Preface

The book covers a wide spectrum of research problems concerning quantum theory of light and experiments using its quantum properties. In reference literature one can find a number of definitions for the term "quantum optics" – from the sphere of phenomena revealing the quantum nature of light to the optics section dealing with statistical properties of emission. Such a contradictory situation reflects a complicated way of notion formation. Difficulties of the classical wave concept of light were the basis for formulating new quantum concepts of light emission, propagation, interaction and a new general concept – field concept of matter in the first quarter of the 20th century. In this sense we can speak about quantum optics whenever optical phenomena are considered from the position of quantum theory. That is especially true for the world of phenomena that can be discussed and understood only within the framework of quantum picture. The possibilities of optical experiments have been broadened fantastically by the invention of laser. Since lasers are quantum optical generators, the domain of experiments with laser emission seems to be close to quantum optics. At the same time, strong electromagnetic fields generated by lasers usually manifest their classical properties, so some analysis is necessary for including the observed phenomena into the field of quantum optics. In classical optics, correlation properties of light connected with the statistical nature of a real experiment were discussed in terms of the conception "coherence". In the experiments of Hanbury Brown–Twiss with quantum detecting of light, the process of receiving an electromagnetic emission was considered a usual random process for the first time. Later on, the whole ideology of probability theory and stochastic processes was applied to optical phenomena using quantum detectors for analyzing the statistical (correlation) properties of electromagnetic fields in optics. It gives substantiation for quantum optics identification as a statistical theory of light.

The editor was faced with different interpretations of quantum optics while analyzing the chapter proposals. Upon examining them, he reached the conclusion that all theoretical and experimental papers were welcome if they contributed to our understanding of light as a quantum phenomenon. The accepted approach was to speak about quantum optics in a wide sense, i.e. different phenomena demonstrating the quantum nature of light together with theoretical constructs applied to them; and in the narrow sense, i.e. the statistical theory of light processes and its incarnation with quantum detecting schemes. The present title of the book reflects its real contents and breadth of topics.

Preface XI

optics. Such considerations are urgent for needs of near-field optics, cavity quantum electrodynamics, and quantum computing. Some new optical phenomena connected with photon localization have drawn scientists' interest in recent years (see, for example Chapter 5). In textbooks the problems of position wave function and measurable quantities locality are compared. The author presents a qualified review together with his original results. Restrictions on photon localization set by the Paley-Wiener theorem and their seeming violation for certain two-dimensional wave packets are

The fourth chapter "Fusion Frames and Dynamics of Open Quantum Systems" by Prof. A. Jamiołkowski deals with the problems of quantum tomography, which is a procedure for reconstructing properties of a quantum object on the basis of experimentally accessible data. The state reconstruction requires identifying the quorum of observables, providing a possibility to determine expectation values of physical quantities for which no measuring apparatuses are available. The problem is discussed in terms of a set of density operators on the Hilbert space of the quantum system states. The main purpose of this contribution is to discuss properties of some Krylov subspaces in a given Hilbert space as natural examples of fusion frames and

The second section of the book under the title "Quantum Phenomena with Laser Radiation" includes three chapters. The fifth chapter "Quantum Optics Phenomena in Synthetic Opal Photonic Crystals" by Prof. V. Moiseyenko and Dr. M. Dergachov opens the section. It contains a useful review of optical phenomena in materials with a space modulation of dielectric constant at distances close to the light wavelengths (so called photonic band-gap structures or photonic crystals). Gaps in their photonic band structure represent frequency regions where electromagnetic waves are forbidden, irrespective of the spatial propagation directions. Since the photon density of states is equal to zero inside the band gaps, emission of light sources embedded in these crystals should be inhibited in these spectral regions. Besides the emission inhibition effect, a number of new optical phenomena in 3D photonic crystals, interesting from the applied point of view, are under intensive study now. The main research directions are the following: effects of light localization, radiation of photonic crystals filled with organic and inorganic luminophores near the edges of photonic band-gaps, radiation of quantum dots in photonic crystal volume, quantum optics phenomena in nano-structured materials based on photonic crystals and nonlinear optical substances, effects of the radiation field amplification in photonic crystals, increase of solar cells efficiency with the use of photonic crystals. Synthetic opal photonic crystals containing nonlinear optical substances give a good chance to observe quantum optics phenomena in spatially nonuniform media where the photon mean free path is close to the light wavelength. The following quantum optics phenomena are considered: luminescence, Raman scattering, and spontaneous parametric down-conversion. Experimental samples were made of nanodisperse globules of silica dioxide synthesized by authors. Results of spectral investigations are presented and discussed.

their applications in reconstructing open quantum system trajectories.

discussed.

The first section titled "Theoretical Fundamentals: Problem of Observables" includes four chapters united in the search for adequate mathematical apparatus for quantum electromagnetic field state description, taking into account experimental research possibilities. The first chapter "Description of Field States with Correlation Functions & Measurements in Quantum Optics" by Dr. S. Lyagushyn and Prof. A. Sokolovsky incorporates the discussion of basic approaches to field investigation in quantum optics. Since measurements with quantum detectors lead to Glauber correlation function and the Glauber-Sudarshan *P*-function is the most consumable tool for practical field description, such functions are regarded as an optimal way for field diagnostics. Then the Bogolyubov reduced description is constructed for a medium consisting of two-level emitters (the Dicke superfluorescence phenomenon) and plasma-field system. In such way, the connection is made using simultaneous correlation functions of field amplitudes for system evolution description, constructing differential equations for them, and coming to a quasiequilibrium statistical operator for system constituents at large times, the statistical operator permitting correlation function calculation. The necessity of considering binary correlation of field is substantiated. Such kinds of electrodynamics in media imply obtaining certain material equations. Various forms of correlation description are presented: oneparticle density matrix, Wigner distribution function, and correlation functions of Glauber type. Correlation functions in the theory of radiation transfer and corresponding equations are considered. A way to field evolution description on the basis of a generating functional and Glauber-Sudarshan distribution connected with it is proposed.

The second chapter is "Nonclassical Features of Superpositions of Coherent and Squeezed States for Electromagnetic Fields in Time-Varying Media" by Prof. Choi Jeong Ryeol. The author is interested in light behavior in media with varying characteristic parameters, the situation promising several interesting applications. A special method of field quantization based on the invariant operator theory is used. Thus deriving quantum solutions for time-dependent Hamiltonians becomes possible. The exact wave functions for the system with time-varying parameters can be derived in Fock, coherent, and squeezed states. Then superpositions of quantum states are considered in the search for nonclassical properties (high-order squeezing, subpoissonian photon statistics, and oscillations in the photon-number distribution). Such analysis is based on the Wigner distribution function, allowing us to know the phase space distribution connected to a simultaneous measurement of position and momentum. The Wigner distribution function is regarded as quasiprobability distribution function and is widely used in explaining intrinsic quantum features that have no classical analogue.

The third chapter "Photon Localization Revisited" by Prof. P. Saari is devoted to the intriguing problem that is traditionally under discussion in literature on quantum optics. Such considerations are urgent for needs of near-field optics, cavity quantum electrodynamics, and quantum computing. Some new optical phenomena connected with photon localization have drawn scientists' interest in recent years (see, for example Chapter 5). In textbooks the problems of position wave function and measurable quantities locality are compared. The author presents a qualified review together with his original results. Restrictions on photon localization set by the Paley-Wiener theorem and their seeming violation for certain two-dimensional wave packets are discussed.

X Preface

breadth of topics.

is proposed.

have no classical analogue.

quantum detecting schemes. The present title of the book reflects its real contents and

The first section titled "Theoretical Fundamentals: Problem of Observables" includes four chapters united in the search for adequate mathematical apparatus for quantum electromagnetic field state description, taking into account experimental research possibilities. The first chapter "Description of Field States with Correlation Functions & Measurements in Quantum Optics" by Dr. S. Lyagushyn and Prof. A. Sokolovsky incorporates the discussion of basic approaches to field investigation in quantum optics. Since measurements with quantum detectors lead to Glauber correlation function and the Glauber-Sudarshan *P*-function is the most consumable tool for practical field description, such functions are regarded as an optimal way for field diagnostics. Then the Bogolyubov reduced description is constructed for a medium consisting of two-level emitters (the Dicke superfluorescence phenomenon) and plasma-field system. In such way, the connection is made using simultaneous correlation functions of field amplitudes for system evolution description, constructing differential equations for them, and coming to a quasiequilibrium statistical operator for system constituents at large times, the statistical operator permitting correlation function calculation. The necessity of considering binary correlation of field is substantiated. Such kinds of electrodynamics in media imply obtaining certain material equations. Various forms of correlation description are presented: oneparticle density matrix, Wigner distribution function, and correlation functions of Glauber type. Correlation functions in the theory of radiation transfer and corresponding equations are considered. A way to field evolution description on the basis of a generating functional and Glauber-Sudarshan distribution connected with it

The second chapter is "Nonclassical Features of Superpositions of Coherent and Squeezed States for Electromagnetic Fields in Time-Varying Media" by Prof. Choi Jeong Ryeol. The author is interested in light behavior in media with varying characteristic parameters, the situation promising several interesting applications. A special method of field quantization based on the invariant operator theory is used. Thus deriving quantum solutions for time-dependent Hamiltonians becomes possible. The exact wave functions for the system with time-varying parameters can be derived in Fock, coherent, and squeezed states. Then superpositions of quantum states are considered in the search for nonclassical properties (high-order squeezing, subpoissonian photon statistics, and oscillations in the photon-number distribution). Such analysis is based on the Wigner distribution function, allowing us to know the phase space distribution connected to a simultaneous measurement of position and momentum. The Wigner distribution function is regarded as quasiprobability distribution function and is widely used in explaining intrinsic quantum features that

The third chapter "Photon Localization Revisited" by Prof. P. Saari is devoted to the intriguing problem that is traditionally under discussion in literature on quantum The fourth chapter "Fusion Frames and Dynamics of Open Quantum Systems" by Prof. A. Jamiołkowski deals with the problems of quantum tomography, which is a procedure for reconstructing properties of a quantum object on the basis of experimentally accessible data. The state reconstruction requires identifying the quorum of observables, providing a possibility to determine expectation values of physical quantities for which no measuring apparatuses are available. The problem is discussed in terms of a set of density operators on the Hilbert space of the quantum system states. The main purpose of this contribution is to discuss properties of some Krylov subspaces in a given Hilbert space as natural examples of fusion frames and their applications in reconstructing open quantum system trajectories.

The second section of the book under the title "Quantum Phenomena with Laser Radiation" includes three chapters. The fifth chapter "Quantum Optics Phenomena in Synthetic Opal Photonic Crystals" by Prof. V. Moiseyenko and Dr. M. Dergachov opens the section. It contains a useful review of optical phenomena in materials with a space modulation of dielectric constant at distances close to the light wavelengths (so called photonic band-gap structures or photonic crystals). Gaps in their photonic band structure represent frequency regions where electromagnetic waves are forbidden, irrespective of the spatial propagation directions. Since the photon density of states is equal to zero inside the band gaps, emission of light sources embedded in these crystals should be inhibited in these spectral regions. Besides the emission inhibition effect, a number of new optical phenomena in 3D photonic crystals, interesting from the applied point of view, are under intensive study now. The main research directions are the following: effects of light localization, radiation of photonic crystals filled with organic and inorganic luminophores near the edges of photonic band-gaps, radiation of quantum dots in photonic crystal volume, quantum optics phenomena in nano-structured materials based on photonic crystals and nonlinear optical substances, effects of the radiation field amplification in photonic crystals, increase of solar cells efficiency with the use of photonic crystals. Synthetic opal photonic crystals containing nonlinear optical substances give a good chance to observe quantum optics phenomena in spatially nonuniform media where the photon mean free path is close to the light wavelength. The following quantum optics phenomena are considered: luminescence, Raman scattering, and spontaneous parametric down-conversion. Experimental samples were made of nanodisperse globules of silica dioxide synthesized by authors. Results of spectral investigations are presented and discussed.

The sixth chapter "Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field" by Prof. S. Roshchupkin et al. describes the great achievements of this group in investigating laser field influence on kinematics and cross-sections of various quantum electrodynamics processes of the both first and second orders in the fine structure constant, such as resonant spontaneous bremstrahlung of an electron scattered by a nucleus, resonant photocreation of electron-positron pair on a nucleus, resonant scattering of a lepton by a lepton, and resonant scattering of a photon by an electron in the field of a pulsed light wave. This scientific direction has been of great interest for many years. Theoretical study of such processes is based on solutions of the Dirac's equation for an electron in the field of a plane electromagnetic wave. Resonant character of strong field influence has common features with resonant interactions in quantum optics. The wide research activities presented in the chapter have resulted in some conclusions to be tested in experiments with accelerators in presence of strong fields.

The seventh chapter "Cold Atoms Experiments: Influence of Laser Intensity Imbalance on Cloud Formation" by Dr. I. Olivares and Dr. F. Aguilar can be regarded as an example showing the possibilities of modern laser experiments. The authors deal with a magneto optical trap with the intent to obtain a cloud of cold atoms. They describe an experiment that proved the stability of the cloud and the optical method to vary the laser intensity of the pump and trap beams. The influence of laser intensity imbalance on cloud formation is investigated and values for the threshold intensity of lasers supporting cloud formation are obtained. The technique of saturated absorption spectroscopy is described. The theoretical analysis is performed in terms of level populations and optical Bloch equations that is conventional for quantum optics.

The book's "geography" – from Estonia to Chile – shows the wide interest for the problems under discussion all over the world. The relative majority of authors from Ukraine reflect both the history of monograph formation and great potential of Ukrainian physics.

I am grateful to InTech's publishing team and especially to Publishing Process Manager Ms. Marina Jozipovic for their constructive approach to the book formation, understanding the authors' problems, and a tolerant attitude for my delays. I would like to thank my older colleagues Prof. A. Sokolovsky and Prof. V. Skalozub for their support at different stages of the Project.

#### **Sergiy Lyagushyn**,

Associate Professor of Theoretical Physics Department of Oles' Honchar Dnipropetrovs'k National University Ukraine
