induction of the reactive MOs for the HOMO and LUMO, that is, DIRH and DIRL,

In our preliminary studies of other biological macromolecular systems, we have also observed the DIRH and DIRL mechanisms (unpublished data). Future theoretical and experimental analyses are amenable to examine the generality of this picture on dynamical rearrangements of electronic structures occurring in the reaction cycles of biological macromolecular systems. In addition, aaRSs are closely relevant to the molecular evolution and the origin of life, which should thus be considered from ab initio QM calculations. So, the present achievements are also related to those evolutional issues, although we do not describe them herein due to

In this chapter, we introduced investigations employing ab initio QM calculations and hybrid ab initio QM/MM MD calculations. For the latter, a catalytic reaction site is considered at ab initio QM level, and the other parts, such as the remainder in protein structures and solvent water molecules, are considered at MM (i.e., classical) level, and thus, we can consider the entire system with reasonable computational costs, to evaluate the electronic structure of the catalytic active site. In both analyses of biological macromolecular systems, we revealed the significant

In the first example, we demonstrated the detailed electronic structures of a crucial functional site, the proximal cluster, in the MBH, which contains multiple transition metals as [4Fe-3S] and is closely related to the ET. We analyzed the effects of the OH ion that was experimentally identified in the proximal cluster, to the ET mechanisms. Thereby, we revealed that the OH ion created the ET path-

In the second example, we investigated dynamical changes of the electronic structures in the catalytic reaction cycles of the LeuRSvalyl-tRNALeu and ValRS threonyl-tRNAVal complexes, employing our hybrid ab initio QM/MM MD calculation system, which is a state-of-the-art theoretical technique to elucidate the functional mechanisms of biological macromolecular systems. As a consequence, we revealed that the dynamical geometrical changes induced the dramatic

Thereby, the reactive MOs, which are positioned energetically far from the Fermi levels in the initial stages of the reaction cycles, are dynamically rearranged, but those MOs become the HOMO and LUMO, as the reaction cycles proceed. Thus,

ways by inducing the delocalization of the LUMO of the proximal cluster. This means that tiny molecular species (e.g., OH) can induce dramatic rearrangements of the electronic structure in the biological macromolecular systems, which thus generates the ET pathways. This is the first work to point out the mechanisms to create the ET pathways in biological macromolecular systems. In this manner, organisms regulate the biological functions employing such a subtle

space limitation (for more details, see the literature [25]).

reconstitutions of the electronic structures in the reaction cycles.

factor but thereby dramatically change their physiological status.

rearrangemets of the electronic structures.

66

This work was partly supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), under contract Nos. 21340108 and 25287099. Computations were performed using computer facilities of the Computer Center for Agriculture, Forestry, and Fisheries Research, MAFF, Japan, and the Supercomputer Center, Institute for Solid State Physics, The University of Tokyo. JK was supported by the Korea Research Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2017H1D3A1A01053094).

#### Conflict of interest

The authors declare to have no conflicting interests.

#### Author details

Jiyoung Kang<sup>1</sup> , Takuya Sumi<sup>2</sup> and Masaru Tateno<sup>2</sup> \*

1 Center for Systems and Translational Brain Sciences, Institute of Human Complexity and Systems Science, System Science Center for Brain and Cognition, Yonsei University, Seoul, Republic of Korea

2 Graduate School of Life Science, University of Hyogo, Kamigori-cho, Hyogo, Japan

\*Address all correspondence to: tateno1611@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[1] Becke AD. Density-functional thermochemistry .3. The role of exact exchange. The Journal of Chemical Physics. 1993;98(7):5648-5652

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Journal of Computational Chemistry.

[10] Field MJ, Bash PA, Karplus M. A combined quantum-mechanical and molecular mechanical potential for molecular-dynamics simulations. Journal of Computational Chemistry.

[11] Lin H, Truhlar DG. QM/MM: What have we learned, where are we, and where do we go from here? Theoretical Chemistry Accounts. 2007;117(2):

[12] Mulholland AJ. Modelling enzyme reaction mechanisms, specificity and catalysis. Drug Discovery Today. 2005;

[13] Ryde U. Combined quantum and molecular mechanics calculations on metalloproteins. Current Opinion in Chemical Biology. 2003;7(1):136-142

[14] Senn HM, Thiel W. QM/MM studies

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[17] Kang J, Hagiwara Y, Tateno M. Biological applications of hybrid quantum mechanics/molecular mechanics calculation. Journal of Biomedicine & Biotechnology. 2012;

[18] Cavasotto CN, Adler NS, Aucar MG. Quantum chemical approaches in

of enzymes. Current Opinion in Chemical Biology. 2007;11(2):182-187

[15] Senn HM, Thiel W. QM/MM methods for biomolecular systems. Angewandte Chemie, International Edition. 2009;48(7):1198-1229

2002;23(1):48-58

1990;11(6):700-733

10(20):1393-1402

3188-3209

2012:11

185-199

[2] Perdew JP, Ernzerhof M, Burke K. Rationale for mixing exact exchange with density functional approximations. The Journal of Chemical Physics. 1996;

[3] Becke AD. Density-functional exchange-energy approximation with correct asymptotic-behavior. Physical Review A. 1988;38(6):3098-3100

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[35] Chengteh Lee WY, Parr RG. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Physical Review B. 1988;37: 785

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[37] Schäfer A, Huber C, Ahlrichs R. Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr. The Journal of Chemical Physics. 1994;100(8): 5829-5835

[38] McLean AD, Chandler GS. Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z=11–18. The Journal of Chemical Physics. 1980;72(10): 5639-5648

[39] Beinert H, Holm RH, Munck E. Iron-sulfur clusters: Nature's modular, multipurpose structures. Science. 1997; 277(5326):653-659

[40] Pelmenschikov V, Kaupp M. Redoxdependent structural transformations of the [4Fe-3S] proximal cluster in O2 tolerant membrane-bound [NiFe] hydrogenase: A DFT study. Journal of the American Chemical Society. 2013; 135(32):11809-11823

[41] Shafaat HS, Rudiger O, Ogata H, Lubitz W. [NiFe] hydrogenases: A common active site for hydrogen metabolism under diverse conditions. Biochimica et Biophysica Acta. 2013; 1827(8–9):986-1002

coli seryl-tRNA synthetase at 2.5 Å. Nature. 1990;347(6290):249-255

DOI: http://dx.doi.org/10.5772/intechopen.83545

Recent Progresses in Ab Initio Electronic Structure Calculation toward…

within Escherichia coli leucyl-tRNA synthetase prevents hydrolytic editing of leucyl-tRNALeu. Biochemistry. 2001;

[58] Nureki O, Vassylyev DG, Tateno M, Shimada A, Nakama T, Fukai S, et al. Enzyme structure with two catalytic sites for double-sieve selection of substrate. Science. 1998;280(5363):

[59] Silvian LF, Wang J, Steitz TA. Insights into editing from an ile-tRNA synthetase structure with tRNAile and mupirocin. Science. 1999;285(5430):

[60] Zhai Y, Martinis SA. Two conserved threonines collaborate in the Escherichia coli leucyl-tRNA synthetase amino acid editing mechanism. Biochemistry. 2005;

[61] Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, et al. General Atomic and Molecular Electronic-Structure System. Journal of Computational Chemistry. 1993;14(11):

[62] Case DA, Cheatham TE, Darden T, Gohlke H, Luo R, Merz KM, et al. The Amber biomolecular simulation programs. Journal of Computational Chemistry. 2005;26(16):1668-1688

[63] Hagiwara Y, Field MJ, Nureki O, Tateno M. Editing mechanism of aminoacyl-trna synthetases operates by a hybrid ribozyme/protein catalyst. Journal of the American Chemical Society. 2010;132(8):2751-2758

[64] Cech TR. Crawling out of the RNA world. Cell. 2009;136(4):599-602

[65] Kang J, Hagiwara Y, Tateno M. Biological applications of hybrid quantum mechanics/molecular mechanics calculation. Journal of Biomedicine & Biotechnology. 2012;

40(18):5376-5381

578-582

1074-1077

1347-1363

2012:11

44(47):15437-15443

[50] Eriani G, Delarue M, Poch O, Gangloff J, Moras D. Partition of tRNA synthetases into two classes based on mutually exclusive sets of sequence motifs. Nature. 1990;347(6289):203-206

[51] Delarue M, Moras D. The aminoacyl-tRNA synthetase family: Modules at work. BioEssays. 1993;

[52] Ibba M, Morgan S, Curnow AW, Pridmore DR, Vothknecht UC, Gardner W, et al. A euryarchaeal lysyl-tRNA synthetase: Resemblance to class I synthetases. Science. 1997;278(5340):

[53] Fukai S, Nureki O, Sekine S, Shimada A, Tao J, Vassylyev DG, et al. Structural basis for double-sieve discrimination of L-valine from Lisoleucine and L-threonine by the complex of tRNAVal and valyl-tRNA synthetase. Cell. 2000;103(5):793-803

[54] Fukunaga R, Fukai S, Ishitani R, Nureki O, Yokoyama S. Crystal structures of the CP1 domain from Thermus thermophilus isoleucyl-tRNA synthetase and its complex with Lvaline. The Journal of Biological Chemistry. 2004;279(9):8396-8402

[55] Fukunaga R, Yokoyama S. Structural basis for non-cognate amino acid discrimination by the valyl-tRNA synthetase editing domain. The Journal of Biological Chemistry. 2005;280(33):

[56] Lincecum TL Jr, Tukalo M, Yaremchuk A, Mursinna RS, Williams AM, Sproat BS, et al. Structural and mechanistic basis of pre- and posttransfer editing by leucyl-tRNA synthetase. Molecular Cell. 2003;11(4):951-963

[57] Mursinna RS, Lincecum TL Jr, Martinis SA. A conserved threonine

15(10):675-687

1119-1122

29937-29945

71

[42] Volbeda A, Darnault C, Parkin A, Sargent F, Armstrong FA, Fontecilla-Camps JC. Crystal structure of the O2 tolerant membrane-bound hydrogenase 1 from Escherichia coli in complex with its cognate cytochrome b. Structure. 2013;21(1):184-190

[43] Kurkin S, George SJ, Thorneley RN, Albracht SP. Hydrogen-induced activation of the [NiFe]-hydrogenase from Allochromatium vinosum as studied by stopped-flow infrared spectroscopy. Biochemistry. 2004; 43(21):6820-6831

[44] Balabin IA, Hu X, Beratan DN. Exploring biological electron transfer pathway dynamics with the pathways plugin for VMD. Journal of Computational Chemistry. 2012;33(8): 906-910

[45] Beratan DN, Betts JN, Onuchic JN. Protein electron transfer rates set by the bridging secondary and tertiary structure. Science. 1991;252(5010): 1285-1288

[46] Hagiwara Y, Kino H, Tateno M. Modulation of electronic structures of bases through DNA recognition of protein. Journal of Physics. Condensed Matter. 2010;22(15):152101

[47] Hagiwara Y, Ohta T, Tateno M. QM/MM hybrid calculation of biological macromolecules using a new interface program connecting QM and MM engines. Journal of Physics. Condensed Matter. 2009;21(6):064234

[48] Berg JM, Tymoczko JL, Stryer L. Biochemistry. 5th ed. New York: W. H. Freeman; 2002. p. 1208

[49] Cusack S, Berthet-Colominas C, Hartlein M, Nassar N, Leberman R. A second class of synthetase structure revealed by X-ray analysis of Escherichia Recent Progresses in Ab Initio Electronic Structure Calculation toward… DOI: http://dx.doi.org/10.5772/intechopen.83545

coli seryl-tRNA synthetase at 2.5 Å. Nature. 1990;347(6290):249-255

[34] Volbeda A, Amara P, Darnault C, Mouesca JM, Parkin A, Roessler MM, et al. X-ray crystallographic and

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Biochimica et Biophysica Acta. 2013;

[42] Volbeda A, Darnault C, Parkin A, Sargent F, Armstrong FA, Fontecilla-Camps JC. Crystal structure of the O2 tolerant membrane-bound hydrogenase 1 from Escherichia coli in complex with its cognate cytochrome b. Structure.

[43] Kurkin S, George SJ, Thorneley RN,

Albracht SP. Hydrogen-induced activation of the [NiFe]-hydrogenase from Allochromatium vinosum as studied by stopped-flow infrared spectroscopy. Biochemistry. 2004;

[44] Balabin IA, Hu X, Beratan DN. Exploring biological electron transfer pathway dynamics with the pathways

Computational Chemistry. 2012;33(8):

[45] Beratan DN, Betts JN, Onuchic JN. Protein electron transfer rates set by the

bridging secondary and tertiary structure. Science. 1991;252(5010):

[46] Hagiwara Y, Kino H, Tateno M. Modulation of electronic structures of bases through DNA recognition of protein. Journal of Physics. Condensed

[47] Hagiwara Y, Ohta T, Tateno M. QM/MM hybrid calculation of biological macromolecules using a new interface program connecting QM and MM engines. Journal of Physics. Condensed

[48] Berg JM, Tymoczko JL, Stryer L. Biochemistry. 5th ed. New York: W. H.

[49] Cusack S, Berthet-Colominas C, Hartlein M, Nassar N, Leberman R. A second class of synthetase structure revealed by X-ray analysis of Escherichia

Matter. 2010;22(15):152101

Matter. 2009;21(6):064234

Freeman; 2002. p. 1208

plugin for VMD. Journal of

1827(8–9):986-1002

2013;21(1):184-190

43(21):6820-6831

906-910

1285-1288

computational studies of the O2-tolerant [NiFe]-hydrogenase 1 from Escherichia coli. Proceedings of the National Academy of Sciences of the United States of America. 2012;109(14):

[35] Chengteh Lee WY, Parr RG. Development of the Colle-Salvetti correlation-energy formula into a

density. Physical Review B. 1988;37:

[36] Schäfer A, Horn H, Ahlrichs R. Fully optimized contracted Gaussian basis sets for atoms Li to Kr. The Journal of Chemical Physics. 1992;97(4):2571

[37] Schäfer A, Huber C, Ahlrichs R. Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr. The Journal of Chemical Physics. 1994;100(8):

[38] McLean AD, Chandler GS. Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z=11–18. The Journal of Chemical Physics. 1980;72(10):

[39] Beinert H, Holm RH, Munck E. Iron-sulfur clusters: Nature's modular, multipurpose structures. Science. 1997;

[40] Pelmenschikov V, Kaupp M. Redoxdependent structural transformations of the [4Fe-3S] proximal cluster in O2 tolerant membrane-bound [NiFe] hydrogenase: A DFT study. Journal of the American Chemical Society. 2013;

[41] Shafaat HS, Rudiger O, Ogata H, Lubitz W. [NiFe] hydrogenases: A common active site for hydrogen metabolism under diverse conditions.

functional of the electron

5305-5310

785

5829-5835

5639-5648

277(5326):653-659

135(32):11809-11823

70

[50] Eriani G, Delarue M, Poch O, Gangloff J, Moras D. Partition of tRNA synthetases into two classes based on mutually exclusive sets of sequence motifs. Nature. 1990;347(6289):203-206

[51] Delarue M, Moras D. The aminoacyl-tRNA synthetase family: Modules at work. BioEssays. 1993; 15(10):675-687

[52] Ibba M, Morgan S, Curnow AW, Pridmore DR, Vothknecht UC, Gardner W, et al. A euryarchaeal lysyl-tRNA synthetase: Resemblance to class I synthetases. Science. 1997;278(5340): 1119-1122

[53] Fukai S, Nureki O, Sekine S, Shimada A, Tao J, Vassylyev DG, et al. Structural basis for double-sieve discrimination of L-valine from Lisoleucine and L-threonine by the complex of tRNAVal and valyl-tRNA synthetase. Cell. 2000;103(5):793-803

[54] Fukunaga R, Fukai S, Ishitani R, Nureki O, Yokoyama S. Crystal structures of the CP1 domain from Thermus thermophilus isoleucyl-tRNA synthetase and its complex with Lvaline. The Journal of Biological Chemistry. 2004;279(9):8396-8402

[55] Fukunaga R, Yokoyama S. Structural basis for non-cognate amino acid discrimination by the valyl-tRNA synthetase editing domain. The Journal of Biological Chemistry. 2005;280(33): 29937-29945

[56] Lincecum TL Jr, Tukalo M, Yaremchuk A, Mursinna RS, Williams AM, Sproat BS, et al. Structural and mechanistic basis of pre- and posttransfer editing by leucyl-tRNA synthetase. Molecular Cell. 2003;11(4):951-963

[57] Mursinna RS, Lincecum TL Jr, Martinis SA. A conserved threonine within Escherichia coli leucyl-tRNA synthetase prevents hydrolytic editing of leucyl-tRNALeu. Biochemistry. 2001; 40(18):5376-5381

[58] Nureki O, Vassylyev DG, Tateno M, Shimada A, Nakama T, Fukai S, et al. Enzyme structure with two catalytic sites for double-sieve selection of substrate. Science. 1998;280(5363): 578-582

[59] Silvian LF, Wang J, Steitz TA. Insights into editing from an ile-tRNA synthetase structure with tRNAile and mupirocin. Science. 1999;285(5430): 1074-1077

[60] Zhai Y, Martinis SA. Two conserved threonines collaborate in the Escherichia coli leucyl-tRNA synthetase amino acid editing mechanism. Biochemistry. 2005; 44(47):15437-15443

[61] Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, et al. General Atomic and Molecular Electronic-Structure System. Journal of Computational Chemistry. 1993;14(11): 1347-1363

[62] Case DA, Cheatham TE, Darden T, Gohlke H, Luo R, Merz KM, et al. The Amber biomolecular simulation programs. Journal of Computational Chemistry. 2005;26(16):1668-1688

[63] Hagiwara Y, Field MJ, Nureki O, Tateno M. Editing mechanism of aminoacyl-trna synthetases operates by a hybrid ribozyme/protein catalyst. Journal of the American Chemical Society. 2010;132(8):2751-2758

[64] Cech TR. Crawling out of the RNA world. Cell. 2009;136(4):599-602

[65] Kang J, Hagiwara Y, Tateno M. Biological applications of hybrid quantum mechanics/molecular mechanics calculation. Journal of Biomedicine & Biotechnology. 2012; 2012:11

[66] Kang J, Kino H, Field MJ, Tateno M. Electronic structure rearrangements in hybrid ribozyme/protein catalysis. Journal of the Physical Society of Japan. 2017;86(4):044801

[67] Weinger JS, Parnell KM, Dorner S, Green R, Strobel SA. Substrate-assisted catalysis of peptide bond formation by the ribosome. Nature Structural and Molecular Biology. 2004;11:1101

[68] Dulic M, Cvetesic N, Zivkovic I, Palencia A, Cusack S, Bertosa B, et al. Kinetic origin of substrate specificity in post-transfer editing by leucyl-tRNA synthetase. Journal of Molecular Biology. 2018;430(1):1-16

[69] Nordin BE, Schimmel P. Transiently misacylated tRNA is a primer for editing of misactivated adenylates by class I aminoacyl-tRNA synthetases. Biochemistry. 2003;42(44):12989-12997

[70] Nordin BE, Schimmel P. Plasticity of recognition of the 3<sup>0</sup> -end of mischarged tRNA by class I aminoacyltRNA Synthetases. The Journal of Biological Chemistry. 2002;277(23): 20510-20517

[71] Kumar S, Das M, Hadad CM, Musier-Forsyth K. Substrate and enzyme functional groups contribute to translational quality control by bacterial prolyl-tRNA synthetase. The Journal of Physical Chemistry B. 2012;116(23): 6991-6999

[72] Waas WF, Schimmel P. Evidence that tRNA synthetase-directed proton transfer stops mistranslation. Biochemistry. 2007;46(43): 12062-12070

[73] Aboelnga MM, Hayward JJ, Gauld JW. Enzymatic post-transfer editing mechanism of E. coli threonyl-tRNA synthetase (ThrRS): A molecular dynamics (MD) and quantum mechanics/molecular mechanics (QM/ MM) investigation. ACS Catalysis. 2017; 7(8):5180-5193

[74] Hussain T, Kruparani SP, Pal B, Dock-Bregeon AC, Dwivedi S, Shekar MR, et al. Post-transfer editing mechanism of a D-aminoacyl-tRNA deacylase-like domain in threonyl-tRNA synthetase from archaea. The EMBO Journal. 2006;25(17):4152-4162

[75] Aboelnga MM, Hayward JJ, Gauld JW. Unraveling the critical role played by Ado7620 OH in the post-transfer editing by archaeal threonyl-tRNA synthetase. The Journal of Physical Chemistry B. 2018;122(3):1092-1101

[76] Ling J, Roy H, Ibba M. Mechanism of tRNA-dependent editing in translational quality control. Proceedings of the National Academy of Sciences. 2007;104(1):72

Section 3

Structural Quantum

Developments

73

[77] Fukunaga R, Yokoyama S. Structural basis for substrate recognition by the editing domain of isoleucyl-tRNA synthetase. Journal of Molecular Biology. 2006;359(4):901-912

[78] Aboelnga MM, Hayward JJ, Gauld JW. A water-mediated and substrateassisted aminoacylation mechanism in the discriminating aminoacyl-tRNA synthetase GlnRS and nondiscriminating GluRS. Physical Chemistry Chemical Physics. 2017; 19(37):25598-25609

[79] Fukui K. Recognition of Stereochemical paths by orbital interaction. Accounts of Chemical Research. 1971;4(2):57

Section 3

## Structural Quantum Developments

[66] Kang J, Kino H, Field MJ, Tateno M. Electronic structure rearrangements in hybrid ribozyme/protein catalysis. Journal of the Physical Society of Japan.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

MM) investigation. ACS Catalysis. 2017;

[74] Hussain T, Kruparani SP, Pal B, Dock-Bregeon AC, Dwivedi S, Shekar MR, et al. Post-transfer editing mechanism of a D-aminoacyl-tRNA deacylase-like domain in threonyl-tRNA synthetase from archaea. The EMBO Journal. 2006;25(17):4152-4162

[75] Aboelnga MM, Hayward JJ, Gauld JW. Unraveling the critical role played

[76] Ling J, Roy H, Ibba M. Mechanism

Proceedings of the National Academy of

[77] Fukunaga R, Yokoyama S. Structural basis for substrate recognition by the editing domain of isoleucyl-tRNA synthetase. Journal of Molecular Biology. 2006;359(4):901-912

[78] Aboelnga MM, Hayward JJ, Gauld JW. A water-mediated and substrateassisted aminoacylation mechanism in the discriminating aminoacyl-tRNA

synthetase GlnRS and nondiscriminating GluRS. Physical Chemistry Chemical Physics. 2017;

[79] Fukui K. Recognition of Stereochemical paths by orbital interaction. Accounts of Chemical

19(37):25598-25609

Research. 1971;4(2):57

of tRNA-dependent editing in translational quality control.

Sciences. 2007;104(1):72

editing by archaeal threonyl-tRNA synthetase. The Journal of Physical Chemistry B. 2018;122(3):1092-1101

OH in the post-transfer

7(8):5180-5193

by Ado7620

[67] Weinger JS, Parnell KM, Dorner S, Green R, Strobel SA. Substrate-assisted catalysis of peptide bond formation by the ribosome. Nature Structural and Molecular Biology. 2004;11:1101

[68] Dulic M, Cvetesic N, Zivkovic I, Palencia A, Cusack S, Bertosa B, et al. Kinetic origin of substrate specificity in post-transfer editing by leucyl-tRNA synthetase. Journal of Molecular Biology. 2018;430(1):1-16

[69] Nordin BE, Schimmel P. Transiently misacylated tRNA is a primer for editing of misactivated adenylates by class I aminoacyl-tRNA synthetases.

Biochemistry. 2003;42(44):12989-12997

[70] Nordin BE, Schimmel P. Plasticity

mischarged tRNA by class I aminoacyltRNA Synthetases. The Journal of Biological Chemistry. 2002;277(23):

[71] Kumar S, Das M, Hadad CM, Musier-Forsyth K. Substrate and enzyme functional groups contribute to translational quality control by bacterial prolyl-tRNA synthetase. The Journal of Physical Chemistry B. 2012;116(23):

[72] Waas WF, Schimmel P. Evidence that tRNA synthetase-directed proton

[73] Aboelnga MM, Hayward JJ, Gauld JW. Enzymatic post-transfer editing mechanism of E. coli threonyl-tRNA synthetase (ThrRS): A molecular dynamics (MD) and quantum

mechanics/molecular mechanics (QM/

transfer stops mistranslation. Biochemistry. 2007;46(43):


of recognition of the 3<sup>0</sup>

20510-20517

6991-6999

12062-12070

72

2017;86(4):044801

Chapter 5

Abstract

1. Introduction

75

and its supercharges, Q^ � and Q� <sup>þ</sup>

terms of two non-mutually adjoint operators [2, 3].

Supersymmetric Quantum

Schemes and Quasi-Exactly

Carlos Villaseñor Mora and Edgar Condori Pozo

We present the general ideas on supersymmetric quantum mechanics (SUSY-

Keywords: supersymmetric quantum mechanics, quasi-exactly solvable potentials

We present the general ideas on supersymmetric quantum mechanics (SUSY-

operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found [1], there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved. In particular, we refer to the factorization of the Hamiltonian in

In this work, we try three main schemes; the first one consists on finding the eigenvalue Schrödinger equation in one dimension using the matrix representation via the appropriate factorization with ladder-like operators and finding the one parameter isospectral equation for this one. In this scheme, the wave function is written as a supermultiplet. Continuing with the Schrödinger model, we extend SUSY to include two-parameter factorizations, which include the SUSY factorization as particular case. As examples, we include the case of the harmonic oscillator and the Pöschl-Teller potentials. Also, we include the steps for the two-dimensional case and apply it to particular cases. The second scheme uses the differential representation in Grassmann numbers, where the wave function can be written as an

, which are defined as matrix or differential

QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian

QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found, there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.

Solvable Potentials

José Socorro García Díaz, Marco A. Reyes,

Mechanics: Two Factorization

#### Chapter 5

## Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable Potentials

José Socorro García Díaz, Marco A. Reyes, Carlos Villaseñor Mora and Edgar Condori Pozo

#### Abstract

We present the general ideas on supersymmetric quantum mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found, there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.

Keywords: supersymmetric quantum mechanics, quasi-exactly solvable potentials

#### 1. Introduction

We present the general ideas on supersymmetric quantum mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, Q^ � and Q� <sup>þ</sup> , which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found [1], there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved. In particular, we refer to the factorization of the Hamiltonian in terms of two non-mutually adjoint operators [2, 3].

In this work, we try three main schemes; the first one consists on finding the eigenvalue Schrödinger equation in one dimension using the matrix representation via the appropriate factorization with ladder-like operators and finding the one parameter isospectral equation for this one. In this scheme, the wave function is written as a supermultiplet. Continuing with the Schrödinger model, we extend SUSY to include two-parameter factorizations, which include the SUSY factorization as particular case. As examples, we include the case of the harmonic oscillator and the Pöschl-Teller potentials. Also, we include the steps for the two-dimensional case and apply it to particular cases. The second scheme uses the differential representation in Grassmann numbers, where the wave function can be written as an

n-dimensional vector or as an expansion in Grassmann variables multiplied by bosonic functions. We apply the scheme in two bosonic variables a particular cosmological model and compare the corresponding solutions found. The third scheme tries on extensions to the SUSY factorization and to the case of quasiexactly solvable potentials; we present a particular case which does not form part of the class of potentials found using Lie algebras.

<sup>H</sup>^ <sup>B</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>p</sup>^<sup>2</sup> <sup>þ</sup> 1 2 ω2

<sup>p</sup> ð Þ <sup>p</sup>^ � <sup>i</sup>ωBq^ , <sup>a</sup>^<sup>þ</sup> <sup>¼</sup> <sup>1</sup>

<sup>H</sup>^ <sup>B</sup> <sup>¼</sup> <sup>ω</sup><sup>B</sup>

b � and ^ b þ

This Hamiltonian can be written in terms of the anti-commutation relation

The symmetric nature of H^ <sup>B</sup> under the interchange of a^� and a^<sup>þ</sup> suggests that these operators satisfy Bose-Einstein statistics, and it is therefore called bosonic.

> b � ; ^ b � n o <sup>¼</sup> ^

and in analogy to (5), we define the corresponding new Hamiltonian as

<sup>2</sup> ^ b þ ; ^ b

<sup>¼</sup> <sup>σ</sup>þ, <sup>σ</sup>� <sup>¼</sup> <sup>1</sup>

that these operators satisfy the Fermi-Dirac statistics, and it is called fermionic.

1 0 � �, <sup>σ</sup><sup>þ</sup> <sup>¼</sup> 0 1

Now, consider both Hamiltonians as a composite system, that is, we consider the superposition of two oscillators, one being bosonic and one fermionic, with energy

¼ ωBnB þ ωFnF þ

<sup>H</sup>^ <sup>F</sup> <sup>¼</sup> <sup>ω</sup><sup>F</sup>

The antisymmetric nature of H^ <sup>F</sup> under the interchange of ^

b þ

respectively, defined as

between these operators as

<sup>a</sup>^� <sup>¼</sup> <sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82254

Now, we build the operators ^

^ b � ; ^ b <sup>þ</sup> n o <sup>¼</sup> <sup>1</sup>; ^

b � and ^ b þ

^ b �

with ½ �¼ <sup>σ</sup>þ; <sup>σ</sup>� <sup>σ</sup>3, <sup>σ</sup>� <sup>¼</sup> 0 0

� �, <sup>σ</sup><sup>3</sup> <sup>¼</sup> 1 0

1 2 � �

ET ¼ ω<sup>B</sup> nB þ

matrices that satisfy all rules defined above, that is,

<sup>¼</sup> <sup>σ</sup>�, ^

0 �1 � �.

<sup>þ</sup> <sup>ω</sup><sup>F</sup> nF � <sup>1</sup>

2 � �

When we demand that both frequencies are the same, ω<sup>B</sup> ¼ ω<sup>F</sup> ¼ ω, we introduce a new symmetry, called supersymmetry (SUSY); we can see that the simultaneous creation of a quantum fermion ð Þ nF ! nF þ 1 causes the destruction of quantum boson ð Þ nB ! nB � 1 and vice versa, in the sense that the total energy is

a^�, a^<sup>þ</sup> changing ½ �; ⇆ f g; , that is,

These operators ^

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> �<sup>i</sup> i 0

ET ¼ EB þ EF

77

ffiffiffiffiffiffiffiffi 2ω<sup>B</sup>

where q is the generalized coordinate and ^ p is the associated momentum, the ^ canonical commutation relation between this quantities being ½ �¼ q^; p^ i. We introduce two new operators, known as the creation and annihilation operators a^þ, a^�,

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

ffiffiffiffiffiffiffiffi 2ω<sup>B</sup>

b þ ; ^ b

admit a matrix representations in terms of Pauli

2

0 0 � �, <sup>σ</sup><sup>1</sup> <sup>¼</sup> 0 1

1 2

Bq^<sup>2</sup> (3)

p ð Þ p^ þ iωBq^ : (4)

<sup>2</sup> <sup>a</sup>^þ; <sup>a</sup>^� � �: (5)

that obey similar rules to operators

� h i: (7)

b � and ^ b þ

ð Þ σ<sup>1</sup> � iσ<sup>2</sup> (8)

1 0 � �,

ð Þ ω<sup>B</sup> � ω<sup>F</sup> : (9)

suggests

<sup>þ</sup> n o <sup>¼</sup> <sup>0</sup>, (6)

To establish the different approaches presented here, we will briefly describe the different main formalisms applied to supersymmetric quantum mechanics, techniques that are now widely used in a rich spectrum of physical problems, covering such diverse fields as particle physics, quantum field theory, quantum gravity, quantum cosmology, and statistical mechanics, to mention some of them:


$$
\hat{\mathbf{Q}}^{-} = \boldsymbol{\Psi}^{\mu} \left[ -\hbar \partial\_{\mathbf{q}^{\mu}} + \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\mu}} \right], \qquad \hat{\mathbf{Q}}^{+} = \overline{\boldsymbol{\Psi}}^{\nu} \left[ -\hbar \partial\_{\mathbf{q}^{\mu}} - \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\nu}} \right], \tag{1}
$$

where S is known as the superpotential function which are related to the physical potential under consideration, when the Hamiltonian density is written as the Hamilton-Jacobi equation, and the algebra for the variables ψ<sup>μ</sup> and ψ<sup>ν</sup> is

$$\{\!\!\!\!\!\/\!\!\/^{\mu}, \overline{\!\!\!\/^{\nu}}\!\!\/) = \eta^{\mu \nu}, \qquad \{\!\!\/\mu^{\mu}, \mu^{\nu}\} = 0, \qquad \{\!\!\/\overline{\mu}^{\mu}, \overline{\!\!\/^{\nu}}\!\!\/^{\nu}\} = 0. \tag{2}$$

There are two forms where the equations in 1D are satisfied: in the literature we find either the matrix representation or the differential operator scheme. However for more than one dimensions, there exist many applications to cosmological models, where the differential representation for the Grassmann variables is widely applied [14–18]. There are few works in more dimensions in the first scheme [19]; we present in this work the main ideas to build the 2D case, where the supercharge operators become 4 � 4 matrices.

#### 2. Factorization method in one dimension: matrix approach

We begin by introducing the main ideas for the one-dimensional quantum harmonic oscillator. The corresponding Hamiltonian is written in operator form as Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

$$
\hat{\mathbf{H}}\_{\text{B}} = \frac{1}{2}\hat{\mathbf{p}}^2 + \frac{1}{2}\omega\_{\text{B}}^2 \hat{\mathbf{q}}^2 \tag{3}
$$

where q is the generalized coordinate and ^ p is the associated momentum, the ^ canonical commutation relation between this quantities being ½ �¼ q^; p^ i. We introduce two new operators, known as the creation and annihilation operators a^þ, a^�, respectively, defined as

$$\hat{a}^- = \frac{1}{\sqrt{2\alpha\_B}}(\hat{p} - i\alpha\_B\hat{q}), \qquad \hat{a}^+ = \frac{1}{\sqrt{2\alpha\_B}}(\hat{p} + i\alpha\_B\hat{q}).\tag{4}$$

This Hamiltonian can be written in terms of the anti-commutation relation between these operators as

$$
\hat{H}\_B = \frac{\alpha\_B}{2} \left\{ \hat{a}^+, \hat{a}^- \right\}. \tag{5}
$$

The symmetric nature of H^ <sup>B</sup> under the interchange of a^� and a^<sup>þ</sup> suggests that these operators satisfy Bose-Einstein statistics, and it is therefore called bosonic.

Now, we build the operators ^ b � and ^ b þ that obey similar rules to operators a^�, a^<sup>þ</sup> changing ½ �; ⇆ f g; , that is,

$$\left\{\hat{\boldsymbol{b}}^{-},\hat{\boldsymbol{b}}^{+}\right\}=\mathbf{1};\quad\left\{\hat{\boldsymbol{b}}^{-},\hat{\boldsymbol{b}}^{-}\right\}=\left\{\hat{\boldsymbol{b}}^{+},\hat{\boldsymbol{b}}^{+}\right\}=\mathbf{0},\tag{6}$$

and in analogy to (5), we define the corresponding new Hamiltonian as

$$
\hat{\mathbf{H}}\_{\rm F} = \frac{\rho\_{\rm F}}{2} \left[ \hat{\mathbf{b}}^{+}, \hat{\mathbf{b}}^{-} \right]. \tag{7}
$$

The antisymmetric nature of H^ <sup>F</sup> under the interchange of ^ b � and ^ b þ suggests that these operators satisfy the Fermi-Dirac statistics, and it is called fermionic.

These operators ^ b � and ^ b þ admit a matrix representations in terms of Pauli matrices that satisfy all rules defined above, that is,

$$
\hat{\boldsymbol{b}}^{-} = \sigma\_{-}, \qquad \hat{\boldsymbol{b}}^{+} = \sigma\_{+}, \qquad \sigma\_{\pm} = \frac{1}{2}(\sigma\_{1} \pm i\sigma\_{2}) \tag{8}
$$

with ½ �¼ <sup>σ</sup>þ; <sup>σ</sup>� <sup>σ</sup>3, <sup>σ</sup>� <sup>¼</sup> 0 0 1 0 � �, <sup>σ</sup><sup>þ</sup> <sup>¼</sup> 0 1 0 0 � �, <sup>σ</sup><sup>1</sup> <sup>¼</sup> 0 1 1 0 � �, <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> �<sup>i</sup> i 0 � �, <sup>σ</sup><sup>3</sup> <sup>¼</sup> 1 0 0 �1 � �.

Now, consider both Hamiltonians as a composite system, that is, we consider the superposition of two oscillators, one being bosonic and one fermionic, with energy ET ¼ EB þ EF

$$\mathbf{E}\_{\rm T} = \rho\_{\rm B} \left( \mathbf{n}\_{\rm B} + \frac{\mathbf{1}}{2} \right) + \rho\_{\rm F} \left( \mathbf{n}\_{\rm F} - \frac{\mathbf{1}}{2} \right) = \rho\_{\rm B} \mathbf{n}\_{\rm B} + \rho\_{\rm F} \mathbf{n}\_{\rm F} + \frac{\mathbf{1}}{2} (\rho\_{\rm B} - \rho\_{\rm F}).\tag{9}$$

When we demand that both frequencies are the same, ω<sup>B</sup> ¼ ω<sup>F</sup> ¼ ω, we introduce a new symmetry, called supersymmetry (SUSY); we can see that the simultaneous creation of a quantum fermion ð Þ nF ! nF þ 1 causes the destruction of quantum boson ð Þ nB ! nB � 1 and vice versa, in the sense that the total energy is

n-dimensional vector or as an expansion in Grassmann variables multiplied by bosonic functions. We apply the scheme in two bosonic variables a particular cosmological model and compare the corresponding solutions found. The third scheme tries on extensions to the SUSY factorization and to the case of quasiexactly solvable potentials; we present a particular case which does not form part

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

To establish the different approaches presented here, we will briefly describe the different main formalisms applied to supersymmetric quantum mechanics, techniques that are now widely used in a rich spectrum of physical problems, covering such diverse fields as particle physics, quantum field theory, quantum gravity, quantum cosmology, and statistical mechanics, to mention some of them:

• In one dimension, SUSY-QM may be considered an equivalent formulation of the Darboux transformation method, which is well-known in mathematics from the original paper of Darboux [4], the book by Ince [5], and the book by Matweev and Salle [6], where the method is widely used in the context of the soliton theory. An essential ingredient of the method is the particular choice of

• Those defined by means of the use of supersymmetry as a square root [11–14], in which the Grassmann variables are auxiliary variables and are not identified as the supersymmetric partners of the bosonic variables. In this formalism, a differential representation is used for the Grassmann variables. Also the

, <sup>Q</sup>^ <sup>þ</sup> <sup>¼</sup> <sup>ψ</sup><sup>ν</sup> �ℏ∂q<sup>ν</sup> � <sup>∂</sup><sup>S</sup>

; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> ημν, <sup>ψ</sup><sup>μ</sup>; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> <sup>0</sup>, <sup>ψ</sup><sup>μ</sup>; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> <sup>0</sup>: (2)

There are two forms where the equations in 1D are satisfied: in the literature we find either the matrix representation or the differential operator scheme. However for more than one dimensions, there exist many applications to cosmological models, where the differential representation for the Grassmann variables is widely applied [14–18]. There are few works in more dimensions in the first scheme [19]; we present in this work the main ideas to build the 2D case, where the supercharge

∂q<sup>ν</sup>

, (1)

a transformation operator in the form of a differential operator which intertwines two Hamiltonian and relates their eigenfunctions. When this approach is applied to quantum theory, it allows to generate a huge family of exactly solvable local potential starting with a given exactly solvable local potential [7]. This technique is also known in the literature as isospectral

of the class of potentials found using Lie algebras.

supercharges for the n-dimensional case read as

∂S ∂q<sup>μ</sup>

2. Factorization method in one dimension: matrix approach

We begin by introducing the main ideas for the one-dimensional quantum harmonic oscillator. The corresponding Hamiltonian is written in operator form as

where S is known as the superpotential function which are related to the physical potential under consideration, when the Hamiltonian density is written as the Hamilton-Jacobi equation, and the algebra for the variables ψ<sup>μ</sup>

<sup>Q</sup>^ � <sup>¼</sup> <sup>ψ</sup><sup>μ</sup> �ℏ∂q<sup>μ</sup> <sup>þ</sup>

ψμ

operators become 4 � 4 matrices.

formalism [7–10].

and ψ<sup>ν</sup> is

76

unaltered. The ground energy state is exact and no degenerate. The degeneration appears from n = 1, where it is double degenerate.

In this way, we have the super-Hamiltonian H^ susy, written as

$$\begin{split} \hat{H}\_{\text{susy}} &= \frac{a\nu}{2} \{\hat{a}^+, \hat{a}^-\} + \frac{a\nu}{2} \left[\hat{b}^+, \hat{b}^-\right] = \frac{a\nu}{2} \{\hat{a}^+, \hat{a}^-\} I + \frac{a\nu}{2} \sigma\_3 = a \begin{pmatrix} \hat{a} - \hat{a}^+ & \mathbf{0} \\ \mathbf{0} & \hat{a}^+ \hat{a}^- \end{pmatrix}, \\ &= \begin{pmatrix} \hat{H}\_- & \mathbf{0} \\ \mathbf{0} & \hat{H}\_+ \end{pmatrix}, \end{split} \tag{10}$$

where I is a 2 � 2 unit matrix and where the two components of <sup>H</sup>^ susy in (10) can be written independently as

$$
\hat{H}\_{+} = \frac{1}{2}\hat{p}^2 + \frac{1}{2}(\alpha^2 q^2 - \alpha) \equiv \alpha \hat{a}^+ \hat{a}^- \tag{11}
$$

The Hamiltonians H^ <sup>þ</sup> and H^ � determine two new potentials,

<sup>p</sup>^<sup>2</sup> <sup>þ</sup> <sup>V</sup><sup>þ</sup> <sup>¼</sup> <sup>1</sup>

<sup>p</sup>^<sup>2</sup> <sup>þ</sup> <sup>V</sup>� <sup>¼</sup> <sup>1</sup>

<sup>V</sup><sup>þ</sup> <sup>¼</sup> <sup>1</sup> 2

(modulo constant ϵ, which is related to some energy eigenvalue) and

quantum equation (17) applied to stationary wave function ui becomes

where Ei are the energy eigenvalues. Considering the transformation

dW dq � <sup>W</sup>^ <sup>2</sup>

family of potentials <sup>V</sup>^ � depending on a free parameter <sup>λ</sup>, the supersymmetric parameter that, to some extent, plays the role of internal time. Following the

2ui

dui dq � �<sup>2</sup>

Then, this equation is the same as the original one, Eq. (21), that is, W is related to an initial solution of the bosonic Hamiltonian. What happens to the isopotential

the question is, what is W if we know the function W? Finding it we can build a ^

, ! <sup>W</sup>^ <sup>¼</sup> <sup>W</sup> <sup>þ</sup>

� dW^ dq ! <sup>¼</sup> <sup>V</sup>� <sup>þ</sup>

�

þ dW^

y qð Þ, where the function y(q) satisfy the linear differential

dui dq � �<sup>2</sup>

dq <sup>¼</sup> 2V^ �,

u2 i <sup>λ</sup> <sup>þ</sup> <sup>Ð</sup> u2

dW^

<sup>i</sup> dq : (22)

dq : (23)

dq and introducing it into (18), we have that

� 1 2 d2 ui

W2 � dW dq � � <sup>¼</sup> <sup>1</sup>

dq � �? Considering that

2V� <sup>¼</sup> W2 <sup>þ</sup>

dq � 2Wy ¼ 1, the solution implies

The family of potentials <sup>V</sup>^ <sup>þ</sup> can be built now as

<sup>V</sup>^ <sup>þ</sup> � Ei <sup>¼</sup> <sup>1</sup>

u2 <sup>i</sup> dq u2 i

<sup>2</sup> <sup>W</sup>^ <sup>2</sup>

y qð Þ¼ <sup>λ</sup> <sup>þ</sup> <sup>Ð</sup>

In a general way, let us now find the general form of the function W. The

2 <sup>p</sup>^<sup>2</sup> <sup>þ</sup> 1 2

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

2 <sup>p</sup>^<sup>2</sup> <sup>þ</sup> 1 <sup>2</sup> <sup>W</sup><sup>2</sup> <sup>þ</sup>

where the potential term V+(q) is related to the superpotential function W(q)

<sup>W</sup><sup>2</sup> � dW

<sup>W</sup><sup>2</sup> � dW

dW

dq � �, (20)

dq , with the same spectrum, except for the ground

dq<sup>2</sup> <sup>þ</sup> V qð Þui <sup>¼</sup> Eiui, (21)

� ui d2 ui dq<sup>2</sup>

¼ 1 2ui d2 ui dq<sup>2</sup> :

2u<sup>2</sup> i

dq � � (18)

dq � �, (19)

<sup>H</sup>^ <sup>þ</sup> <sup>¼</sup> <sup>1</sup> 2

DOI: http://dx.doi.org/10.5772/intechopen.82254

<sup>H</sup>^ � <sup>¼</sup> <sup>1</sup> 2

state, which is related to the energy potential Vþ.

via the Riccati equation

<sup>2</sup> W2 <sup>þ</sup> dW

W qð Þ¼� dln u½ � <sup>i</sup>ð Þ <sup>q</sup>

<sup>V</sup>�ð Þ¼ <sup>q</sup> <sup>1</sup>

equation dy

79

V qð Þ� Ei <sup>¼</sup> <sup>1</sup>

<sup>2</sup> <sup>W</sup><sup>2</sup> <sup>þ</sup> dW

procedure <sup>W</sup>^ <sup>¼</sup> <sup>W</sup> <sup>þ</sup> <sup>1</sup>

2

dq � � <sup>¼</sup> <sup>V</sup><sup>þ</sup> <sup>þ</sup> dW

<sup>V</sup>� <sup>¼</sup> <sup>1</sup>

$$
\hat{H}\_{-}=\frac{1}{2}\hat{p}^{2}+\frac{1}{2}\left(\rho^{2}q^{2}+\rho\right)\equiv\rho\hat{a}^{-}\hat{a}^{+}.\tag{12}
$$

From Eqs. (18) and (19), we can see that <sup>H</sup>^ <sup>þ</sup> and <sup>H</sup>^ � are the same representation of one Hamiltonian with a constant shifting ω in the energy spectrum.

The question is, what are the generators for this SUSY Hamiltonian? The answer is, considering that the degeneration is the result of the simultaneous destruction (creation) of quantum boson and the creation (destruction) of quantum fermion, the corresponding generators for this symmetry must be written as a^�^ b þ (or a^þ^ b � ). Therefore we introduce the following generators, called supercharges Q^ � and Q^ <sup>þ</sup> , defined as

$$
\hat{\mathbf{Q}}^{-} = \sqrt{2a}\hat{a}^{-}\hat{b}^{+} = \sqrt{2a}\begin{pmatrix} \mathbf{0} & \hat{a}^{-} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}, \qquad \hat{\mathbf{Q}}^{+} = \sqrt{2a}\hat{a}^{+}\hat{b}^{-} = \sqrt{2a}\begin{pmatrix} \mathbf{0} & \mathbf{0} \\ \hat{a}^{+} & \mathbf{0} \end{pmatrix}, \tag{13}
$$

implying that

$$
\hat{\mathbf{H}}\_{\text{susy}} = \frac{1}{2} \left\{ \hat{\mathbf{Q}}^{+} , \hat{\mathbf{Q}}^{-} \right\} \tag{14}
$$

and satisfying the following relations

$$\left\{\hat{\mathbf{Q}}^{-},\hat{\mathbf{Q}}^{-}\right\} = \left\{\hat{\mathbf{Q}}^{+},\hat{\mathbf{Q}}^{+}\right\} = \mathbf{0}; \quad \left[\hat{\mathbf{Q}}^{-},\hat{\mathbf{H}}\_{\text{susy}}\right] = \left[\hat{\mathbf{Q}}^{+},\hat{\mathbf{H}}\_{\text{susy}}\right] = \mathbf{0}.\tag{15}$$

We can generalize this procedure for a certain function W(q), and at this point, we can define two new operators A^ � and A^ <sup>þ</sup> with a property similar to (4),

$$
\hat{\mathbf{A}}^{-} = \frac{1}{\sqrt{2\alpha}} (\hat{\mathbf{p}} - \mathrm{i}\omega \mathbf{W}(\mathbf{q})), \qquad \hat{\mathbf{A}}^{+} = \frac{\mathbf{1}}{\sqrt{2\alpha}} (\hat{\mathbf{p}} + \mathrm{i}\omega \mathbf{W}(\mathbf{q})).\tag{16}
$$

In order to obtain the general solutions, we can use an arbitrary potential in Eq. (3), that is,

$$
\hat{\mathbf{H}}\_{\mathbf{B}} = \frac{1}{2}\hat{\mathbf{p}}^2 + \mathbf{V}(\mathbf{q}).\tag{17}
$$

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

The Hamiltonians H^ <sup>þ</sup> and H^ � determine two new potentials,

$$\hat{\mathbf{H}}\_{+} = \frac{1}{2}\hat{\mathbf{p}}^{2} + \mathbf{V}\_{+} = \frac{1}{2}\hat{\mathbf{p}}^{2} + \frac{1}{2}\left(\mathbf{W}^{2} - \frac{\mathbf{d}\mathbf{W}}{\mathbf{dq}}\right) \tag{18}$$

$$
\hat{\mathbf{H}}\_{-} = \frac{1}{2}\hat{\mathbf{p}}^2 + \mathbf{V}\_{-} = \frac{1}{2}\hat{\mathbf{p}}^2 + \frac{1}{2}\left(\mathbf{W}^2 + \frac{\mathbf{dW}}{\mathbf{dq}}\right),\tag{19}
$$

where the potential term V+(q) is related to the superpotential function W(q) via the Riccati equation

$$\mathbf{V}\_{+} = \frac{1}{2} \left( \mathbf{W}^{2} - \frac{\mathbf{d} \mathbf{W}}{\mathbf{dq}} \right), \tag{20}$$

(modulo constant ϵ, which is related to some energy eigenvalue) and <sup>V</sup>� <sup>¼</sup> <sup>1</sup> <sup>2</sup> W2 <sup>þ</sup> dW dq � � <sup>¼</sup> <sup>V</sup><sup>þ</sup> <sup>þ</sup> dW dq , with the same spectrum, except for the ground state, which is related to the energy potential Vþ.

In a general way, let us now find the general form of the function W. The quantum equation (17) applied to stationary wave function ui becomes

$$-\frac{1}{2}\frac{d^2\mathbf{u}\_i}{d\mathbf{q}^2} + \mathbf{V}(\mathbf{q})\mathbf{u}\_i = \mathbf{E}\_i\mathbf{u}\_{i\flat} \tag{21}$$

where Ei are the energy eigenvalues. Considering the transformation W qð Þ¼� dln u½ � <sup>i</sup>ð Þ <sup>q</sup> dq and introducing it into (18), we have that

$$\mathbf{V(q)} - \mathbf{E\_i} = \frac{\mathbf{1}}{2} \left( \mathbf{W^2} - \frac{\mathbf{dw}}{\mathbf{dq}} \right) = \left( \frac{\mathbf{1}}{2 \mathbf{u\_i}} \frac{\mathbf{du\_i}}{\mathbf{dq}} \right)^2 - \frac{\left( \frac{\mathbf{du\_i}}{\mathbf{dq}} \right)^2 - \mathbf{u\_i} \frac{\mathbf{d^2 u\_i}}{\mathbf{dq}^2}}{2 \mathbf{u\_i^2}} = \frac{\mathbf{1}}{2 \mathbf{u\_i}} \frac{\mathbf{d^2 u\_i}}{\mathbf{dq}^2}.$$

Then, this equation is the same as the original one, Eq. (21), that is, W is related to an initial solution of the bosonic Hamiltonian. What happens to the isopotential <sup>V</sup>�ð Þ¼ <sup>q</sup> <sup>1</sup> <sup>2</sup> <sup>W</sup><sup>2</sup> <sup>þ</sup> dW dq � �? Considering that

$$2\mathbf{V}\_{-}=\mathbf{W}^{2}+\frac{\mathbf{d}\mathbf{W}}{\mathbf{d}\mathbf{q}}\equiv\hat{\mathbf{W}}^{2}+\frac{\mathbf{d}\hat{\mathbf{W}}}{\mathbf{d}\mathbf{q}}=2\hat{\mathbf{V}}\_{-},$$

the question is, what is W if we know the function W? Finding it we can build a ^ family of potentials <sup>V</sup>^ � depending on a free parameter <sup>λ</sup>, the supersymmetric parameter that, to some extent, plays the role of internal time. Following the procedure <sup>W</sup>^ <sup>¼</sup> <sup>W</sup> <sup>þ</sup> <sup>1</sup> y qð Þ, where the function y(q) satisfy the linear differential equation dy dq � 2Wy ¼ 1, the solution implies

$$\mathbf{y}(\mathbf{q}) = \frac{\lambda + \int \mathbf{u}\_i^2 \mathbf{d} \mathbf{q}}{\mathbf{u}\_i^2}, \quad \rightarrow \quad \hat{\mathbf{W}} = \mathcal{W} + \frac{\mathbf{u}\_i^2}{\lambda + \int \mathbf{u}\_i^2 \mathbf{d} \mathbf{q}}. \tag{22}$$

The family of potentials <sup>V</sup>^ <sup>þ</sup> can be built now as

$$
\hat{\mathbf{V}}\_{+} - \mathbf{E}\_{i} = \frac{1}{2} \left( \hat{\mathbf{W}}^{2} - \frac{\mathbf{d} \hat{\mathbf{W}}}{\mathbf{dq}} \right) = \mathbf{V}\_{-} + \frac{\mathbf{d} \hat{\mathbf{W}}}{\mathbf{dq}} \ . \tag{23}
$$

unaltered. The ground energy state is exact and no degenerate. The degeneration

¼ ω

<sup>2</sup> <sup>a</sup>^þ; <sup>a</sup>^� � �<sup>I</sup> <sup>þ</sup> <sup>ω</sup>

where I is a 2 � 2 unit matrix and where the two components of <sup>H</sup>^ susy in (10) can

From Eqs. (18) and (19), we can see that <sup>H</sup>^ <sup>þ</sup> and <sup>H</sup>^ � are the same representation

The question is, what are the generators for this SUSY Hamiltonian? The answer is, considering that the degeneration is the result of the simultaneous destruction (creation) of quantum boson and the creation (destruction) of quantum fermion,

Therefore we introduce the following generators, called supercharges Q^ � and Q^ <sup>þ</sup>

, <sup>Q</sup>^ <sup>þ</sup> <sup>¼</sup> ffiffiffiffiffiffi

; H^ susy h i

<sup>2</sup><sup>ω</sup> <sup>p</sup> <sup>a</sup>^þ^ b �

<sup>¼</sup> <sup>Q</sup>^ <sup>þ</sup>

ffiffiffiffiffiffi

; H^ susy h i

<sup>p</sup>^<sup>2</sup> <sup>þ</sup> V qð Þ: (17)

<sup>2</sup><sup>ω</sup> <sup>p</sup> ð Þ <sup>p</sup>^ <sup>þ</sup> <sup>i</sup>ωW qð Þ : (16)

of one Hamiltonian with a constant shifting ω in the energy spectrum.

the corresponding generators for this symmetry must be written as a^�^

<sup>H</sup>^ susy <sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>0</sup>; <sup>Q</sup>^ �

we can define two new operators A^ � and A^ <sup>þ</sup> with a property similar to (4),

<sup>2</sup><sup>ω</sup> <sup>p</sup> ð Þ <sup>p</sup>^ � <sup>i</sup>ωW qð Þ , <sup>A</sup>^ <sup>þ</sup> <sup>¼</sup> <sup>1</sup>

<sup>H</sup>^ <sup>B</sup> <sup>¼</sup> <sup>1</sup> 2

<sup>2</sup> <sup>Q</sup>^ <sup>þ</sup> ; <sup>Q</sup>^ � n o

We can generalize this procedure for a certain function W(q), and at this point,

In order to obtain the general solutions, we can use an arbitrary potential in

<sup>2</sup> <sup>σ</sup><sup>3</sup> <sup>¼</sup> <sup>ω</sup> <sup>a</sup>^ � <sup>a</sup>^<sup>þ</sup> <sup>0</sup>

<sup>q</sup><sup>2</sup> � <sup>ω</sup> � � � <sup>ω</sup>a^<sup>þ</sup>a^� (11)

<sup>q</sup><sup>2</sup> <sup>þ</sup> <sup>ω</sup> � � � <sup>ω</sup>a^�a^<sup>þ</sup>: (12)

0 a^þa^�

b þ

<sup>2</sup><sup>ω</sup> <sup>p</sup> 0 0 a^<sup>þ</sup> 0 � �

<sup>¼</sup> ffiffiffiffiffiffi

(or a^þ^ b � ).

,

(13)

(14)

¼ 0: (15)

,

(10)

!

In this way, we have the super-Hamiltonian H^ susy, written as

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

appears from n = 1, where it is double degenerate.

2 ^ b þ ; ^ b � h i

<sup>H</sup>^ <sup>þ</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>p</sup>^<sup>2</sup> <sup>þ</sup> 1 <sup>2</sup> <sup>ω</sup><sup>2</sup>

<sup>H</sup>^ � <sup>¼</sup> <sup>1</sup> 2 <sup>p</sup>^<sup>2</sup> <sup>þ</sup> 1 <sup>2</sup> <sup>ω</sup><sup>2</sup>

,

<sup>2</sup> <sup>a</sup>^þ; <sup>a</sup>^� � � <sup>þ</sup> <sup>ω</sup>

!

<sup>¼</sup> <sup>H</sup>^ � <sup>0</sup> <sup>0</sup> <sup>H</sup>^ <sup>þ</sup>

be written independently as

<sup>H</sup>^ susy <sup>¼</sup> <sup>ω</sup>

defined as

<sup>Q</sup>^ � <sup>¼</sup> ffiffiffiffiffiffi

implying that

Q^ � ; <sup>Q</sup>^ � n o

Eq. (3), that is,

78

<sup>2</sup><sup>ω</sup> <sup>p</sup> <sup>a</sup>^�^ b þ

<sup>¼</sup> ffiffiffiffiffiffi

and satisfying the following relations

<sup>¼</sup> <sup>Q</sup>^ <sup>þ</sup> ; <sup>Q</sup>^ <sup>þ</sup> n o

ffiffiffiffiffiffi

<sup>A</sup>^ � <sup>¼</sup> <sup>1</sup>

<sup>2</sup><sup>ω</sup> <sup>p</sup> <sup>0</sup> <sup>a</sup>^� 0 0 � � Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Finally

$$
\hat{\mathbf{u}} = \mathbf{g}(\lambda) \frac{\mathbf{u}\_i}{\lambda + \int \mathbf{u}\_i^2 \mathbf{d} \mathbf{q}} \tag{24}
$$

where

<sup>a</sup>� <sup>¼</sup> <sup>1</sup> ffiffi 2 p

DOI: http://dx.doi.org/10.5772/intechopen.82254

<sup>b</sup>� <sup>¼</sup> <sup>1</sup> ffiffi 2 p

<sup>V</sup><sup>þ</sup> <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> <sup>V</sup>þ1 xð Þþ <sup>V</sup>þ<sup>2</sup> <sup>y</sup>

1 2 d2

u2

<sup>W</sup>^ ¼ � <sup>1</sup> u1 du1 dx þ

> u1 du1

In the same manner, we have that

<sup>Z</sup>^ ¼ � <sup>1</sup> u2 du2 dy þ

dy and I2 <sup>¼</sup> <sup>Ð</sup>

the isopotential, using the new potential W, as ^

1 <sup>2</sup> <sup>W</sup>^ <sup>2</sup>

For the other coordinate, we have

u2 su2

<sup>V</sup>^ <sup>þ</sup><sup>1</sup>ð Þ¼ <sup>x</sup>; <sup>λ</sup><sup>1</sup>

<sup>y</sup> <sup>¼</sup> <sup>u</sup>�<sup>2</sup>

<sup>1</sup> ð Þ <sup>x</sup> <sup>E</sup><sup>1</sup> <sup>þ</sup> <sup>Ð</sup>

where Wp ¼ � <sup>1</sup>

with Zp ¼ � <sup>1</sup>

81

and, using separation variables, we get

and V x; <sup>y</sup> � � <sup>¼</sup> W xð Þþ Z y� �.

d

d

The Riccati equation (20) is written in 2D as

dx <sup>þ</sup> W xð Þ � �, a<sup>þ</sup> <sup>¼</sup> <sup>1</sup>

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

dx <sup>þ</sup> Z y� � � �, b<sup>þ</sup> <sup>¼</sup> <sup>1</sup>

� � <sup>¼</sup> <sup>1</sup>

2 W2

> Z<sup>2</sup> y � � � dZ

Following the same steps as in the 1D case, we find that the solutions (22)

are the same in this case. So, the general solution for W is ^ <sup>W</sup>^ <sup>¼</sup> <sup>W</sup> <sup>þ</sup> <sup>1</sup>

u2 <sup>1</sup> dx.

u2 <sup>2</sup> dy.

u2 1 <sup>λ</sup><sup>1</sup> <sup>þ</sup> <sup>Ð</sup> u2

u2 2 <sup>λ</sup><sup>2</sup> <sup>þ</sup> <sup>Ð</sup> u2

� <sup>W</sup>^ <sup>0</sup> � � <sup>¼</sup> <sup>V</sup>þð Þ� <sup>x</sup> 2u1

V1ð Þ� <sup>x</sup> <sup>1</sup>

V2 y � � � <sup>1</sup> 2

In general, we find that each potential Vþ<sup>i</sup> satisfies

and we can find the isopotential as W ¼ � <sup>1</sup>

dx and <sup>I</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup>

<sup>2</sup> <sup>W</sup><sup>2</sup> � dW dx � � <sup>þ</sup>

ð Þ� <sup>x</sup> dW

dx2 uið Þþ <sup>x</sup> <sup>V</sup>þiuið Þ¼ <sup>x</sup> Eiuið Þ <sup>x</sup> , <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, (35)

d

d

du1 dx λ<sup>1</sup> þ I1

þ

u4 1 ð Þ λ<sup>1</sup> þ I1

u1 du1

<sup>1</sup>ð Þ <sup>x</sup> dx � �. The general solution for the superpotential W x ^ ð Þ is

<sup>1</sup> dx <sup>¼</sup> Wp <sup>þ</sup>

<sup>2</sup> dy <sup>¼</sup> Zp <sup>þ</sup>

On the other hand, using the Riccati equation, we can build a generalization for

ffiffi 2 <sup>p</sup> � <sup>d</sup>

ffiffi 2 <sup>p</sup> � <sup>d</sup>

> 1 2

dx � � <sup>¼</sup> C0 (33)

dy � � ¼ �C0: (34)

dx , when u1 is known.

<sup>Z</sup><sup>2</sup> � dZ

dy � �, (32)

y xð Þ, with

<sup>2</sup> : (38)

dx ½ � Lnð Þ <sup>λ</sup><sup>1</sup> <sup>þ</sup> I1 (36)

dy ½ � Lnð Þ <sup>λ</sup><sup>2</sup> <sup>þ</sup> I2 (37)

dx <sup>þ</sup> W xð Þ � � (30)

dx <sup>þ</sup> Z yð Þ � � (31)

is the isospectral solution of the Schrödinger-like equation for the new family potential (23), with the condition gð Þ¼ <sup>λ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ λð Þ <sup>þ</sup> <sup>1</sup> <sup>p</sup> , which in the limit

λ ! �∞, gð Þ¼ λ λ, u^<sup>i</sup> ! ui:

This λ parameter is included not for factorization reasons; in particular, in quantum cosmology the wave functions are still nonnormalizable, and λ is used as a decoherence parameter embodying a sort of quantum cosmological dissipation (or damping) distance.

#### 2.1 Two-dimensional case

We use Witten's idea [20] to find the supersymmetric supercharge operators Q � and Q <sup>þ</sup> that generate the super-Hamiltonian Hsusy. Using Eqs. (13)–(15), we can generalize the one-dimensional factorization scheme. We define the twodimensional Hamiltonian as

$$
\hat{\mathbf{H}}\_{\rm B}(\mathbf{x}, \mathbf{y}) = \frac{1}{2}\hat{\mathbf{p}}\_{\mathbf{x}}^{2} + \frac{1}{2}\hat{\mathbf{p}}\_{\mathbf{y}}^{2} + \mathbf{V}(\mathbf{x}) + \mathbf{V}(\mathbf{y}), \tag{25}
$$

where the Schrödinger-like equation can be obtained as the bosonic sector of this super-Hamiltonian in the superspace, i.e., when all fermionic fields are set equal to zero (classical limit).

In two dimensions, the supercharges are defined by the tensorial products

$$\mathbf{Q}^{-} = \sqrt{2}\mathbf{d}\big|^{-} \otimes \sigma\_{+}, \qquad \mathbf{Q}^{+} = \sqrt{2}\mathbf{d}\big|^{+} \otimes \sigma\_{-} \tag{26}$$

with

$$d\begin{vmatrix} \\ \\ \end{vmatrix}^{-} = \begin{pmatrix} a^- & \mathbf{0} \\ \mathbf{0} & b^- \end{pmatrix}, \qquad d\begin{vmatrix} \\ \\ \end{vmatrix}^+ = \begin{pmatrix} a^+ & \mathbf{0} \\ \mathbf{0} & b^+ \end{pmatrix},\tag{27}$$

where σ� are the same as in (8). From Eq. (26) we have that the supercharges are 4 � 4 matrices

$$
\hat{\mathbf{Q}}^{+} = \sqrt{2} \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ a^{+} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & b^{+} & \mathbf{0} & \mathbf{0} \end{bmatrix} \qquad \hat{\mathbf{Q}}^{-} = \sqrt{2} \begin{bmatrix} \mathbf{0} & \mathbf{0} & a^{-} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & b^{-} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{28}
$$

where the super-Hamiltonian, (14), can be written as

$$\mathbf{H}\_{\text{susy}} = \begin{pmatrix} a^-a^+ & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & b^-b^+ & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & a^+a^- & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & b^+b^- \end{pmatrix} = \begin{pmatrix} H\_-^1(\mathbf{x}) & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & H\_-^1(\mathbf{y}) & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & H\_+^2(\mathbf{x}) & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & H\_+^2(\mathbf{y}) \end{pmatrix},\tag{29}$$

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

where

Finally

damping) distance.

2.1 Two-dimensional case

dimensional Hamiltonian as

zero (classical limit).

with

Hsusy ¼

80

are 4 � 4 matrices

<sup>Q</sup>^ <sup>þ</sup> <sup>¼</sup> ffiffi 2 p

0

BBB@

<sup>u</sup>^ <sup>¼</sup> <sup>g</sup>ð Þ<sup>λ</sup> ui <sup>λ</sup> <sup>þ</sup> <sup>Ð</sup> u2

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

potential (23), with the condition gð Þ¼ <sup>λ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

is the isospectral solution of the Schrödinger-like equation for the new family

λ ! �∞, gð Þ¼ λ λ, u^<sup>i</sup> ! ui:

We use Witten's idea [20] to find the supersymmetric supercharge operators Q � and Q <sup>þ</sup> that generate the super-Hamiltonian Hsusy. Using Eqs. (13)–(15), we can

where the Schrödinger-like equation can be obtained as the bosonic sector of this super-Hamiltonian in the superspace, i.e., when all fermionic fields are set equal to

In two dimensions, the supercharges are defined by the tensorial products

<sup>p</sup> <sup>d</sup> � <sup>⊗</sup> <sup>σ</sup>þ; <sup>Q</sup> <sup>þ</sup> <sup>¼</sup> ffiffi

; d

where σ� are the same as in (8). From Eq. (26) we have that the supercharges

� � � � þ

<sup>Q</sup>^ � <sup>¼</sup> ffiffi 2 p

H1

0

BBBB@

0 H<sup>1</sup>

0 0 H<sup>2</sup>

<sup>y</sup> þ V xð Þþ V y

2 <sup>p</sup> <sup>d</sup>

<sup>¼</sup> <sup>a</sup><sup>þ</sup> <sup>0</sup> 0 b<sup>þ</sup> � �

0 0 a� 0 00 0 b� 00 0 0 00 0 0

�ð Þ x 000

000 H<sup>2</sup>

�ð Þy 0 0

þð Þ x 0

þð Þy

1

CCCCA , (29)

� � � þ

generalize the one-dimensional factorization scheme. We define the two-

2 p^2 <sup>x</sup> þ 1 2 p^2

� � �

0 b� � �

> > 1

CCCA ¼

<sup>H</sup>^ <sup>B</sup> <sup>x</sup>; <sup>y</sup> � � <sup>¼</sup> <sup>1</sup>

<sup>Q</sup> � <sup>¼</sup> ffiffi 2

� � � �

<sup>d</sup> � <sup>¼</sup> <sup>a</sup>� <sup>0</sup>

0 0 00 0 0 00 a<sup>þ</sup> 0 00 0 b<sup>þ</sup> 0 0

where the super-Hamiltonian, (14), can be written as

a�a<sup>þ</sup> 000 0 b�b<sup>þ</sup> 0 0 0 0 aþa� 0 000 b<sup>þ</sup>b�

This λ parameter is included not for factorization reasons; in particular, in quantum cosmology the wave functions are still nonnormalizable, and λ is used as a decoherence parameter embodying a sort of quantum cosmological dissipation (or

<sup>i</sup> dq (24)

� �, (25)

⊗ σ� (26)

, (27)

(28)

λ λð Þ <sup>þ</sup> <sup>1</sup> <sup>p</sup> , which in the limit

$$a^- = \frac{1}{\sqrt{2}} \left( \frac{d}{dx} + \mathcal{W}(\mathbf{x}) \right), \qquad a^+ = \frac{1}{\sqrt{2}} \left( -\frac{d}{dx} + \mathcal{W}(\mathbf{x}) \right) \tag{30}$$

$$b^- = \frac{1}{\sqrt{2}} \left( \frac{d}{d\mathbf{x}} + Z(\mathbf{y}) \right), \qquad b^+ = \frac{1}{\sqrt{2}} \left( -\frac{d}{d\mathbf{x}} + Z(\mathbf{y}) \right) \tag{31}$$

and V x; <sup>y</sup> � � <sup>¼</sup> W xð Þþ Z y� �. The Riccati equation (20) is written in 2D as

$$\mathbf{V}\_{+}\left(\mathbf{x},\mathbf{y}\right) = \mathbf{V}\_{+}\mathbf{1}\left(\mathbf{x}\right) + \mathbf{V}\_{+2}\left(\mathbf{y}\right) = \frac{1}{2}\left(\mathbf{W}^{2} - \frac{\mathbf{dW}}{\mathbf{dx}}\right) + \frac{1}{2}\left(\mathbf{Z}^{2} - \frac{\mathbf{dZ}}{\mathbf{dy}}\right),\tag{32}$$

and, using separation variables, we get

$$\mathbf{V}\_1(\mathbf{x}) - \frac{1}{2} \left( \mathbf{W}^2(\mathbf{x}) - \frac{\mathbf{d}\mathbf{W}}{\mathbf{d}\mathbf{x}} \right) = \mathbf{C}\_0 \tag{33}$$

$$\mathbf{V}\_2(\mathbf{y}) - \frac{1}{2} \left( \mathbf{Z}^2(\mathbf{y}) - \frac{\mathbf{d}\mathbf{Z}}{\mathbf{dy}} \right) = -\mathbf{C}\_0. \tag{34}$$

In general, we find that each potential Vþ<sup>i</sup> satisfies

$$\frac{1}{2}\frac{d^2}{dx^2}\mathbf{u}\_{\mathbf{i}}(\mathbf{x}) + \mathbf{V}\_{+\mathbf{i}}\mathbf{u}\_{\mathbf{i}}(\mathbf{x}) = \mathbf{E}\_{\mathbf{i}}\mathbf{u}\_{\mathbf{i}}(\mathbf{x}), \qquad \mathbf{i} = \mathbf{1}, \mathbf{2}, \tag{35}$$

and we can find the isopotential as W ¼ � <sup>1</sup> u1 du1 dx , when u1 is known.

Following the same steps as in the 1D case, we find that the solutions (22) are the same in this case. So, the general solution for W is ^ <sup>W</sup>^ <sup>¼</sup> <sup>W</sup> <sup>þ</sup> <sup>1</sup> y xð Þ, with <sup>y</sup> <sup>¼</sup> <sup>u</sup>�<sup>2</sup> <sup>1</sup> ð Þ <sup>x</sup> <sup>E</sup><sup>1</sup> <sup>þ</sup> <sup>Ð</sup> u2 <sup>1</sup>ð Þ <sup>x</sup> dx � �. The general solution for the superpotential W x ^ ð Þ is

$$\hat{\mathbf{W}} = -\frac{\mathbf{1}}{\mathbf{u}\_1} \frac{\mathbf{du}\_1}{\mathbf{dx}} + \frac{\mathbf{u}\_1^2}{\lambda\_1 + \int \mathbf{u}\_1^2 \mathbf{dx}} = \mathbf{W}\_\mathbf{P} + \frac{\mathbf{d}}{\mathbf{dx}} [\mathbf{Ln}(\lambda\_1 + \mathbf{I}\_1)] \tag{36}$$

where Wp ¼ � <sup>1</sup> u1 du1 dx and <sup>I</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> u2 <sup>1</sup> dx. In the same manner, we have that

$$\hat{Z} = -\frac{\mathbf{1}}{\mathbf{u}\_2} \frac{\mathbf{du}\_2}{\mathbf{dy}} + \frac{\mathbf{u}\_2^2}{\lambda\_2 + \int \mathbf{u}\_2^2 \mathbf{d}\mathbf{y}} = Z\_\mathbf{p} + \frac{\mathbf{d}}{\mathbf{dy}} [\mathbf{L}\mathbf{n}(\lambda\_2 + \mathbf{I}\_2)] \tag{37}$$

with Zp ¼ � <sup>1</sup> u2 su2 dy and I2 <sup>¼</sup> <sup>Ð</sup> u2 <sup>2</sup> dy.

On the other hand, using the Riccati equation, we can build a generalization for the isopotential, using the new potential W, as ^

$$
\hat{\mathbf{V}}\_{+1}(\mathbf{x}, \boldsymbol{\lambda}\_1) = \frac{1}{2} \left( \hat{\mathbf{W}}^2 - \hat{\mathbf{W}}^\prime \right) = \mathbf{V}\_+(\mathbf{x}) - \frac{2\mathbf{u}\_1 \frac{d\mathbf{u}\_1}{d\mathbf{x}}}{\lambda\_1 + \mathbf{I}\_1} + \frac{\mathbf{u}\_1^4}{\left(\lambda\_1 + \mathbf{I}\_1\right)^2}.\tag{38}
$$

For the other coordinate, we have

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

$$
\hat{\mathcal{V}}\_{+2}(\mathbf{y}, \lambda\_2) = \frac{1}{2} \left( \hat{Z}^2 - \frac{d\hat{Z}}{dy} \right) = \mathcal{V}\_{+}(\mathbf{y}) - \frac{2u\_2 \frac{du\_1}{dy}}{\lambda\_2 + I\_2} + \frac{u\_2^4}{\left(\lambda\_2 + I\_2\right)^2}.\tag{39}
$$

The general solutions for u^<sup>i</sup> depend on the initial solutions to the original Schrödinger equations in the variables (x,y), that is, u1 ¼ u1ð Þ x , u2 ¼ u2 y � �, being

$$
\hat{u}\_1(\mathbf{x}, \boldsymbol{\lambda}\_1) = \mathbf{C}\_1(\boldsymbol{\lambda}\_1) \frac{\boldsymbol{u}\_1}{\boldsymbol{\lambda}\_1 + I\_1}, \qquad \hat{u}\_2(\boldsymbol{y}, \boldsymbol{\lambda}\_2) = \mathbf{C}\_2(\boldsymbol{\lambda}\_2) \frac{\boldsymbol{u}\_2}{\boldsymbol{\lambda}\_2 + I\_2}, \tag{40}
$$

where the variables Cið Þ λ<sup>i</sup> have the same properties that gð Þλ obtained in the 1D case.

#### 2.2 Application to cosmological Taub model

The Wheeler-DeWitt equation for the cosmological Taub model is given by

$$\frac{\partial^2 \Psi}{\partial a^2} - \frac{\partial^2 \Psi}{\partial \beta^2} + e^{4a} V(\beta) \Psi = 0 \tag{41}$$

3. Differential approach: Grassmann variables

DOI: http://dx.doi.org/10.5772/intechopen.82254

<sup>Q</sup>^ � <sup>¼</sup> <sup>ψ</sup><sup>μ</sup> <sup>P</sup><sup>μ</sup> <sup>þ</sup> <sup>i</sup>

ψμ, ψ<sup>ν</sup> variables in terms of the Grassmann numbers, as

HS <sup>¼</sup> <sup>1</sup>

numbers, which have the property of θ<sup>i</sup>

<sup>Ψ</sup> <sup>¼</sup> <sup>A</sup><sup>þ</sup> <sup>þ</sup> <sup>B</sup>νθ<sup>ν</sup> <sup>þ</sup>

<sup>2</sup> <sup>Q</sup>^ <sup>þ</sup>

<sup>Ψ</sup> <sup>¼</sup> <sup>A</sup><sup>þ</sup> <sup>þ</sup> <sup>B</sup>0θ<sup>0</sup> <sup>þ</sup> <sup>B</sup>1θ<sup>1</sup> <sup>þ</sup> <sup>A</sup>�θ<sup>0</sup>θ<sup>1</sup>

1 2 ϵμνλC<sup>λ</sup>

; <sup>Q</sup>^ � n o <sup>¼</sup> <sup>H</sup><sup>0</sup> <sup>þ</sup>

nian, □ is the d'Alembertian in three dimension when we have three bosonic independent coordinates, and U q<sup>μ</sup> ð Þ is the potential energy in consideration. The superspace for three-dimensional model becomes q1; q2; q3; θ<sup>0</sup>; θ<sup>1</sup>

where the variables θ<sup>i</sup> are the coordinate in the fermionic space, as the Grassmann

<sup>θ</sup>μθν <sup>þ</sup> <sup>A</sup>�θ<sup>0</sup>θ<sup>1</sup>

where the indices <sup>μ</sup>, <sup>ν</sup>, <sup>λ</sup> values are 0, 1, and 2 and <sup>A</sup>�,B<sup>ν</sup> and <sup>C</sup><sup>λ</sup> are bosonic functions which depend on the bosonic coordinates q<sup>μ</sup> and not on the Grassmann numbers. Here, the wave function representation structure is set in terms of 2<sup>n</sup>

<sup>θ</sup><sup>j</sup> ¼ �θ<sup>j</sup>

θi

θ2

<sup>Ψ</sup> <sup>¼</sup> <sup>A</sup><sup>þ</sup> <sup>þ</sup> <sup>B</sup>0θ<sup>0</sup>, 1 dimension (50)

<sup>ψ</sup><sup>μ</sup> <sup>¼</sup> <sup>η</sup>μν <sup>∂</sup>

tion for the fermionic operator ^

changing the function W ! <sup>∂</sup><sup>S</sup>

(similar to Eq. (6))

Hamiltonian is written as

representation

83

The supersymmetric scheme has the particularity of being very restrictive, because there are many constraint equations applied to the wave function. So, in this work and in others, we found that there exist a tendency for supersymmetric vacua to remain close to their semiclassical limits, because the exact solutions found are also the lowest-order WKB-like approximations and do not correspond to the

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

Maintaining the structure of Eqs. (13)–(16), taking the differential representa-

� �, <sup>Q</sup>^ <sup>þ</sup> <sup>¼</sup> <sup>ψ</sup><sup>ν</sup> <sup>P</sup><sup>ν</sup> � <sup>i</sup>

<sup>ψ</sup>μ; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> ημν, <sup>ψ</sup>μ; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> <sup>0</sup>, <sup>ψ</sup>μ; <sup>ψ</sup><sup>ν</sup> f g <sup>¼</sup> <sup>0</sup>: (47)

<sup>∂</sup>θ<sup>ν</sup> , <sup>ψ</sup><sup>ν</sup> <sup>¼</sup> <sup>θ</sup><sup>ν</sup>

ℏ 2 ∂2 S

where ημν is a diagonal constant matrix, its dimensions depending on the independent bosonic variables that appear in the bosonic Hamiltonian. Now the super-

where <sup>H</sup><sup>0</sup> <sup>¼</sup> □ <sup>þ</sup> U q<sup>μ</sup> ð Þ is the quantum version of the classical bosonic Hamilto-

where S is known as the superpotential functions which are related to the physical potential under consideration, when the Hamiltonian density is written as the Hamilton-Jacobi equation, and the following algebra for the variables ψ<sup>μ</sup> and ψ<sup>ν</sup>

These rules are satisfied when we use a differential representation for these

<sup>b</sup> \$ <sup>ψ</sup><sup>μ</sup> for convenience in the calculations, and

<sup>∂</sup>q<sup>μ</sup>, the supercharges for the n-dimensional case read as

∂S ∂q<sup>ν</sup>

� �, (46)

, (48)

<sup>∂</sup>qμ∂q<sup>ν</sup> <sup>ψ</sup>μ; <sup>ψ</sup><sup>ν</sup> ½ �, (49)

; θ<sup>2</sup> � �,

, and the wave function has the

, 2 dimensions (51)

, 3 dimensions (52)

full quantum solutions found previously for particular models [14–18].

∂S ∂q<sup>μ</sup>

where Vð Þ¼ <sup>β</sup> <sup>1</sup> <sup>3</sup> <sup>e</sup>�8<sup>β</sup> � 4e�2<sup>β</sup> � �. These equations can be separated using x1 ¼ 4α � 8β and x2 ¼ 4α � 2β, rendering

$$-\frac{\partial^2 \mathbf{f}\_1(\mathbf{x}\_1)}{\partial \mathbf{x}\_1^2} + \frac{1}{144} e^{\mathbf{x}\_1} \mathbf{f}\_1(\mathbf{x}\_1) = \frac{\alpha^2}{4} f\_1(\mathbf{x}\_1), \qquad -\frac{\partial^2 \mathbf{f}\_2(\mathbf{x}\_2)}{\partial \mathbf{x}\_2^2} + \frac{1}{9} \mathbf{e}^{\mathbf{x}\_2} \mathbf{f}\_2(\mathbf{x}\_2) = \alpha^2 \mathbf{f}\_2(\mathbf{x}\_2), \tag{42}$$

where the parameter ω is the separation constant. These equations possess the solutions

$$\mathbf{f}\_1 = \mathbf{K}\_{\mathrm{iso}} \left( \frac{\mathbf{1}}{\mathbf{6}} \mathbf{e}^{\frac{\mathbf{v}\_1}{\mathbf{7}}} \right), \qquad \mathbf{f}\_2 = \mathbf{L}\_{2\mathrm{iso}} \left( \frac{\mathbf{2}}{\mathbf{3}} \mathbf{e}^{\frac{\mathbf{v}\_1}{\mathbf{7}}} \right) + \mathbf{K}\_{2\mathrm{iso}} \left( \frac{\mathbf{2}}{\mathbf{3}} \mathbf{e}^{\frac{\mathbf{v}\_2}{\mathbf{7}}} \right) \tag{43}$$

where K (or I) is the modified Bessel function of imaginary order and the function L is defined as

$$L\_{2ia} = \frac{\pi i}{2\sinh\left(2a\sigma\right)} \left(I\_{2ia} + I\_{-2ia}\right).$$

Using Eqs. (38) and (39), we obtain the isopotential for this model

$$\begin{split} \hat{\mathbf{V}}(\mathbf{x}\_{1}) &= \mathbf{V}\_{+}(\mathbf{x}\_{1}) - \frac{2\mathbf{K}\_{iio}\mathbf{K}\_{iio}'}{\lambda\_{1} + \mathbf{I}\_{1}} + \frac{\mathbf{K}\_{iio}^{4}}{\left(\lambda\_{1} + \mathbf{I}\_{1}\right)^{2}}, \\ \hat{\mathbf{V}}(\mathbf{x}\_{2}) &= \mathbf{V}\_{+}(\mathbf{x}\_{2}) - \frac{2(\mathbf{L}\_{2io} + \mathbf{K}\_{2io})\left(\mathbf{L}\_{2io} + \mathbf{K}\_{2io}\right)'}{\lambda\_{2} + \mathbf{I}\_{2}} + \frac{\left(\mathbf{L}\_{2io} + \mathbf{K}\_{2io}\right)^{4}}{\left(\lambda\_{2} - \mathbf{I}\_{2}\right)^{2}}. \end{split} \tag{44}$$

Using Eq. (40) we can obtain general solutions for the functions f <sup>1</sup> and f <sup>2</sup> in the following way

$$\hat{f}\_1 = \frac{C\_1 K\_{i\alpha} \left(\frac{1}{6} \boldsymbol{e}^{\frac{\pi}{2}}\right)}{\lambda\_1 + I\_1}, \qquad \hat{f}\_2 = \frac{C\_2 \left[L\_{2i\alpha} \left(\frac{2}{3} \boldsymbol{e}^{\frac{\pi}{2}}\right) + K\_{2i\alpha} \left(\frac{2}{3} \boldsymbol{e}^{\frac{\pi}{2}}\right)\right]}{\lambda\_2 + I\_2}. \tag{45}$$

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

#### 3. Differential approach: Grassmann variables

<sup>V</sup>^ <sup>þ</sup>2ð Þ¼ <sup>y</sup>; <sup>λ</sup><sup>2</sup>

u^1ð Þ¼ x; λ<sup>1</sup> C1ð Þ λ<sup>1</sup>

where Vð Þ¼ <sup>β</sup> <sup>1</sup>

f <sup>1</sup>ð Þ x1 ∂x<sup>2</sup> 1 þ 1 144 e

function L is defined as

V x ^ ð Þ¼ <sup>1</sup> <sup>V</sup>þð Þ� x1

V x ^ ð Þ¼ <sup>2</sup> <sup>V</sup>þð Þ� x2

C1Ki<sup>ω</sup> <sup>1</sup> 6 e x1 2 � �

λ<sup>1</sup> þ I<sup>1</sup>

following way

82

^<sup>f</sup> <sup>1</sup> <sup>¼</sup>

� ∂2

solutions

1 <sup>2</sup> <sup>Z</sup>^<sup>2</sup>

2.2 Application to cosmological Taub model

x1 ¼ 4α � 8β and x2 ¼ 4α � 2β, rendering

f1 ¼ Ki<sup>ω</sup>

x1 f1ð Þ¼ x1

1 6 e x1 2 � � ω2

<sup>L</sup>2i<sup>ω</sup> <sup>¼</sup> <sup>π</sup><sup>i</sup>

2Ki<sup>ω</sup>K<sup>0</sup> iω λ<sup>1</sup> þ I1

� dZ^ dy

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

u1 λ<sup>1</sup> þ I<sup>1</sup>

∂2 Ψ <sup>∂</sup>α<sup>2</sup> � <sup>∂</sup><sup>2</sup>

¼ Vþð Þ� y

The general solutions for u^<sup>i</sup> depend on the initial solutions to the original Schrödinger equations in the variables (x,y), that is, u1 ¼ u1ð Þ x , u2 ¼ u2 y

2u<sup>2</sup> du<sup>2</sup> dy λ<sup>2</sup> þ I<sup>2</sup>

, u^2ð Þ¼ y; λ<sup>2</sup> C2ð Þ λ<sup>2</sup>

<sup>3</sup> <sup>e</sup>�8<sup>β</sup> � 4e�2<sup>β</sup> � �. These equations can be separated using

f <sup>2</sup>ð Þ x2 ∂x<sup>2</sup> 2 þ 1

2 3 e x2 2 � �

ð Þ I2i<sup>ω</sup> þ I�2i<sup>ω</sup> :

þ K2i<sup>ω</sup>

2 3 e x2 2 � �

<sup>þ</sup> ð Þ L2i<sup>ω</sup> <sup>þ</sup> K2i<sup>ω</sup> <sup>4</sup> ð Þ λ<sup>2</sup> � I2

<sup>þ</sup> <sup>K</sup>2i<sup>ω</sup> <sup>2</sup> 3 e x2 2

h i � �

λ<sup>2</sup> þ I<sup>2</sup>

<sup>2</sup> :

where the variables Cið Þ λ<sup>i</sup> have the same properties that gð Þλ obtained in the 1D case.

The Wheeler-DeWitt equation for the cosmological Taub model is given by

<sup>4</sup> <sup>f</sup> <sup>1</sup>ð Þ x1 , � <sup>∂</sup><sup>2</sup>

where the parameter ω is the separation constant. These equations possess the

, f <sup>2</sup> ¼ L2i<sup>ω</sup>

2sinh 2ð Þ ωπ

K4 iω ð Þ λ<sup>1</sup> þ I1

2 Lð Þ <sup>2</sup>i<sup>ω</sup> þ K2i<sup>ω</sup> ð Þ L2i<sup>ω</sup> þ K2i<sup>ω</sup> <sup>0</sup> λ<sup>2</sup> þ I2

2 ,

Using Eq. (40) we can obtain general solutions for the functions f <sup>1</sup> and f <sup>2</sup> in the

C<sup>2</sup> L2i<sup>ω</sup> <sup>2</sup>

3 e x2 2 � �

Using Eqs. (38) and (39), we obtain the isopotential for this model

þ

, ^<sup>f</sup> <sup>2</sup> <sup>¼</sup>

where K (or I) is the modified Bessel function of imaginary order and the

Ψ <sup>∂</sup>β<sup>2</sup> <sup>þ</sup> <sup>e</sup> þ

u4 2 ð Þ λ<sup>2</sup> þ I<sup>2</sup>

> u2 λ<sup>2</sup> þ I<sup>2</sup>

<sup>4</sup>αVð Þ <sup>β</sup> <sup>Ψ</sup> <sup>¼</sup> <sup>0</sup> (41)

<sup>9</sup> ex2 <sup>f</sup> <sup>2</sup>ð Þ¼ x2 <sup>ω</sup><sup>2</sup>

<sup>2</sup> : (39)

, (40)

� �, being

f2ð Þ x2 ,

(42)

(43)

(44)

: (45)

!

The supersymmetric scheme has the particularity of being very restrictive, because there are many constraint equations applied to the wave function. So, in this work and in others, we found that there exist a tendency for supersymmetric vacua to remain close to their semiclassical limits, because the exact solutions found are also the lowest-order WKB-like approximations and do not correspond to the full quantum solutions found previously for particular models [14–18].

Maintaining the structure of Eqs. (13)–(16), taking the differential representation for the fermionic operator ^ <sup>b</sup> \$ <sup>ψ</sup><sup>μ</sup> for convenience in the calculations, and changing the function W ! <sup>∂</sup><sup>S</sup> <sup>∂</sup>q<sup>μ</sup>, the supercharges for the n-dimensional case read as

$$
\hat{\mathbf{Q}}^{-} = \boldsymbol{\Psi}^{\mu} \left[ \mathbf{P}\_{\mu} + \mathbf{i} \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\mu}} \right], \qquad \hat{\mathbf{Q}}^{+} = \overline{\boldsymbol{\Psi}}^{\nu} \left[ \mathbf{P}\_{\nu} - \mathbf{i} \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\nu}} \right], \tag{46}
$$

where S is known as the superpotential functions which are related to the physical potential under consideration, when the Hamiltonian density is written as the Hamilton-Jacobi equation, and the following algebra for the variables ψ<sup>μ</sup> and ψ<sup>ν</sup> (similar to Eq. (6))

$$\{\psi^{\mu}, \overline{\psi}^{\nu}\} = \eta^{\mu \nu}, \qquad \{\psi^{\mu}, \psi^{\nu}\} = 0, \qquad \{\overline{\psi}^{\mu}, \overline{\psi}^{\nu}\} = 0. \tag{47}$$

These rules are satisfied when we use a differential representation for these ψμ, ψ<sup>ν</sup> variables in terms of the Grassmann numbers, as

$$
\eta^{\mu} = \eta^{\mu\nu} \frac{\partial}{\partial \theta^{\nu}}, \qquad \qquad \qquad \overline{\eta^{\nu}} = \theta^{\nu}, \tag{48}
$$

where ημν is a diagonal constant matrix, its dimensions depending on the independent bosonic variables that appear in the bosonic Hamiltonian. Now the super-Hamiltonian is written as

$$H\_S = \frac{1}{2} \left\{ \hat{\boldsymbol{Q}}^+, \hat{\boldsymbol{Q}}^- \right\} = \mathcal{H}\_0 + \frac{\hbar}{2} \frac{\partial^2 \mathcal{S}}{\partial q^\mu \partial q^\nu} [\overline{\boldsymbol{\eta}}^\mu, \boldsymbol{\upmu}^\nu], \tag{49}$$

where <sup>H</sup><sup>0</sup> <sup>¼</sup> □ <sup>þ</sup> U q<sup>μ</sup> ð Þ is the quantum version of the classical bosonic Hamiltonian, □ is the d'Alembertian in three dimension when we have three bosonic independent coordinates, and U q<sup>μ</sup> ð Þ is the potential energy in consideration.

The superspace for three-dimensional model becomes q1; q2; q3; θ<sup>0</sup>; θ<sup>1</sup> ; θ<sup>2</sup> � �, where the variables θ<sup>i</sup> are the coordinate in the fermionic space, as the Grassmann numbers, which have the property of θ<sup>i</sup> <sup>θ</sup><sup>j</sup> ¼ �θ<sup>j</sup> θi , and the wave function has the representation

$$
\Psi = \mathcal{A}\_{+} + \mathcal{B}\_{0} \theta^{0}, \qquad \mathbf{1} \text{ dimension} \tag{50}
$$

$$\Psi = \mathcal{A}\_{+} + \mathcal{B}\_{0}\theta^{0} + \mathcal{B}\_{1}\theta^{1} + \mathcal{A}\_{-}\theta^{0}\theta^{1}, \qquad \text{2 dimensions} \tag{51}$$

$$
\Psi = \mathcal{A}\_{+} + \mathcal{B}\_{\nu}\theta^{\nu} + \frac{1}{2}\epsilon\_{\mu\nu\lambda}\mathcal{C}^{\lambda}\theta^{\mu}\theta^{\nu} + \mathcal{A}\_{-}\theta^{0}\theta^{1}\theta^{2}, \qquad \text{3 dimensions} \tag{52}
$$

where the indices <sup>μ</sup>, <sup>ν</sup>, <sup>λ</sup> values are 0, 1, and 2 and <sup>A</sup>�, <sup>B</sup><sup>ν</sup> and <sup>C</sup><sup>λ</sup> are bosonic functions which depend on the bosonic coordinates q<sup>μ</sup> and not on the Grassmann numbers. Here, the wave function representation structure is set in terms of 2<sup>n</sup>

#### Panorama of Contemporary Quantum Mechanics - Concepts and Applications

components, for n independent bosonic coordinates, with half of the terms coming from the bosonic (fermionic) contribution into the wave function.

It is well-known that the physical states are determined by the applications of the supercharges Q^ � and Q^ <sup>þ</sup> on the wave functions, that is,

$$
\hat{\mathbf{Q}}\_{-}\hat{\mathbf{A}}=\mathbf{0}, \qquad \hat{\mathbf{Q}}\_{+}\hat{\mathbf{A}}=\mathbf{0},\tag{53}
$$

is well-known [21], and here we present the main ideas. Let Ψ<sup>1</sup> and Ψ<sup>2</sup> be two functions that depend on Grassmann numbers; the product < Ψ1, Ψ<sup>2</sup> > is defined as

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

<sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup>0<sup>θ</sup> <sup>∗</sup> <sup>0</sup> <sup>þ</sup> <sup>θ</sup><sup>1</sup>

<sup>0</sup> <sup>B</sup><sup>0</sup> <sup>þ</sup> <sup>B</sup><sup>∗</sup>

By demanding that j j <sup>Ψ</sup> <sup>2</sup> does not diverge when <sup>∣</sup>q0∣, <sup>∣</sup>q1<sup>∣</sup> ! <sup>∞</sup>, only the contri-

Although most of the SUSY partners of 1D Schrödinger problems have been found [1], there are still some unveiled aspects of the factorization procedure. We have shown this for the simple harmonic oscillator in previous works [2, 3] and will proceed here in the same way for the problem of the modified Pöschl-Teller potential. The factorization operators depend on two supersymmetric type parameters, which when the operator product is inverted, allow us to define a new SL operator,

The Hamiltonian of a particle in a modified Pöschl-Teller potential is [1, 22]

!

where α >0 and the integer m is greater than 0. To shorten the algebraic equa-

The eigenvalue problem may be solved using the Infeld and Hull's (IH) factor-

<sup>m</sup>�<sup>n</sup> <sup>¼</sup> ð Þ Hmþ<sup>1</sup> <sup>þ</sup> <sup>ϵ</sup><sup>m</sup>þ<sup>1</sup> <sup>ψ</sup> <sup>m</sup>

d

<sup>m</sup>�<sup>n</sup> <sup>¼</sup> ð Þ Hmþ<sup>1</sup> <sup>þ</sup> <sup>ϵ</sup><sup>m</sup> <sup>ψ</sup> <sup>m</sup>

<sup>m</sup> ¼ k xð Þ ; m ∓

and where k xð Þ¼ ; <sup>m</sup> <sup>α</sup><sup>m</sup> tanh <sup>α</sup>x; also <sup>ϵ</sup><sup>m</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup>m2, and <sup>n</sup> is the eigenvalue index,

ð Þ m � n 2

dx<sup>2</sup> � <sup>α</sup><sup>2</sup>m mð Þ <sup>þ</sup> <sup>1</sup> cosh <sup>2</sup>αx

<sup>i</sup> <sup>d</sup>θi, Cð Þ <sup>θ</sup>i⋯θ<sup>r</sup> <sup>∗</sup> <sup>¼</sup> <sup>θ</sup> <sup>∗</sup>

mdθ <sup>∗</sup>

θ <sup>∗</sup> <sup>1</sup> ,

Ψ ¼ EΨ , (62)

<sup>m</sup>�n, (63)

<sup>m</sup>�n, (64)

dx (65)

, n ¼ 0; 1; 2… < m: (66)

<sup>i</sup> θi⋯θmθ <sup>∗</sup>

<sup>i</sup> <sup>θ</sup><sup>i</sup> come from

<sup>θ</sup> <sup>∗</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup>0<sup>θ</sup> <sup>∗</sup> <sup>0</sup>θ<sup>1</sup>

dθ ¼ 0, which act as a filter, we obtain that

�A�:

<sup>1</sup> <sup>B</sup><sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>∗</sup>

θ ∗

θ ∗

<sup>r</sup> ⋯θ <sup>∗</sup> <sup>i</sup> C<sup>∗</sup> ,

<sup>i</sup> dθ<sup>i</sup> ¼ 1.

mdθm⋯dθ <sup>∗</sup>

�∑<sup>i</sup> θ ∗ <sup>i</sup> θ<sup>i</sup> Πidθ <sup>∗</sup>

<sup>þ</sup>A<sup>þ</sup> <sup>þ</sup> <sup>B</sup><sup>∗</sup>

< Ψ1, Ψ<sup>2</sup> > ¼

ð

DOI: http://dx.doi.org/10.5772/intechopen.82254

e �∑i<sup>θ</sup> <sup>∗</sup>

and using that Ð

<sup>Ψ</sup><sup>1</sup> <sup>θ</sup> <sup>∗</sup> ð Þ ð Þ <sup>∗</sup> <sup>Ψ</sup><sup>2</sup> <sup>θ</sup> <sup>∗</sup> ð Þ <sup>e</sup>

and the integral over the Grassmann numbers isÐ

In 2D, the main contributions to the term e�∑<sup>i</sup>

�∑<sup>i</sup> θiθ <sup>∗</sup>

<sup>θ</sup>d<sup>θ</sup> <sup>¼</sup> 1, and <sup>Ð</sup>

j j <sup>Ψ</sup> <sup>2</sup> <sup>¼</sup> <sup>A</sup><sup>∗</sup>

<sup>i</sup> <sup>θ</sup><sup>i</sup> <sup>¼</sup> <sup>e</sup>

bution with the exponential e�2S will remain.

4. Beyond SUSY factorization

which includes the original QM problem.

<sup>2</sup><sup>μ</sup> ¼ 1.

A<sup>þ</sup> <sup>m</sup>þ<sup>1</sup>A�

> A� mA<sup>þ</sup> <sup>m</sup> ψ <sup>m</sup>

where the IH raising/lowering operators are given by

<sup>m</sup>�n, En ¼ �ϵ<sup>m</sup>�<sup>n</sup> ¼ �α<sup>2</sup>

A<sup>∓</sup>

tions, we shall set <sup>ℏ</sup><sup>2</sup>

<sup>Ψ</sup><sup>n</sup> <sup>¼</sup> <sup>ψ</sup> <sup>m</sup>

85

izations [23],

Hmþ<sup>1</sup><sup>Ψ</sup> ¼ � <sup>ℏ</sup><sup>2</sup>

2μ d2

<sup>m</sup>þ<sup>1</sup> <sup>ψ</sup> <sup>m</sup>

where we use the usual representation for the momentum P<sup>μ</sup> ¼ �<sup>i</sup> <sup>∂</sup> <sup>∂</sup>q<sup>μ</sup>. Considering the 2D case, the last second equation gives

$$\theta^0 : \left[\frac{\partial A\_+}{\partial q^0} - A\_+ \frac{\partial \mathbb{S}}{\partial q^0}\right] = \mathbf{0},\tag{54}$$

$$\theta^1: \left[\frac{\partial A\_+}{\partial q^1} - A\_+ \frac{\partial \mathbb{S}}{\partial q^1}\right] = 0,\tag{55}$$

$$\partial^0 \theta^1: \left[\frac{\partial B\_1}{\partial q^0} - B\_1 \frac{\partial \mathcal{S}}{\partial q^0}\right] - \left[\frac{\partial B\_0}{\partial q^1} - B\_0 \frac{\partial \mathcal{S}}{\partial q^1}\right] = \mathbf{0}.\tag{56}$$

From (54)–(55), we obtain the relation <sup>∂</sup>A<sup>þ</sup> <sup>∂</sup>q<sup>μ</sup> � A<sup>þ</sup> ∂S <sup>∂</sup>q<sup>μ</sup> ¼ 0 with the solution <sup>A</sup><sup>þ</sup> <sup>¼</sup> <sup>a</sup>þe<sup>S</sup>:

On the other hand, the first equation in (53) gives

$$\theta^0 : \left[ \frac{\partial A\_-}{\partial q^1} + A\_- \frac{\partial S}{\partial q^1} \right] = 0,\tag{57}$$

$$\theta^1: \left[\frac{\partial A\_-}{\partial q^0} + A\_- \frac{\partial \mathbb{S}}{\partial q^0}\right] = \mathbf{0},\tag{58}$$

$$\left[\text{free term}: -\left[\frac{\partial B\_0}{\partial q^0} + B\_0 \frac{\partial S}{\partial q^0}\right] + \left[\frac{\partial B\_1}{\partial q^1} + B\_1 \frac{\partial S}{\partial q^1}\right] = 0.\right] \tag{59}$$

The free term equation is written as ημν <sup>∂</sup>μB<sup>ν</sup> <sup>þ</sup> <sup>B</sup>ν∂μ<sup>S</sup> <sup>¼</sup> 0, and taking the ansatz <sup>B</sup><sup>μ</sup> <sup>¼</sup> <sup>e</sup>�<sup>S</sup>∂ν<sup>f</sup> <sup>þ</sup> <sup>q</sup><sup>μ</sup> ð Þ, Eq. (56) is fulfilled, so we obtain for the free term,

$$
\Box \mathbf{f}\_{+} + 2\eta^{\mu\nu} \nabla\_{\mu} \mathbf{S} \nabla\_{\nu} \mathbf{f}\_{+} = \mathbf{0},\tag{60}
$$

with the solution to <sup>f</sup> <sup>þ</sup> <sup>¼</sup> h q<sup>1</sup> � <sup>q</sup><sup>2</sup> ð Þ, with h an arbitrary function depending of its argument. However, this function f must depend on the potential under consideration.

Also, Eqs. (57) and (58) are written as

$$\frac{\partial \mathbf{A}\_{-}}{\partial \mathbf{q}^{\mu}} + \mathbf{A}\_{-} \frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\mu}} = \mathbf{0}, \qquad \frac{\mathbf{1}}{\mathbf{A}\_{-}} \frac{\partial \mathbf{A}\_{-}}{\partial \mathbf{q}^{\mu}} = -\frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\mu}} \quad \rightarrow \quad \frac{\partial \mathbf{L} \mathbf{n} \mathbf{A}\_{-}}{\partial \mathbf{q}^{\mu}} = -\frac{\partial \mathbf{S}}{\partial \mathbf{q}^{\mu}} \tag{61}$$

whose solution is A� <sup>¼</sup> <sup>a</sup>�e�S. In this way, all functions entering the wave function are

$$\mathbf{A}\_{\pm} = \mathbf{a}\_{\pm} \mathbf{e}^{\pm \mathbf{S}}, \qquad \mathbf{B}\_0 = \mathbf{e}^{-\mathbf{S}} \partial\_0(\mathbf{f}\_+), \qquad \mathbf{B}\_1 = \mathbf{e}^{-\mathbf{S}} \partial\_1(\mathbf{f}\_+).$$

#### 3.1 The unnormalized probability density

To obtain the wave function probability density j j <sup>Ψ</sup> <sup>2</sup> in this supersymmetric fashion, we need first to integrate over the Grassmann variables θ<sup>i</sup> . This procedure is well-known [21], and here we present the main ideas. Let Ψ<sup>1</sup> and Ψ<sup>2</sup> be two functions that depend on Grassmann numbers; the product < Ψ1, Ψ<sup>2</sup> > is defined as

$$<\Psi\_1, \Psi\_2> = \int \left(\Psi\_1(\theta^\*)\right)^\* \Psi\_2(\theta^\*) \, e^{-\sum\_i \theta\_i^\* \cdot \theta\_i} \Pi\_i d\theta\_i^\* \, d\theta\_i, \qquad \left(\mathcal{C}\theta\_i \cdots \theta\_r\right)^\* = \theta\_r^\* \cdots \theta\_i^\* \, \mathcal{C}^\*,$$

and the integral over the Grassmann numbers isÐ θ ∗ <sup>i</sup> θi⋯θmθ <sup>∗</sup> mdθ <sup>∗</sup> mdθm⋯dθ <sup>∗</sup> <sup>i</sup> dθ<sup>i</sup> ¼ 1. In 2D, the main contributions to the term e�∑<sup>i</sup> θ ∗ <sup>i</sup> <sup>θ</sup><sup>i</sup> come from

$$e^{-\sum\_{i} \theta\_i^\* \theta\_i} = e^{-\sum\_{i} \theta\_i \theta\_i^\*} = \mathbf{1} + \theta^0 \theta^{\*0} + \theta^1 \theta^{\*1} + \theta^0 \theta^{\*0} \theta^1 \theta^{\*1},$$

and using that Ð <sup>θ</sup>d<sup>θ</sup> <sup>¼</sup> 1, and <sup>Ð</sup> dθ ¼ 0, which act as a filter, we obtain that

$$|\Psi|^2 = \mathcal{A}\_+^\* \mathcal{A}\_+ + \mathcal{B}\_0^\* \mathcal{B}\_0 + \mathcal{B}\_1^\* \mathcal{B}\_1 + \mathcal{A}\_-^\* \mathcal{A}\_-.$$

By demanding that j j <sup>Ψ</sup> <sup>2</sup> does not diverge when <sup>∣</sup>q0∣, <sup>∣</sup>q1<sup>∣</sup> ! <sup>∞</sup>, only the contribution with the exponential e�2S will remain.

#### 4. Beyond SUSY factorization

components, for n independent bosonic coordinates, with half of the terms coming

It is well-known that the physical states are determined by the applications of

Ψ ¼ 0, (53)

¼ 0, (54)

¼ 0, (55)

<sup>∂</sup>q<sup>μ</sup> ¼ 0 with the solution

¼ 0, (57)

¼ 0, (58)

<sup>∂</sup>q<sup>μ</sup> ¼ � <sup>∂</sup><sup>S</sup>

¼ 0: (59)

<sup>∂</sup>q<sup>μ</sup> (61)

. This procedure

∂S ∂q<sup>1</sup>

¼ 0: (56)

<sup>∂</sup>q<sup>μ</sup>. Consid-

<sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>, <sup>Q</sup>^ <sup>þ</sup>

where we use the usual representation for the momentum P<sup>μ</sup> ¼ �<sup>i</sup> <sup>∂</sup>

<sup>∂</sup>q<sup>0</sup> � <sup>A</sup><sup>þ</sup>

<sup>∂</sup>q<sup>1</sup> � <sup>A</sup><sup>þ</sup>

<sup>∂</sup>q<sup>1</sup> <sup>þ</sup> <sup>A</sup>�

<sup>∂</sup>q<sup>0</sup> <sup>þ</sup> <sup>A</sup>�

∂S ∂q<sup>0</sup>

∂S ∂q<sup>0</sup>

∂S ∂q<sup>0</sup>

∂S ∂q<sup>1</sup>

� <sup>∂</sup>B<sup>0</sup>

<sup>∂</sup>q<sup>1</sup> � <sup>B</sup><sup>0</sup>

<sup>∂</sup>q<sup>μ</sup> � A<sup>þ</sup>

∂S ∂q<sup>1</sup>

∂S ∂q<sup>0</sup>

þ

The free term equation is written as ημν <sup>∂</sup>μB<sup>ν</sup> <sup>þ</sup> <sup>B</sup>ν∂μ<sup>S</sup> <sup>¼</sup> 0, and taking the ansatz

with the solution to <sup>f</sup> <sup>þ</sup> <sup>¼</sup> h q<sup>1</sup> � <sup>q</sup><sup>2</sup> ð Þ, with h an arbitrary function depending of its argument. However, this function f must depend on the potential under consideration.

whose solution is A� <sup>¼</sup> <sup>a</sup>�e�S. In this way, all functions entering the wave

To obtain the wave function probability density j j <sup>Ψ</sup> <sup>2</sup> in this supersymmetric

∂B<sup>1</sup> <sup>∂</sup>q<sup>1</sup> <sup>þ</sup> <sup>B</sup><sup>1</sup>

□f<sup>þ</sup> <sup>þ</sup> <sup>2</sup>ημν∇μS∇νf<sup>þ</sup> <sup>¼</sup> <sup>0</sup>, (60)

<sup>∂</sup>q<sup>μ</sup> ! <sup>∂</sup>LnA�

, B0 <sup>¼</sup> <sup>e</sup>�<sup>S</sup>∂0ð Þ <sup>f</sup><sup>þ</sup> , B1 <sup>¼</sup> <sup>e</sup>�<sup>S</sup>∂1ð Þ <sup>f</sup><sup>þ</sup> :

∂S

∂S ∂q<sup>1</sup>

from the bosonic (fermionic) contribution into the wave function.

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

<sup>θ</sup><sup>0</sup> : <sup>∂</sup>A<sup>þ</sup>

<sup>θ</sup><sup>1</sup> : <sup>∂</sup>A<sup>þ</sup>

<sup>θ</sup><sup>0</sup> : <sup>∂</sup>A�

<sup>θ</sup><sup>1</sup> : <sup>∂</sup>A�

<sup>∂</sup>q<sup>0</sup> <sup>þ</sup> <sup>B</sup><sup>0</sup>

<sup>B</sup><sup>μ</sup> <sup>¼</sup> <sup>e</sup>�<sup>S</sup>∂ν<sup>f</sup> <sup>þ</sup> <sup>q</sup><sup>μ</sup> ð Þ, Eq. (56) is fulfilled, so we obtain for the free term,

A�

fashion, we need first to integrate over the Grassmann variables θ<sup>i</sup>

<sup>∂</sup>A� <sup>∂</sup>q<sup>μ</sup> ¼ � <sup>∂</sup><sup>S</sup>

<sup>∂</sup>q<sup>0</sup> � <sup>B</sup><sup>1</sup>

the supercharges Q^ � and Q^ <sup>þ</sup> on the wave functions, that is,

Q^ �

ering the 2D case, the last second equation gives

<sup>θ</sup><sup>0</sup>θ<sup>1</sup> : <sup>∂</sup>B<sup>1</sup>

From (54)–(55), we obtain the relation <sup>∂</sup>A<sup>þ</sup>

free term : � <sup>∂</sup>B<sup>0</sup>

Also, Eqs. (57) and (58) are written as

<sup>∂</sup>q<sup>μ</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>

∂S

<sup>A</sup>� <sup>¼</sup> <sup>a</sup>�e�<sup>S</sup>

3.1 The unnormalized probability density

On the other hand, the first equation in (53) gives

<sup>A</sup><sup>þ</sup> <sup>¼</sup> <sup>a</sup>þe<sup>S</sup>:

<sup>∂</sup>A� <sup>∂</sup>q<sup>μ</sup> <sup>þ</sup> <sup>A</sup>�

function are

84

Although most of the SUSY partners of 1D Schrödinger problems have been found [1], there are still some unveiled aspects of the factorization procedure. We have shown this for the simple harmonic oscillator in previous works [2, 3] and will proceed here in the same way for the problem of the modified Pöschl-Teller potential. The factorization operators depend on two supersymmetric type parameters, which when the operator product is inverted, allow us to define a new SL operator, which includes the original QM problem.

The Hamiltonian of a particle in a modified Pöschl-Teller potential is [1, 22]

$$H\_{m+1}\Psi = \left(-\frac{\hbar^2}{2\mu}\frac{d^2}{d\mathbf{x}^2} - \frac{a^2m(m+1)}{\cosh^2\alpha\mathbf{x}}\right)\Psi = E\Psi\ ,\tag{62}$$

where α >0 and the integer m is greater than 0. To shorten the algebraic equations, we shall set <sup>ℏ</sup><sup>2</sup> <sup>2</sup><sup>μ</sup> ¼ 1.

The eigenvalue problem may be solved using the Infeld and Hull's (IH) factorizations [23],

$$A\_{m+1}^{+}A\_{m+1}^{-}\boldsymbol{\psi}\_{m-n}^{m} = (H\_{m+1} + \epsilon\_{m+1})\boldsymbol{\psi}\_{m-n}^{m},\tag{63}$$

$$A\_m^- A\_m^+ \psi\_{m-n}^m = (H\_{m+1} + \epsilon\_m) \varphi\_{m-n}^m. \tag{64}$$

where the IH raising/lowering operators are given by

$$A\_m^{\mp} = k(\varkappa, m) \mp \frac{d}{d\varkappa} \tag{65}$$

and where k xð Þ¼ ; <sup>m</sup> <sup>α</sup><sup>m</sup> tanh <sup>α</sup>x; also <sup>ϵ</sup><sup>m</sup> <sup>¼</sup> <sup>α</sup><sup>2</sup>m2, and <sup>n</sup> is the eigenvalue index,

$$\Psi\_n = \psi\_{m-n}^m, \quad E\_n = -\epsilon\_{m-n} = -a^2(m-n)^2, \quad n = 0, 1, 2... < m. \tag{66}$$

Beginning with the zeroth-order eigenfunctions, the eigenfunctions can be found by successive applications of the raising operator, which only increases the value of the upper index. That is,

$$
\psi\_{\ell}^{\ell}(\mathbf{x}) = \sqrt{\frac{a\Gamma\left(\ell + \frac{1}{2}\right)}{\sqrt{\pi}\Gamma\left(\ell'\right)}} \cosh^{-\ell} a\mathbf{x}.\tag{67}
$$

determine the γ1; γ<sup>2</sup> ð Þ parameter space. When γ<sup>1</sup> ¼ γ<sup>2</sup> ¼ 0 we recover the original IH

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

Now we invert the first-order operators' product, keeping in mind Eq. (64),

m � � � <sup>α</sup><sup>2</sup>

<sup>m</sup>�<sup>n</sup> � <sup>B</sup><sup>∗</sup>

<sup>m</sup> ð Þ x . This new SL operator is isospectral to the original PT problem. The zeroth-order

> 1 þ γ<sup>1</sup> Ð x

one sets γ<sup>2</sup> ¼ 0, moving along the horizontal axis. In this case, L becomes

ð Þ� <sup>α</sup><sup>x</sup> <sup>2</sup>S<sup>2</sup>

1þγ<sup>1</sup> Ð x <sup>0</sup> sech2<sup>λ</sup>

ð Þþ <sup>α</sup><sup>x</sup> <sup>2</sup>S<sup>2</sup>

The zeroth-order eigenfunction is defined by B�ϕ<sup>0</sup> ¼ 0, that is,

<sup>ϕ</sup><sup>0</sup> <sup>¼</sup> sech<sup>λ</sup>

1 þ γ<sup>1</sup> Ð x <sup>0</sup> sech<sup>2</sup><sup>λ</sup>

We may recover the original QM problem when γ<sup>1</sup> ¼ γ<sup>2</sup> ¼ 0, the origin of the two-parameter space. Moreover, the SUSY partner of the PT problem arises when

ð Þ αx

ð Þ αx

dx<sup>2</sup> <sup>þ</sup> V x <sup>~</sup> ð Þ " #Φ<sup>n</sup> <sup>¼</sup> EnΦnð Þ <sup>x</sup> (78)

dx <sup>þ</sup> <sup>V</sup><sup>0</sup> <sup>þ</sup> <sup>ϵ</sup><sup>m</sup> � <sup>η</sup>mβ<sup>0</sup>

<sup>m</sup> � <sup>β</sup><sup>0</sup> m ηm � �: (73)

<sup>m</sup> <sup>ψ</sup> <sup>m</sup>�<sup>1</sup> <sup>m</sup>�n, (75)

� �Φ<sup>0</sup> <sup>¼</sup> 0 which gives

<sup>0</sup> sech<sup>2</sup>ð Þ <sup>m</sup>þ<sup>1</sup> ð Þ <sup>α</sup><sup>y</sup> dy : (76)

<sup>1</sup>ð Þ� αx 4αλ tanh ð Þ αx S1ð Þ αx (77)

αy dy, and <sup>ω</sup>ð Þ¼ <sup>x</sup> 1. These in turn

<sup>1</sup>ð Þþ αx 4αλ tanh ð Þ αx S1ð Þ αx : (79)

ð Þ <sup>α</sup><sup>y</sup> dy : (80)

ð Þ αx (74)

m mð Þ <sup>þ</sup> <sup>1</sup> sech2

dx þ βmη<sup>m</sup>

ð Þ αx

sech<sup>m</sup>þ<sup>1</sup>

4.2 Reversing the operator product: new Sturm-Liouville operator

η0 m ηm d

m � � <sup>1</sup> <sup>þ</sup> <sup>η</sup>�<sup>2</sup>

<sup>Φ</sup><sup>n</sup> <sup>¼</sup> <sup>ϕ</sup><sup>m</sup>

Then we can define a new Sturm-Liouville (SL) eigenvalue problem

raising/lowering operators.

DOI: http://dx.doi.org/10.5772/intechopen.82254

B∗

LΦ<sup>n</sup> þ ωð Þ x EnΦ<sup>n</sup> ¼ 0, where

dx <sup>η</sup>�<sup>2</sup> m d dx � � <sup>þ</sup> <sup>ϵ</sup><sup>m</sup> � <sup>β</sup><sup>2</sup>

<sup>L</sup> <sup>¼</sup> <sup>d</sup>

<sup>L</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

dx<sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

define a SUSY PT problem

<sup>V</sup><sup>~</sup> ¼ �α<sup>2</sup>

87

mBm ¼ � <sup>d</sup><sup>2</sup>

with the weight function <sup>ω</sup>ð Þ¼ <sup>x</sup> <sup>η</sup>�<sup>2</sup>

4.3 Regions in the two-parameter space

eigenfunction is easily found by solving <sup>B</sup>ϕ<sup>0</sup> <sup>¼</sup> <sup>d</sup>

Φ<sup>0</sup> ¼ ηmð Þ� x

λ λð Þ <sup>þ</sup> <sup>1</sup> sech<sup>2</sup>

� d2

where the partner SUSY potentials are given by

λ λð Þ <sup>þ</sup> <sup>1</sup> sech<sup>2</sup>

where <sup>λ</sup> <sup>¼</sup> <sup>m</sup> <sup>þ</sup> 1, with <sup>S</sup>1ð Þ¼ <sup>α</sup><sup>x</sup> <sup>γ</sup><sup>1</sup> sech2<sup>λ</sup>

dx<sup>2</sup> <sup>þ</sup> <sup>2</sup>

We repeatedly apply the creation operator A� <sup>s</sup>þ<sup>1</sup> <sup>ψ</sup><sup>s</sup> <sup>ℓ</sup> <sup>¼</sup> <sup>ψ</sup>sþ<sup>1</sup> <sup>ℓ</sup> . Note that from (63) and (64), A� mA<sup>þ</sup> <sup>m</sup> and A<sup>þ</sup> mA� <sup>m</sup> give different Hamiltonian operators.

#### 4.1 Two-parameter factorization of the Pöschl-Teller Hamiltonian

Following our previous work [2, 3], we define two non-mutually adjoint first-order operators,

$$B\_m = \eta\_m^{-1} \frac{d}{d\mathbf{x}} + \beta\_m, \qquad B\_m^\* = -\eta\_m \frac{d}{d\mathbf{x}} + \beta\_m. \tag{68}$$

where <sup>β</sup><sup>m</sup> and <sup>η</sup><sup>m</sup> are functions of <sup>x</sup>, and we require that Bmþ<sup>1</sup>B<sup>∗</sup> <sup>m</sup>þ<sup>1</sup> ¼ Hmþ<sup>1</sup>þ ϵ<sup>m</sup>þ1. Then β<sup>m</sup>þ<sup>1</sup> and η<sup>m</sup>þ<sup>1</sup> are the solutions of

$$-\frac{\eta'}{\eta} + \frac{\beta}{\eta} - \beta\eta = 0, \qquad \frac{\beta'}{\eta} + \beta^2 = -\frac{a^2 m(m+1)}{\cosh^2 a \infty} + \epsilon. \tag{69}$$

By multiplying the first equation by β=η and adding, we have that

$$
\left(\frac{\beta\_{m+1}}{\eta\_{m+1}}\right) + \left(\frac{\beta\_{m+1}}{\eta\_{m+1}}\right)^2 = -\frac{a^2m(m+1)}{\cosh^2 a\infty} + \epsilon\_{m+1}.\tag{70}
$$

This Riccati equation was found in [24]; it has the solution β=η ¼ D tanh αx, with <sup>ϵ</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> , and two possible values for D, D ¼ αð Þ m þ 1 , � αm. If we simply set η<sup>m</sup> ! 1, we recover the factorization (63).

The constant ϵ is usually related to the lowest energy eigenvalue, but here the two different values come from the index asymmetry in the factorizations (63) and (64). Following Ref. [24], we solve for D ¼ αð Þ m þ 1 .

The general solution to the pair of coupled equations (69) is

$$\eta\_{m+1}(\mathbf{x}) = \left[ 1 + \frac{\chi\_2 \operatorname{sech}^{2(m+1)} \alpha \mathbf{x}}{\left( 1 + \chi\_1 \int\_0^\chi \operatorname{sech}^{2(m+1)} \alpha \mathbf{y} \, d\mathbf{y} \right)^2} \right]^{-1/2},\tag{71}$$

and

$$\beta\_{m+1}(\mathbf{x}) = \left[ a(m+1) \tanh \alpha \mathbf{x} + \frac{\gamma\_1 \operatorname{sech}^{2(m+1)} \alpha \mathbf{x}}{1 + \gamma\_1 \int\_0^{\infty} \operatorname{sech}^{2(m+1)} \alpha \mathbf{y} \, \, d\mathbf{y}} \right] \times \eta\_{m+1}(\mathbf{x}), \tag{72}$$

where <sup>γ</sup><sup>1</sup> has to satisfy <sup>∣</sup>γ1∣<2αΓð Þ <sup>m</sup> <sup>þ</sup> <sup>3</sup>=<sup>2</sup> <sup>=</sup> ffiffiffi π p ð Þ Γð Þ m þ 1 . The corresponding condition on <sup>γ</sup><sup>2</sup> involves transcendental functions, but one may use <sup>γ</sup><sup>2</sup> <sup>&</sup>gt; � <sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>1</sup> to Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

determine the γ1; γ<sup>2</sup> ð Þ parameter space. When γ<sup>1</sup> ¼ γ<sup>2</sup> ¼ 0 we recover the original IH raising/lowering operators.

#### 4.2 Reversing the operator product: new Sturm-Liouville operator

Now we invert the first-order operators' product, keeping in mind Eq. (64),

$$B\_m^\* B\_m = -\frac{d^2}{d\mathfrak{x}^2} + 2\frac{\eta\_m'}{\eta\_m}\frac{d}{d\mathfrak{x}} + \left(V\_0 + \epsilon\_m - \eta\_m \beta\_m' - \frac{\beta\_m'}{\eta\_m}\right). \tag{73}$$

Then we can define a new Sturm-Liouville (SL) eigenvalue problem LΦ<sup>n</sup> þ ωð Þ x EnΦ<sup>n</sup> ¼ 0, where

$$\mathcal{L} = \frac{d}{d\mathbf{x}} \left[ \eta\_m^{-2} \frac{d}{d\mathbf{x}} \right] + \left( \epsilon\_m - \beta\_m^2 \right) \left( \mathbf{1} + \eta\_m^{-2} \right) - a^2 m (m+1) \operatorname{sech}^2(\alpha \mathbf{x}) \tag{74}$$

$$\Phi\_n = \phi\_{m-n}^m \equiv B\_m^\* \,\mu\_{m-n}^{m-1},\tag{75}$$

with the weight function <sup>ω</sup>ð Þ¼ <sup>x</sup> <sup>η</sup>�<sup>2</sup> <sup>m</sup> ð Þ x .

This new SL operator is isospectral to the original PT problem. The zeroth-order eigenfunction is easily found by solving <sup>B</sup>ϕ<sup>0</sup> <sup>¼</sup> <sup>d</sup> dx þ βmη<sup>m</sup> � �Φ<sup>0</sup> <sup>¼</sup> 0 which gives

$$\Phi\_0 = \eta\_m(\mathbf{x}) \times \frac{\text{sech}^{m+1}(a\mathbf{x})}{\mathbf{1} + \gamma\_1 \int\_0^\infty \text{sech}^{2(m+1)}(a\mathbf{y}) d\mathbf{y}}.\tag{76}$$

#### 4.3 Regions in the two-parameter space

We may recover the original QM problem when γ<sup>1</sup> ¼ γ<sup>2</sup> ¼ 0, the origin of the two-parameter space. Moreover, the SUSY partner of the PT problem arises when one sets γ<sup>2</sup> ¼ 0, moving along the horizontal axis. In this case, L becomes

$$\mathcal{L} = \frac{d^2}{d\mathbf{x}^2} + a^2 \lambda(\lambda + 1) \operatorname{sech}^2(a\mathbf{x}) - 2\mathbf{S}\_1^2(a\mathbf{x}) - 4a\lambda \tanh\left(a\mathbf{x}\right)\mathbf{S}\_1(a\mathbf{x})\tag{77}$$

where <sup>λ</sup> <sup>¼</sup> <sup>m</sup> <sup>þ</sup> 1, with <sup>S</sup>1ð Þ¼ <sup>α</sup><sup>x</sup> <sup>γ</sup><sup>1</sup> sech2<sup>λ</sup> ð Þ αx 1þγ<sup>1</sup> Ð x <sup>0</sup> sech2<sup>λ</sup> αy dy, and <sup>ω</sup>ð Þ¼ <sup>x</sup> 1. These in turn define a SUSY PT problem

$$\left[-\frac{d^2}{d\mathbf{x}^2} + \tilde{V}(\mathbf{x})\right] \Phi\_n = E\_n \Phi\_n(\mathbf{x})\tag{78}$$

where the partner SUSY potentials are given by

$$\tilde{V} = -a^2 \lambda (\lambda + \mathbf{1}) \operatorname{sech}^2(a\mathbf{x}) + 2\mathbf{S}\_1^2(a\mathbf{x}) + 4a\lambda \tanh\left(a\mathbf{x}\right) \mathbf{S}\_1(a\mathbf{x}).\tag{79}$$

The zeroth-order eigenfunction is defined by B�ϕ<sup>0</sup> ¼ 0, that is,

$$\phi\_0 = \frac{\text{sech}^\circ(a\mathbf{x})}{\mathbf{1} + \chi\_1 \int\_0^\chi \text{sech}^{2\ell}(a\mathbf{y}) \, d\mathbf{y}}.\tag{80}$$

Beginning with the zeroth-order eigenfunctions, the eigenfunctions can be found by successive applications of the raising operator, which only increases the

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup><sup>Γ</sup> <sup>ℓ</sup> <sup>þ</sup> <sup>1</sup> 2 � � ffiffiffi <sup>π</sup> <sup>p</sup> <sup>Γ</sup>ð Þ <sup>ℓ</sup>

> > <sup>s</sup>þ<sup>1</sup> <sup>ψ</sup><sup>s</sup>

<sup>m</sup> ¼ �η<sup>m</sup>

<sup>þ</sup> <sup>β</sup><sup>2</sup> ¼ � <sup>α</sup><sup>2</sup>m mð Þ <sup>þ</sup> <sup>1</sup>

¼ � <sup>α</sup><sup>2</sup>m mð Þ <sup>þ</sup> <sup>1</sup> cosh <sup>2</sup>αx

, and two possible values for D, D ¼ αð Þ m þ 1 , � αm. If we simply set

<sup>0</sup> sech<sup>2</sup>ð Þ <sup>m</sup>þ<sup>1</sup> <sup>α</sup>ydy � �<sup>2</sup>

<sup>0</sup> sech<sup>2</sup>ð Þ <sup>m</sup>þ<sup>1</sup> <sup>α</sup>y dy

π p

cosh <sup>2</sup>αx

3 7 5

�1=2

ð Þ Γð Þ m þ 1 . The corresponding

, (71)

� η<sup>m</sup>þ<sup>1</sup>ð Þ x , (72)

<sup>1</sup> to

<sup>m</sup> give different Hamiltonian operators.

<sup>ℓ</sup> <sup>¼</sup> <sup>ψ</sup>sþ<sup>1</sup>

d

cosh �ℓαx: (67)

<sup>ℓ</sup> . Note that from (63)

dx <sup>þ</sup> <sup>β</sup>m, (68)

<sup>m</sup>þ<sup>1</sup> ¼ Hmþ<sup>1</sup>þ

þ ϵ: (69)

þ ϵ<sup>m</sup>þ<sup>1</sup>: (70)

s

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

4.1 Two-parameter factorization of the Pöschl-Teller Hamiltonian

Following our previous work [2, 3], we define two non-mutually adjoint

dx <sup>þ</sup> <sup>β</sup>m, B<sup>∗</sup>

η

This Riccati equation was found in [24]; it has the solution β=η ¼ D tanh αx,

The constant ϵ is usually related to the lowest energy eigenvalue, but here the two different values come from the index asymmetry in the factorizations (63) and

where <sup>β</sup><sup>m</sup> and <sup>η</sup><sup>m</sup> are functions of <sup>x</sup>, and we require that Bmþ<sup>1</sup>B<sup>∗</sup>

By multiplying the first equation by β=η and adding, we have that

<sup>η</sup> � βη <sup>¼</sup> <sup>0</sup>, <sup>β</sup><sup>0</sup>

<sup>þ</sup> <sup>β</sup><sup>m</sup>þ<sup>1</sup> η<sup>m</sup>þ<sup>1</sup> � �<sup>2</sup>

The general solution to the pair of coupled equations (69) is

<sup>η</sup><sup>m</sup>þ<sup>1</sup>ð Þ¼ <sup>x</sup> <sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> sech<sup>2</sup>ð Þ <sup>m</sup>þ<sup>1</sup> <sup>α</sup><sup>x</sup> 1 þ γ<sup>1</sup> Ð x

> 1 þ γ<sup>1</sup> Ð x

condition on <sup>γ</sup><sup>2</sup> involves transcendental functions, but one may use <sup>γ</sup><sup>2</sup> <sup>&</sup>gt; � <sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

" #

value of the upper index. That is,

mA<sup>þ</sup>

first-order operators,

<sup>m</sup> and A<sup>þ</sup>

and (64), A�

with <sup>ϵ</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup>

and

86

ψℓ <sup>ℓ</sup>ð Þ¼ x

We repeatedly apply the creation operator A�

mA�

Bm <sup>¼</sup> <sup>η</sup>�<sup>1</sup> m d

ϵ<sup>m</sup>þ1. Then β<sup>m</sup>þ<sup>1</sup> and η<sup>m</sup>þ<sup>1</sup> are the solutions of

β<sup>m</sup>þ<sup>1</sup> η<sup>m</sup>þ<sup>1</sup> � �

η<sup>m</sup> ! 1, we recover the factorization (63).

(64). Following Ref. [24], we solve for D ¼ αð Þ m þ 1 .

2 6 4

<sup>β</sup><sup>m</sup>þ<sup>1</sup>ð Þ¼ <sup>x</sup> <sup>α</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> tanh <sup>α</sup><sup>x</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup> sech<sup>2</sup>ð Þ <sup>m</sup>þ<sup>1</sup> <sup>α</sup><sup>x</sup>

where <sup>γ</sup><sup>1</sup> has to satisfy <sup>∣</sup>γ1∣<2αΓð Þ <sup>m</sup> <sup>þ</sup> <sup>3</sup>=<sup>2</sup> <sup>=</sup> ffiffiffi

� η0 η þ β

#### 5. Quasi-exactly solvable potentials

In exactly solvable problems, the whole spectrum is found analytically, but the vast majority of problems have to be solved numerically. A new possibility arises with the class of QES potentials, where a subset of the spectrum may be found analytically [25–27]. QES potentials have been studied using the Lie algebraic method [25]: Manning [28], Razavy [29], and Ushveridze [30] potentials belong to this class (see also [31]). These are double-well potentials, which received much attention due to their applications in theoretical and experimental problems. Furthermore, hyperbolic-type potentials are found in many physical applications, like the Rosen-Morse potential [32], Dirac-type hyperbolic potentials [33], bidimensional quantum dot [34], Scarf-type entangled states [35], etc. QES potentials' classification has been given by Turbiner [25] and Ushveridze [30].

aj ¼ 16ζjð Þ 2j � σ þ η ð Þ j � n � 1

5.2 Symmetric solutions for V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup>ð Þ <sup>x</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82254

2 � � dψ

<sup>d</sup>β<sup>2</sup> þ �αβ βð Þþ � <sup>1</sup> <sup>β</sup> � <sup>1</sup>

<sup>E</sup> <sup>¼</sup> <sup>1</sup>

<sup>3</sup> <sup>þ</sup> <sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> ð Þ� <sup>α</sup><sup>2</sup>

�<sup>3</sup> � <sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> ð Þ� <sup>α</sup><sup>2</sup>

1

<sup>þ</sup> �αβ<sup>2</sup>

dβ þ 1 4

and to ensure that <sup>ψ</sup>ð Þ <sup>x</sup> vanishes as <sup>x</sup> ! �∞, let <sup>ψ</sup>ð Þ¼ <sup>x</sup> <sup>e</sup>�<sup>α</sup>

� � � � df

get

β βð Þ � 1

β βð Þ � <sup>1</sup> <sup>d</sup><sup>2</sup>

Q<sup>N</sup>

and the roots satisfy

fð Þ¼ β f <sup>0</sup>

equations

89

d2 ψ <sup>d</sup>β<sup>2</sup> <sup>þ</sup> <sup>β</sup> � <sup>1</sup>

requiring <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>V0, we obtain [41]

f

∑ N i6¼j

2 β<sup>i</sup> � β<sup>j</sup>

bj ¼ �4j j<sup>ð</sup> <sup>þ</sup> <sup>1</sup> � <sup>σ</sup> <sup>þ</sup> <sup>2</sup>ζÞ þ ð Þ <sup>2</sup><sup>n</sup> <sup>þ</sup> <sup>1</sup> <sup>ð</sup>2ð Þþ <sup>n</sup> � <sup>σ</sup> <sup>3</sup>Þ þ ζ ζð Þ � <sup>2</sup><sup>η</sup> <sup>þ</sup> <sup>4</sup><sup>n</sup> : (84)

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

To find the even solutions to Eq. (81) with <sup>k</sup> <sup>¼</sup> 0, let us set <sup>β</sup>ð Þ¼ <sup>x</sup> cosh <sup>2</sup>ð Þ <sup>x</sup> , to

may not include square-integrable solutions to the Razavy potential [38–40]. By

2

We shall look for rank N polynomial solutions: fð Þ¼ β f <sup>0</sup> for N ¼ 0, or

N i¼1

<sup>i</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>1</sup> <sup>β</sup><sup>i</sup> � <sup>1</sup>

<sup>V</sup><sup>0</sup> is found to depend on the order of the polynomial, <sup>V</sup><sup>0</sup> <sup>¼</sup> 2 2ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> for even solutions, and solutions with different N cannot be scaled one into the other due to the sinh <sup>4</sup>ð Þ <sup>x</sup> dependence of the potential function. The highest solution order is n ¼ 2N, and we use subindexes f g N; n to label eigenvalues/eigenfunctions. For N ¼ 0, fð Þ¼ β 1, we get V<sup>0</sup> ¼ 2, E0,<sup>0</sup> ¼ 1, and the (unnormalized) ground-

state eigenfunction <sup>ψ</sup>0,0ð Þ¼ <sup>x</sup> <sup>e</sup>� cosh <sup>2</sup>ð Þ <sup>x</sup> . For <sup>N</sup> <sup>¼</sup> 2, <sup>f</sup>ð Þ¼ <sup>β</sup> <sup>f</sup> <sup>0</sup> <sup>β</sup> � <sup>β</sup><sup>1</sup> ð Þ <sup>β</sup> � <sup>β</sup><sup>2</sup> ð Þ, equating to zero the coefficients of the polynomial Pð Þ β , we get the coupled

<sup>2</sup> <sup>¼</sup> <sup>0</sup>

þ � <sup>α</sup><sup>2</sup>

<sup>4</sup> <sup>þ</sup> <sup>α</sup> 4 þ E 2

� �

4 þ

þ β1β<sup>2</sup>

9α 2 þ E 2

> α2 <sup>4</sup> � <sup>α</sup> 2 � �

¼ 0:

� �

α2 <sup>4</sup> � <sup>5</sup><sup>α</sup>

> 3α 2

� �

4 þ

4 þ 5α 4 þ E <sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> <sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> ð Þþ <sup>β</sup>1β<sup>2</sup> � <sup>α</sup><sup>2</sup>

� �

in Eq. (86). Sometimes the N ¼ 0 solution is not even considered [35].

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup> <sup>4</sup> <sup>∑</sup>

β2 <sup>i</sup> � β<sup>i</sup>

<sup>2</sup><sup>E</sup> � <sup>2</sup>V0β<sup>2</sup> <sup>þ</sup> <sup>4</sup>V0<sup>β</sup> � <sup>2</sup>V<sup>0</sup>

<sup>d</sup><sup>β</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup><sup>β</sup>

<sup>i</sup>¼<sup>1</sup> <sup>β</sup> � <sup>β</sup><sup>i</sup> ð Þ for <sup>N</sup> <sup>&</sup>gt;0, the <sup>β</sup><sup>i</sup> being the roots of the resulting polynomial

β<sup>i</sup> � 1 � 4N � �

2

The highest power of β in Eq. (86) fix α to α ¼ 4N þ 2. The energy eigenvalues

<sup>4</sup> � αβ

� � <sup>¼</sup> 0 (85)

<sup>2</sup> <sup>þ</sup> <sup>α</sup> 4 þ E <sup>2</sup> � <sup>α</sup><sup>2</sup> 4

� <sup>4</sup>N<sup>2</sup> � � (87)

¼ 0, i ¼ 1, 2, …, n: (88)

¼ 0

¼ 0

(89)

� �<sup>f</sup> <sup>¼</sup> <sup>0</sup>:

<sup>2</sup><sup>β</sup>fð Þ <sup>β</sup> . Previous works

(86)

Here we show that the Lie algebraic procedure may impose strict restrictions on the solutions: we shall construct here analytical solutions for the Razavy-type potential V xð Þ¼ <sup>V</sup><sup>0</sup> sin <sup>h</sup><sup>4</sup> ð Þ� <sup>x</sup> <sup>k</sup> sin <sup>h</sup><sup>2</sup> ð Þ <sup>x</sup> based on the polynomial solutions of the related confluent Heun equation (CHE) and show that in that case the energy eigenvalues diverge when k ! �1, a feature solely of the procedure. We shall also show that other QES potentials may be found that do not belong to any of the potentials found using the Lie algebraic method.

#### 5.1 A Razavy-type QES potential

Let us consider Schrödinger's problem for the Razavy-type potential V xð Þ¼ <sup>V</sup><sup>0</sup> sin <sup>h</sup><sup>4</sup> ð Þ� <sup>x</sup> <sup>k</sup> sin <sup>h</sup><sup>2</sup> ð Þ <sup>x</sup> ,

$$\frac{-\hbar^2}{2\mu}\frac{d^2\psi(\mathbf{x})}{d\mathbf{x}^2} + V\_0 \left(\sinh^4(\lambda \mathbf{x}) - k\sin h^2(\lambda \mathbf{x})\right)\psi(\mathbf{x}) = E\psi(\mathbf{x}).\tag{81}$$

For simplicity, we set μ ¼ ℏ ¼ λ ¼ 1 [35, 36].

Here the potential function is the hyperbolic Razavy potential V xð Þ¼ <sup>1</sup> <sup>2</sup> ð Þ <sup>ζ</sup> cosh 2ð Þ� <sup>x</sup> <sup>M</sup> <sup>2</sup> , with <sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>ζ<sup>2</sup> , where M energy levels are found if M is a positive integer [29]. It may also be viewed as the Ushveridze potential V xð Þ¼ <sup>2</sup>ξ<sup>2</sup> sinh <sup>4</sup>ð Þþ <sup>x</sup> <sup>2</sup>ξ ξ½ � � <sup>2</sup>ð Þ� <sup>γ</sup> <sup>þ</sup> <sup>δ</sup> <sup>2</sup><sup>ℓ</sup> sinh <sup>2</sup>ð Þþ <sup>x</sup> <sup>2</sup> <sup>δ</sup> � <sup>1</sup> 4 <sup>δ</sup> � <sup>3</sup> 4 csch2 ð Þ� <sup>x</sup> <sup>2</sup>ðγ� <sup>1</sup> 4Þ <sup>γ</sup> � <sup>3</sup> 4 sech<sup>2</sup> ð Þ <sup>x</sup> , when <sup>γ</sup> <sup>¼</sup> <sup>1</sup> <sup>4</sup> and <sup>δ</sup> <sup>¼</sup> <sup>3</sup> 4, or vice versa [30], which is QES if <sup>ℓ</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, <sup>⋯</sup> (with <sup>δ</sup><sup>≥</sup> <sup>1</sup> 4). El-Jaick et al. showed that it is also QES if ℓ ¼half-integer and <sup>γ</sup>, <sup>δ</sup> <sup>¼</sup> <sup>1</sup> <sup>4</sup> , <sup>3</sup> <sup>4</sup> [37].

In the case of the Razavy potential, the solutions obtained by Finkel et al. are

$$\mathcal{Y}\_{\sigma\eta}(\mathbf{x}, E\_R) \propto (\sinh \mathbf{x})^{\frac{1}{\mathfrak{N}}(1-\sigma-\eta)} (\cosh \mathbf{x})^{\frac{1}{\mathfrak{N}}(1-\sigma+\eta)} e^{-\frac{\zeta}{2}\cosh \left(2\mathbf{x}\right)} \sum\_{j=0}^{n} \frac{\hat{P}\_j^{\sigma\eta}(E\_R)}{\left(2j + \frac{\eta-\sigma+1}{2}\right)!} \cosh^{\mathfrak{N}}(\mathbf{x}) \tag{82}$$

with the parameters ð Þ¼ � σ; η ð Þ 1; 0 or 0ð Þ ; �1 , the energy eigenvalues being the roots of the polynomials Pση <sup>j</sup>þ<sup>1</sup>ð Þ ER , satisfying the three-term recursive relations

$$
\hat{P}\_{j+1}^{\sigma\eta} = (E\_R - b\_j)\hat{P}\_j^{\sigma\eta}(E\_R) - a\_j \hat{P}\_{j-1}^{\sigma\eta}(E\_R), \qquad j \ge 0 \tag{83}
$$

with ER ¼ 2E, and

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

$$\begin{aligned} a\_j &= \mathbf{1} \mathsf{6} \zeta j (2j - \sigma + \eta)(j - n - 1) \\ b\_j &= -4j(j + \mathbf{1} - \sigma + 2\zeta) + (2n + \mathbf{1})(2(n - \sigma) + \mathbf{3}) + \zeta(\zeta - 2\eta + 4n) . \end{aligned} \tag{84}$$

### 5.2 Symmetric solutions for V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup>ð Þ <sup>x</sup>

5. Quasi-exactly solvable potentials

potential V xð Þ¼ <sup>V</sup><sup>0</sup> sin <sup>h</sup><sup>4</sup>

V xð Þ¼ <sup>V</sup><sup>0</sup> sin <sup>h</sup><sup>4</sup>

V xð Þ¼ <sup>1</sup>

<sup>γ</sup> � <sup>3</sup> 4 sech<sup>2</sup>

88

and <sup>γ</sup>, <sup>δ</sup> <sup>¼</sup> <sup>1</sup>

�ℏ<sup>2</sup> 2μ

<sup>ℓ</sup> <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, <sup>⋯</sup> (with <sup>δ</sup><sup>≥</sup> <sup>1</sup>

<sup>4</sup> , <sup>3</sup> <sup>4</sup> [37].

ψσηð Þ <sup>x</sup>; ER <sup>∝</sup>ð Þ sinh <sup>x</sup> <sup>1</sup>

roots of the polynomials Pση

with ER ¼ 2E, and

d2 ψð Þ x

<sup>2</sup> ð Þ <sup>ζ</sup> cosh 2ð Þ� <sup>x</sup> <sup>M</sup> <sup>2</sup>

ð Þ <sup>x</sup> , when <sup>γ</sup> <sup>¼</sup> <sup>1</sup>

2

P^ση

<sup>j</sup>þ<sup>1</sup> <sup>¼</sup> ER � bj

5.1 A Razavy-type QES potential

In exactly solvable problems, the whole spectrum is found analytically, but the vast majority of problems have to be solved numerically. A new possibility arises with the class of QES potentials, where a subset of the spectrum may be found analytically [25–27]. QES potentials have been studied using the Lie algebraic method [25]: Manning [28], Razavy [29], and Ushveridze [30] potentials belong to this class (see also [31]). These are double-well potentials, which received much attention due to their applications in theoretical and experimental problems. Furthermore, hyperbolic-type potentials are found in many physical applications, like

bidimensional quantum dot [34], Scarf-type entangled states [35], etc. QES poten-

the related confluent Heun equation (CHE) and show that in that case the energy eigenvalues diverge when k ! �1, a feature solely of the procedure. We shall also show that other QES potentials may be found that do not belong to any of the

Here we show that the Lie algebraic procedure may impose strict restrictions on

ð Þ <sup>x</sup> based on the polynomial solutions of

ð Þ <sup>λ</sup><sup>x</sup> <sup>ψ</sup>ð Þ¼ <sup>x</sup> <sup>E</sup>ψð Þ <sup>x</sup> : (81)

4 <sup>δ</sup> � <sup>3</sup>

4). El-Jaick et al. showed that it is also QES if ℓ ¼half-integer

<sup>2</sup> cosh 2ð Þ <sup>x</sup> ∑ n j¼0

<sup>j</sup>þ<sup>1</sup>ð Þ ER , satisfying the three-term recursive relations

4, or vice versa [30], which is QES if

, where M energy levels are found if M

ð Þ� <sup>x</sup> <sup>2</sup>ðγ� <sup>1</sup>

cosh <sup>2</sup><sup>j</sup>

4Þ

ð Þ x

(82)

4 csch2

> P^ση <sup>j</sup> ð Þ ER <sup>2</sup><sup>j</sup> <sup>þ</sup> <sup>η</sup>�σþ<sup>1</sup> 2 !

<sup>j</sup>�<sup>1</sup>ð Þ ER , j≥<sup>0</sup> (83)

the Rosen-Morse potential [32], Dirac-type hyperbolic potentials [33],

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

ð Þ� <sup>x</sup> <sup>k</sup> sin <sup>h</sup><sup>2</sup>

potentials found using the Lie algebraic method.

ð Þ� <sup>x</sup> <sup>k</sup> sin <sup>h</sup><sup>2</sup> ð Þ <sup>x</sup> ,

For simplicity, we set μ ¼ ℏ ¼ λ ¼ 1 [35, 36].

<sup>2</sup>ξ<sup>2</sup> sinh <sup>4</sup>ð Þþ <sup>x</sup> <sup>2</sup>ξ ξ½ � � <sup>2</sup>ð Þ� <sup>γ</sup> <sup>þ</sup> <sup>δ</sup> <sup>2</sup><sup>ℓ</sup> sinh <sup>2</sup>ð Þþ <sup>x</sup> <sup>2</sup> <sup>δ</sup> � <sup>1</sup>

tials' classification has been given by Turbiner [25] and Ushveridze [30].

the solutions: we shall construct here analytical solutions for the Razavy-type

Let us consider Schrödinger's problem for the Razavy-type potential

dx<sup>2</sup> <sup>þ</sup> <sup>V</sup><sup>0</sup> sinh <sup>4</sup>ð Þ� <sup>λ</sup><sup>x</sup> <sup>k</sup> sin <sup>h</sup><sup>2</sup>

Here the potential function is the hyperbolic Razavy potential

<sup>4</sup> and <sup>δ</sup> <sup>¼</sup> <sup>3</sup>

ð Þ <sup>1</sup>�σ�<sup>η</sup> ð Þ cosh <sup>x</sup> <sup>1</sup>

P^ση

, with <sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>ζ<sup>2</sup>

is a positive integer [29]. It may also be viewed as the Ushveridze potential V xð Þ¼

In the case of the Razavy potential, the solutions obtained by Finkel et al. are

with the parameters ð Þ¼ � σ; η ð Þ 1; 0 or 0ð Þ ; �1 , the energy eigenvalues being the

<sup>j</sup> ð Þ� ER ajP^ση

2 ð Þ <sup>1</sup>�σþ<sup>η</sup> e �ζ

To find the even solutions to Eq. (81) with <sup>k</sup> <sup>¼</sup> 0, let us set <sup>β</sup>ð Þ¼ <sup>x</sup> cosh <sup>2</sup>ð Þ <sup>x</sup> , to get

$$\beta(\beta-1)\frac{d^2\nu}{d\beta^2} + \left(\beta-\frac{1}{2}\right)\frac{d\nu}{d\beta} + \frac{1}{4}\left[2E - 2V\_0\beta^2 + 4V\_0\beta - 2V\_0\right] = 0\tag{85}$$

and to ensure that <sup>ψ</sup>ð Þ <sup>x</sup> vanishes as <sup>x</sup> ! �∞, let <sup>ψ</sup>ð Þ¼ <sup>x</sup> <sup>e</sup>�<sup>α</sup> <sup>2</sup><sup>β</sup>fð Þ <sup>β</sup> . Previous works may not include square-integrable solutions to the Razavy potential [38–40]. By requiring <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>V0, we obtain [41]

$$\beta(\beta - 1)\frac{d^2f}{d\beta^2} + \left[ -a\beta(\beta - 1) + \left(\beta - \frac{1}{2}\right) \right] \frac{df}{d\beta} + \left[ \frac{a^2\beta}{4} - \frac{a\beta}{2} + \frac{a}{4} + \frac{E}{2} - \frac{a^2}{4} \right] f = 0. \tag{86}$$

We shall look for rank N polynomial solutions: fð Þ¼ β f <sup>0</sup> for N ¼ 0, or fð Þ¼ β f <sup>0</sup> Q<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>β</sup> � <sup>β</sup><sup>i</sup> ð Þ for <sup>N</sup> <sup>&</sup>gt;0, the <sup>β</sup><sup>i</sup> being the roots of the resulting polynomial in Eq. (86). Sometimes the N ¼ 0 solution is not even considered [35].

The highest power of β in Eq. (86) fix α to α ¼ 4N þ 2. The energy eigenvalues and the roots satisfy

$$E = \frac{1}{2} \left[ a^2 + a \left( 4 \sum\_{i=1}^{N} \beta\_i - 1 - 4N \right) - 4N^2 \right] \tag{87}$$

$$\sum\_{i \neq j}^{N} \frac{2}{\beta\_i - \beta\_j} + \frac{-a\beta\_i^2 + (a+1)\beta\_i - \frac{1}{2}}{\beta\_i^2 - \beta\_i} = 0, \qquad i = 1, 2, \dots, n. \tag{88}$$

<sup>V</sup><sup>0</sup> is found to depend on the order of the polynomial, <sup>V</sup><sup>0</sup> <sup>¼</sup> 2 2ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> for even solutions, and solutions with different N cannot be scaled one into the other due to the sinh <sup>4</sup>ð Þ <sup>x</sup> dependence of the potential function. The highest solution order is n ¼ 2N, and we use subindexes f g N; n to label eigenvalues/eigenfunctions.

For N ¼ 0, fð Þ¼ β 1, we get V<sup>0</sup> ¼ 2, E0,<sup>0</sup> ¼ 1, and the (unnormalized) groundstate eigenfunction <sup>ψ</sup>0,0ð Þ¼ <sup>x</sup> <sup>e</sup>� cosh <sup>2</sup>ð Þ <sup>x</sup> . For <sup>N</sup> <sup>¼</sup> 2, <sup>f</sup>ð Þ¼ <sup>β</sup> <sup>f</sup> <sup>0</sup> <sup>β</sup> � <sup>β</sup><sup>1</sup> ð Þ <sup>β</sup> � <sup>β</sup><sup>2</sup> ð Þ, equating to zero the coefficients of the polynomial Pð Þ β , we get the coupled equations

$$\frac{a^2}{4} - \frac{5a}{2} = 0$$

$$3 + (\beta\_1 + \beta\_2) \left( -\frac{a^2}{4} + \frac{3a}{2} \right) + \left( -\frac{a^2}{4} + \frac{9a}{2} + \frac{E}{2} \right) = 0$$

$$-3 - (\beta\_1 + \beta\_2) \left( -\frac{a^2}{4} + \frac{5a}{4} + \frac{E}{2} + 1 \right) + \beta\_1 \beta\_2 \left( \frac{a^2}{4} - \frac{a}{2} \right) = 0$$

$$\frac{1}{2} (\beta\_1 + \beta\_2) + \beta\_1 \beta\_2 \left( -\frac{a^2}{4} + \frac{a}{4} + \frac{E}{2} \right) = 0.$$

Solving these, we find that V<sup>0</sup> ¼ 50, and the three possible eigenvalues, E2,<sup>0</sup> ¼ 2:6301, E2, <sup>2</sup> ¼ 19:0121, and E2,<sup>4</sup> ¼ 43:2490.

#### 5.3 Antisymmetric solutions

In order to find antisymmetric solutions to Eq. (86), we set fð Þ¼ β sinh ð Þ x gð Þ β , to obtain

$$\begin{split} \beta[\beta - 1] \frac{d^2 \mathbf{g}}{dx^2} + \left[ -a\beta^2 + (a+2)\beta - \frac{1}{2} \right] \frac{d \mathbf{g}}{dx} \\ + \left[ \left( -a + \frac{a^2}{4} \right) \beta + \left( -\frac{a^2}{4} + \frac{a}{4} + \frac{E}{2} + \frac{1}{4} \right) \right] \mathbf{g} = \mathbf{0}. \end{split} \tag{90}$$

<sup>þ</sup> <sup>α</sup>2<sup>β</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82254

We now find that <sup>V</sup><sup>0</sup> <sup>¼</sup> 2 2ð Þ <sup>N</sup>þ<sup>1</sup> <sup>2</sup>

energy eigenvalues found are

cosmological implications [18].

g

get the CHE

Figure 1.

91

eigenvalues are shown in dotted lines.

β βð Þ � <sup>1</sup> <sup>d</sup><sup>2</sup>

eigenfunctions are plotted in Figure 1.

<sup>þ</sup> <sup>β</sup> <sup>α</sup><sup>2</sup>

<sup>d</sup>β<sup>2</sup> þ �αβ<sup>2</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>2</sup> <sup>β</sup> � <sup>1</sup>

� � dg

<sup>4</sup> ð Þ� <sup>1</sup> <sup>þ</sup> <sup>c</sup> <sup>α</sup> � �

E2, <sup>5</sup> ¼ 9:53574. The eigenfunctions are plotted in Figure 1.

<sup>4</sup> ð Þ� <sup>1</sup> <sup>þ</sup> <sup>k</sup> αβ

<sup>E</sup> <sup>¼</sup> <sup>9</sup> � ð Þ� <sup>1</sup> <sup>þ</sup> <sup>k</sup>

For the case with N ¼ 2, choosing k ¼ 4, the energy eigenvalues are E2,<sup>0</sup> ¼ �3:74456, E2, <sup>2</sup> ¼ 1:00000, and E2,<sup>4</sup> ¼ 7:74456. The corresponding

2

<sup>þ</sup> <sup>α</sup> 4 þ E <sup>2</sup> � <sup>α</sup><sup>2</sup>

For N ¼ 0 we get that α ¼ 4=ð Þ 1 þ k and E<sup>1</sup> ¼ 6=ð Þ� 1 þ k 1=2, such that if k>11, we may find negative energy eigenvalues. For N ¼ 2, α ¼ 12=ð Þ 1 þ k , if we set k ¼ 5, the energy eigenvalues found are E2, <sup>1</sup> ¼ �7:11693, E2, <sup>3</sup> ¼ 1:08119, and

Note that in this case ð Þ E<sup>1</sup> � E<sup>0</sup> =E<sup>0</sup> ¼ 0:0052, and it is not possible to distinguish these eigenvalue's lines from each other in Figure 1 for antisymmetric

Left: the three even eigenfunctions (narrow solid lines) found analytically for k ¼ 4 and N ¼ 2, together with the corresponding eigenvalues (dashed lines). Right: the three odd eigenfunctions (narrow solid lines) found analytically for k ¼ 5 and N ¼ 2, together with the corresponding eigenvalues (dashed lines). The unsolved

<sup>2</sup> <sup>þ</sup> <sup>α</sup> 4 þ E <sup>2</sup> � <sup>α</sup><sup>2</sup>

E0,<sup>0</sup> ¼ 1=ð Þ 1 þ k , and no negative energy eigenvalues may exist. For N ¼ 1 the two

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

meaning that for k> 3=2 we will have negative eigenvalues. Note that for N >0, it is always possible to find a zero-energy ground state, a feature that may have

Now, to find the antisymmetric eigenfunctions, we set fð Þ¼ β sinh ð Þ x gð Þ β , to

dβ

� � � � <sup>g</sup> <sup>¼</sup> <sup>0</sup> :

<sup>4</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup> � �<sup>f</sup> <sup>¼</sup> <sup>0</sup>: (93)

<sup>1</sup>þ<sup>k</sup> , <sup>k</sup> varying freely. For example, if <sup>N</sup> <sup>¼</sup> 0,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup> <sup>2</sup> <sup>þ</sup> <sup>36</sup> <sup>q</sup>

<sup>4</sup> ð Þþ <sup>1</sup> <sup>þ</sup> <sup>c</sup>

1 4 (95)

<sup>1</sup> <sup>þ</sup> <sup>k</sup> (94)

This CHE can be solved in power series: gð Þ¼ β g<sup>0</sup> if N ¼ 0, or gð Þ¼ β g<sup>0</sup> Q<sup>N</sup> <sup>i</sup>¼<sup>1</sup> <sup>β</sup> � <sup>β</sup><sup>i</sup> ð Þ for <sup>N</sup> <sup>&</sup>gt;0. Then, <sup>α</sup> <sup>¼</sup> <sup>4</sup>ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> , and

$$E = \frac{1}{2} \left[ a^2 + a \left( 4 \sum\_{i=1}^{N} \beta\_i - 1 - 4N \right) - 4N^2 - 4N - 1 \right]. \tag{91}$$

Here, <sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> , and all even and odd solutions have different V0. The maximum solutions order is n ¼ 2N þ 1. For example, for N ¼ 3 we get α ¼ 16, V<sup>0</sup> ¼ 128, and

$$(\beta\_1 + \beta\_2 + \beta\_3) \left(3a - \frac{a^2}{4}\right) + \left(-\frac{a^2}{4} + \frac{13a}{4} + \frac{E}{2} - \frac{49}{4}\right) = 0$$

$$(\beta\_1 + \beta\_2 + \beta\_3) \left(\frac{a^2}{4} - \frac{9a}{4} - \frac{E}{2} - \frac{25}{4}\right) + (\beta\_1 \beta\_2 + \beta\_2 \beta\_3 + \beta\_3 \beta\_1) \left(\frac{a^2}{4} - 2a\right) - \frac{15}{2} = 0$$

$$3(\beta\_1 + \beta\_2 + \beta\_3) + (\beta\_1 \beta\_2 + \beta\_2 \beta\_3 + \beta\_3 \beta\_1) \left(-\frac{a^2}{4} + \frac{5a}{4} + \frac{9}{4} + \frac{E}{2}\right) + \beta\_1 \beta\_2 \beta\_3 \left(-\frac{a^2}{4} + a\right) = 0$$

$$-\frac{1}{2}(\beta\_1 \beta\_2 + \beta\_2 \beta\_3 + \beta\_3 \beta\_1) - \beta\_1 \beta\_2 \beta\_3 \left(\frac{a^2}{4} - \frac{a}{4} - \frac{E}{2} - \frac{1}{4}\right) = 0. \tag{92}$$

We find four eigenvalues, E3, <sup>1</sup> ¼ 12:8152, E3,<sup>3</sup> ¼ 40:4568, E3, <sup>5</sup> ¼ 75:7246, and E3, <sup>7</sup> ¼ 117:003.

### 6. The potential function V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup>ð Þ� <sup>x</sup> <sup>k</sup> sinh <sup>2</sup>ð Þ <sup>x</sup> � �

Now we apply our analysis to the problem with V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup> <sup>ð</sup> ð Þ� <sup>x</sup> <sup>k</sup> sinh <sup>2</sup>ð ÞÞ <sup>x</sup> , which is a symmetric double well if <sup>k</sup>>0. To find even solutions, we set again <sup>β</sup>ð Þ¼ <sup>x</sup> cosh <sup>2</sup>ð Þ <sup>x</sup> and ψ βð Þ¼ <sup>e</sup>�<sup>α</sup> <sup>2</sup><sup>β</sup>fð Þ <sup>β</sup> , with <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>V0,

$$
\beta(\beta - 1) \frac{d^2 f}{d\beta^2} + \left[ -a\beta(\beta - 1) + \left(\beta - \frac{1}{2}\right) \right] \frac{df}{d\beta}
$$

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

$$+\left[\frac{a^2\beta}{4}(\mathbf{1}+k)-\frac{a\beta}{2}+\frac{a}{4}+\frac{E}{2}-\frac{a^2}{4}(\mathbf{1}+k)\right]f=\mathbf{0}.\tag{93}$$

We now find that <sup>V</sup><sup>0</sup> <sup>¼</sup> 2 2ð Þ <sup>N</sup>þ<sup>1</sup> <sup>2</sup> <sup>1</sup>þ<sup>k</sup> , <sup>k</sup> varying freely. For example, if <sup>N</sup> <sup>¼</sup> 0, E0,<sup>0</sup> ¼ 1=ð Þ 1 þ k , and no negative energy eigenvalues may exist. For N ¼ 1 the two energy eigenvalues found are

$$E = \frac{9 - (\mathbf{1} + k) \pm \sqrt{\left(\mathbf{1} + k\right)^2 + 36}}{\mathbf{1} + k} \tag{94}$$

meaning that for k> 3=2 we will have negative eigenvalues. Note that for N >0, it is always possible to find a zero-energy ground state, a feature that may have cosmological implications [18].

For the case with N ¼ 2, choosing k ¼ 4, the energy eigenvalues are E2,<sup>0</sup> ¼ �3:74456, E2, <sup>2</sup> ¼ 1:00000, and E2,<sup>4</sup> ¼ 7:74456. The corresponding eigenfunctions are plotted in Figure 1.

Now, to find the antisymmetric eigenfunctions, we set fð Þ¼ β sinh ð Þ x gð Þ β , to get the CHE

$$\begin{split} \beta(\beta - 1) \frac{d^2 \mathbf{g}}{d\rho^2} + \left[ -a\rho^2 + (a+2)\beta - \frac{1}{2} \right] \frac{d \mathbf{g}}{d\beta} \\ &+ \left[ \beta \left( \frac{a^2}{4} (\mathbf{1} + c) - a \right) + \left( \frac{a}{4} + \frac{E}{2} - \frac{a^2}{4} (\mathbf{1} + c) + \frac{1}{4} \right) \right] \mathbf{g} = \mathbf{0} \end{split} \tag{95}$$

For N ¼ 0 we get that α ¼ 4=ð Þ 1 þ k and E<sup>1</sup> ¼ 6=ð Þ� 1 þ k 1=2, such that if k>11, we may find negative energy eigenvalues. For N ¼ 2, α ¼ 12=ð Þ 1 þ k , if we set k ¼ 5, the energy eigenvalues found are E2, <sup>1</sup> ¼ �7:11693, E2, <sup>3</sup> ¼ 1:08119, and E2, <sup>5</sup> ¼ 9:53574. The eigenfunctions are plotted in Figure 1.

Note that in this case ð Þ E<sup>1</sup> � E<sup>0</sup> =E<sup>0</sup> ¼ 0:0052, and it is not possible to distinguish these eigenvalue's lines from each other in Figure 1 for antisymmetric

#### Figure 1.

Solving these, we find that V<sup>0</sup> ¼ 50, and the three possible eigenvalues,

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

In order to find antisymmetric solutions to Eq. (86), we set fð Þ¼ β sinh ð Þ x gð Þ β ,

2

<sup>β</sup> þ � <sup>α</sup><sup>2</sup>

β<sup>i</sup> � 1 � 4N � �

maximum solutions order is n ¼ 2N þ 1. For example, for N ¼ 3 we get α ¼ 16,

þ � <sup>α</sup><sup>2</sup>

4 þ 5α 4 þ 9 4 þ E 2

We find four eigenvalues, E3, <sup>1</sup> ¼ 12:8152, E3,<sup>3</sup> ¼ 40:4568, E3, <sup>5</sup> ¼ 75:7246, and

4 þ

� �

α2 <sup>4</sup> � <sup>α</sup> <sup>4</sup> � <sup>E</sup> 2 � 1 4

<sup>2</sup><sup>β</sup>fð Þ <sup>β</sup> , with <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup>V0,

� � � � df

2

dβ

� �

4 � �

> <sup>2</sup> � <sup>25</sup> 4

<sup>2</sup> <sup>β</sup>1β<sup>2</sup> <sup>þ</sup> <sup>β</sup>2β<sup>3</sup> <sup>þ</sup> <sup>β</sup>3β<sup>1</sup> <sup>ð</sup> Þ � <sup>β</sup>1β2β<sup>3</sup>

6. The potential function V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup>ð Þ� <sup>x</sup> <sup>k</sup> sinh <sup>2</sup>ð Þ <sup>x</sup> � �

Now we apply our analysis to the problem with V xð Þ¼ <sup>V</sup><sup>0</sup> sinh <sup>4</sup> <sup>ð</sup> ð Þ� <sup>x</sup> <sup>k</sup> sinh <sup>2</sup>ð ÞÞ <sup>x</sup> , which is a symmetric double well if <sup>k</sup>>0. To find even solutions, we set

<sup>d</sup>β<sup>2</sup> þ �αβ βð Þþ � <sup>1</sup> <sup>β</sup> � <sup>1</sup>

� �

� � � �

dx

<sup>4</sup> <sup>þ</sup> <sup>α</sup> 4 þ E 2 þ 1 4

g ¼ 0:

¼ 0

<sup>4</sup> � <sup>2</sup><sup>α</sup> � �

<sup>þ</sup> <sup>β</sup>1β2β<sup>3</sup> � <sup>α</sup><sup>2</sup>

¼ 0:

� <sup>4</sup>N<sup>2</sup> � <sup>4</sup><sup>N</sup> � <sup>1</sup>

, and all even and odd solutions have different V0. The

13α 4 þ E <sup>2</sup> � <sup>49</sup> 4

<sup>þ</sup> <sup>β</sup>1β<sup>2</sup> <sup>þ</sup> <sup>β</sup>2β<sup>3</sup> <sup>þ</sup> <sup>β</sup>3β<sup>1</sup> ð Þ <sup>α</sup><sup>2</sup>

� �

(90)

: (91)

� 15 <sup>2</sup> <sup>¼</sup> <sup>0</sup>

<sup>4</sup> <sup>þ</sup> <sup>α</sup> � �

¼ 0

(92)

E2,<sup>0</sup> ¼ 2:6301, E2, <sup>2</sup> ¼ 19:0121, and E2,<sup>4</sup> ¼ 43:2490.

dx<sup>2</sup> þ �αβ<sup>2</sup> <sup>þ</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>2</sup> <sup>β</sup> � <sup>1</sup>

4 � �

This CHE can be solved in power series: gð Þ¼ β g<sup>0</sup> if N ¼ 0, or

<sup>i</sup>¼<sup>1</sup> <sup>β</sup> � <sup>β</sup><sup>i</sup> ð Þ for <sup>N</sup> <sup>&</sup>gt;0. Then, <sup>α</sup> <sup>¼</sup> <sup>4</sup>ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> , and

N i¼1

þ �<sup>α</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

<sup>2</sup> <sup>α</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup> <sup>4</sup> <sup>∑</sup>

<sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> ð Þ <sup>3</sup><sup>α</sup> � <sup>α</sup><sup>2</sup>

� �

<sup>4</sup> � <sup>9</sup><sup>α</sup> <sup>4</sup> � <sup>E</sup>

<sup>3</sup> <sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> ð Þþ <sup>β</sup>1β<sup>2</sup> <sup>þ</sup> <sup>β</sup>2β<sup>3</sup> <sup>þ</sup> <sup>β</sup>3β<sup>1</sup> <sup>ð</sup> Þ � <sup>α</sup><sup>2</sup>

� � dg

5.3 Antisymmetric solutions

β β½ � � <sup>1</sup> <sup>d</sup><sup>2</sup>

Q<sup>N</sup>

Here, <sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>8</sup>ð Þ <sup>N</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup>

<sup>β</sup><sup>1</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> ð Þ <sup>α</sup><sup>2</sup>

� 1

again <sup>β</sup>ð Þ¼ <sup>x</sup> cosh <sup>2</sup>ð Þ <sup>x</sup> and ψ βð Þ¼ <sup>e</sup>�<sup>α</sup>

β βð Þ � 1

d2 f

g

<sup>E</sup> <sup>¼</sup> <sup>1</sup>

to obtain

gð Þ¼ β g<sup>0</sup>

V<sup>0</sup> ¼ 128, and

E3, <sup>7</sup> ¼ 117:003.

90

Left: the three even eigenfunctions (narrow solid lines) found analytically for k ¼ 4 and N ¼ 2, together with the corresponding eigenvalues (dashed lines). Right: the three odd eigenfunctions (narrow solid lines) found analytically for k ¼ 5 and N ¼ 2, together with the corresponding eigenvalues (dashed lines). The unsolved eigenvalues are shown in dotted lines.

#### Panorama of Contemporary Quantum Mechanics - Concepts and Applications

eigenvalues, implying quasi-degenerate eigenstates. A similar effect is seen in the symmetric case.

### 6.1 The case with k ¼ �1

As was seen in Section VI, the ground-state energy diverges as 1=ð Þ 1 þ k and as k ! �1, and this also happens to all higher-order even eigenvalues (see Eq. (94)). This is a strange behavior, since it is clear that the potential function has a rather simple functional form for any value of k: a single or double well with infinite barriers. We can see that this is only a characteristic due to the analytical solution procedure, coming from the fact that the potential strength V<sup>0</sup> is also divergent when k ! �1.

#### 6.2 Unclassified QES potentials

Finally, we would like to emphasize that there should be other potential functions which may not be classified form the Lie algebraic method [25].

Indeed, let us consider Schrödinger's problem with the potential function

$$V(x) = \frac{a^2}{2}\cosh^2(x) - \frac{3a}{2}\cosh(x) + \frac{a}{\cosh(x)}.\tag{96}$$

For this problem, the ground-state eigenfunction and eigenvalue are given by

$$\psi = \psi\_0 e^{-a \cosh(x)} \cosh(x), \qquad E = \frac{a^2 - 1}{2} \tag{97}$$

Author details

Edgar Condori Pozo

93

José Socorro García Díaz\*, Marco A. Reyes, Carlos Villaseñor Mora and

Guanajuato-Campus León, León, Guanajuato, Mexico

\*Address all correspondence to: socorro@fisica.ugto.mx

provided the original work is properly cited.

Departamento de Física, División de Ciencias e Ingenierías, Universidad de

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable…

DOI: http://dx.doi.org/10.5772/intechopen.82254

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

while this particular problem does not belong to the class of potentials found using the Lie algebraic method. Similar potentials may be found which do not belong to that class, leaving space for further developments.

#### Acknowledgements

This work was partially supported by CONACYT 179881 grants and PROMEP grants UGTO-CA-3. This work is part of the collaboration within the Instituto Avanzado de Cosmología. E. Condori-Pozo is supported by a CONACYT graduate fellowship.

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable… DOI: http://dx.doi.org/10.5772/intechopen.82254

#### Author details

eigenvalues, implying quasi-degenerate eigenstates. A similar effect is seen in the

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

As was seen in Section VI, the ground-state energy diverges as 1=ð Þ 1 þ k and as k ! �1, and this also happens to all higher-order even eigenvalues (see Eq. (94)). This is a strange behavior, since it is clear that the potential function has a rather simple functional form for any value of k: a single or double well with infinite barriers. We can see that this is only a characteristic due to the analytical solution procedure, coming from the fact that the potential strength V<sup>0</sup> is also divergent

Finally, we would like to emphasize that there should be other potential func-

3α 2

For this problem, the ground-state eigenfunction and eigenvalue are given by

while this particular problem does not belong to the class of potentials found using the Lie algebraic method. Similar potentials may be found which do not

This work was partially supported by CONACYT 179881 grants and PROMEP grants UGTO-CA-3. This work is part of the collaboration within the Instituto Avanzado de Cosmología. E. Condori-Pozo is supported by a CONACYT graduate

�<sup>α</sup> cosh ð Þ <sup>x</sup> cosh ð Þ <sup>x</sup> , E <sup>¼</sup> <sup>α</sup><sup>2</sup> � <sup>1</sup>

cosh ð Þþ <sup>x</sup> <sup>α</sup>

cosh ð Þ <sup>x</sup> : (96)

<sup>2</sup> (97)

Indeed, let us consider Schrödinger's problem with the potential function

ð Þ� x

tions which may not be classified form the Lie algebraic method [25].

cosh <sup>2</sup>

symmetric case.

when k ! �1.

6.1 The case with k ¼ �1

6.2 Unclassified QES potentials

Acknowledgements

fellowship.

92

V xð Þ¼ <sup>α</sup><sup>2</sup> 2

ψ ¼ ψ0e

belong to that class, leaving space for further developments.

José Socorro García Díaz\*, Marco A. Reyes, Carlos Villaseñor Mora and Edgar Condori Pozo Departamento de Física, División de Ciencias e Ingenierías, Universidad de Guanajuato-Campus León, León, Guanajuato, Mexico

\*Address all correspondence to: socorro@fisica.ugto.mx

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### *Edited by Trong T. Tuong*

This book is devoted to recent developments in quantum mechanics. After an Introductory chapter, Chapter 2 describes the cooperative spontaneous lasing mechanism in gas in three level systems and their possible quantum retardation effects. Chapter 3 is concerned with the evolution of states of large quantum particle systems via marginal correlation operators. Chapter 4 studies the effects of electronic transfer using ab initio quantum calculation methods to access biological macromolecular system behaviors. Chapter 5 concentrates on new features of supersymmetric quantum mechanics using the adjunction of boson-fermion symmetry. The book will be of interest to graduate and Ph.D students as well as scientists from various backgrounds who are concerned with quantum effects.

Published in London, UK © 2019 IntechOpen © sequential5 / iStock

Panorama of Contemporary Quantum Mechanics - Concepts and Applications

Panorama of Contemporary

Quantum Mechanics

Concepts and Applications

*Edited by Trong T. Tuong*