Preface

This short book provides readers with some basic concepts of single photons and the latest developments in the field. The book reviews the most important method of generating single photons in experiments and it also introduces an emerging field of single-photon isolation.

The first section provides a very simple picture of the concept of evolution of the photon" from the classical to the quantum formalisms, and the quantization of the light field as a description of the photon. The next section focuses on recent experiments exploiting optical nonlinear processes to generate and manipulate single photons. In the last section, nonreciprocal quantum devices are designed to isolate the backscattering of single photons.

The book aims to present some basic knowledge of single photons to readers in a quick read but with not many details.

> **Keyu Xia** Professor, Nanjing University, Jiangsu, China

Section 1

Basic Concept

**1**

Section 1 Basic Concept

**Chapter 1**

**Abstract**

Do We Manipulate Photons or

Structured Light?

*ChandraSekhar Roychoudhuri*

spin, angular momentum, etc.

**1. Introduction**

**3**

Diffractive EM Waves to Generate

In the domain of light emissions, quantum mechanics has been an immensely successful guiding tool for us. In the propagation of light and optical instrument design, Huygens-Fresnel diffraction integral (HFDI) (or its advanced versions) and Maxwell's wave equation are continuing to be the essential guiding tools for optical scientists and engineers. In fact, most branches of optical science and engineering, like optical instrument design, image processing, Fourier optics, Holography, etc., cannot exist without using the foundational postulates behind the Huygens-Fresnel diffraction integral. Further, the field of structured light is also growing where phases and the state of polarizations are manipulated usually with suitable classical macro-devices to create wave fronts that restructured through light-matter interactions through these devices. Mathematical modeling of generating such complex wave fronts generally follows classical concepts and classical macro tools of physical optics. Some of these complex light beams can impart mechanical angular momentum and spin-like properties to material particles inserted inside these structured beams because of their electromagnetic dipolar properties and/or structural anisotropy. Does that mean these newly structured beams have acquired new quantum properties without being generated through quantum devices and quantum transitions? In this chapter, we bridge the classical and quantum formalism by defining a hybrid photon (HP). HP is a quantum of energy, hν, at the initial moment of emission. It then immediately evolves into a classical time-finite wave packet, still transporting the original energy, hν, with a classical carrier frequency ν (oscillation of the E-vector). This chapter will raise enquiring questions whether all these observed "quantum-like" behaviors are manifestations of the joint properties of interacting material particles with classical EM waves or are causal implications of the existence of propagation of "indivisible light quanta" with exotic properties like

**Keywords:** structured light, hybrid photon, non-interaction of light (NIW),

Structured light is a matured applied field of study. It has been steadily inventing many new tools and techniques to manipulate and study, from nanoparticles to molecules to atoms. Other chapters of this book have described these developments.

Huygens principle, photoelectric effect, semiclassical model

## **Chapter 1**

## Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?

*ChandraSekhar Roychoudhuri*

## **Abstract**

In the domain of light emissions, quantum mechanics has been an immensely successful guiding tool for us. In the propagation of light and optical instrument design, Huygens-Fresnel diffraction integral (HFDI) (or its advanced versions) and Maxwell's wave equation are continuing to be the essential guiding tools for optical scientists and engineers. In fact, most branches of optical science and engineering, like optical instrument design, image processing, Fourier optics, Holography, etc., cannot exist without using the foundational postulates behind the Huygens-Fresnel diffraction integral. Further, the field of structured light is also growing where phases and the state of polarizations are manipulated usually with suitable classical macro-devices to create wave fronts that restructured through light-matter interactions through these devices. Mathematical modeling of generating such complex wave fronts generally follows classical concepts and classical macro tools of physical optics. Some of these complex light beams can impart mechanical angular momentum and spin-like properties to material particles inserted inside these structured beams because of their electromagnetic dipolar properties and/or structural anisotropy. Does that mean these newly structured beams have acquired new quantum properties without being generated through quantum devices and quantum transitions? In this chapter, we bridge the classical and quantum formalism by defining a hybrid photon (HP). HP is a quantum of energy, hν, at the initial moment of emission. It then immediately evolves into a classical time-finite wave packet, still transporting the original energy, hν, with a classical carrier frequency ν (oscillation of the E-vector). This chapter will raise enquiring questions whether all these observed "quantum-like" behaviors are manifestations of the joint properties of interacting material particles with classical EM waves or are causal implications of the existence of propagation of "indivisible light quanta" with exotic properties like spin, angular momentum, etc.

**Keywords:** structured light, hybrid photon, non-interaction of light (NIW), Huygens principle, photoelectric effect, semiclassical model

## **1. Introduction**

Structured light is a matured applied field of study. It has been steadily inventing many new tools and techniques to manipulate and study, from nanoparticles to molecules to atoms. Other chapters of this book have described these developments. The purpose of this chapter is to promote the development of out-of-box enquiring questions in physics leveraging the topic of structured light. It is a complex thinking and analytical process to describe a physical phenomenon simply based upon reproducible experimental data. This is because experimental data generation requires detector and deductee to undergo some physical transformation in their relevant parameters after exchanging some energy guided by some allowed force of interaction between them. Since we cannot directly observe the details of the physical interaction process, we cannot be certain from the properties of the measured data as to which property belongs to the detector and which belongs to the deductee. We have not been addressing this important enquiring question explicitly in physics. The field of structured light is a good test optical phenomenon to explore this enquiring question.

transformation, which becomes our data. If the detecting element is inherently quantum mechanical in nature, then the amount of energy Δ*Emn* ¼ *hνmn* is absorbed by the detector out of the EM waves and will correspond to the specific quantum transition. However, all light-matter interactions are frequency dependent since all materials are individual dipolar atoms and molecules or their assemblies in solid or liquid states. X-rays and *γ*-rays do not interact with quantum mechanical Siphotoelectric detectors or classical photothermal detectors. We need appropriate materials where X-rays can stimulate the electrons in the inner shells of atoms and

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?*

*DOI: http://dx.doi.org/10.5772/intechopen.88849*

The strength of the *evidence-based science* lies with the corroboration of evidences with a suitable mathematical model. The model must help us to *visualize the interaction processes* that give rise to measurable data (evidence). This is the foundation of our causal approach to explore the laws of nature. This causal approach allows us to keep refining both the measurements and the modeling as we keep integrating diverse observations into a broader and well-validated theory. This is how our scientific advances have been continuing for centuries. Therefore, let us explore the physical process steps behind the generic detection/interaction processes

1.All measured *data* are some quantitative *physical transformations* in some

2.All physical transformations must be triggered by some physical interaction (stimulation) process, followed by energy exchange between the detector and the deductee. Discouraging the visualization of such invisible interaction processes has been the key mystifying reason behind our "working" theories,

3.All energy exchange must be guided (allowed) by some specific and allowed force of interaction existing between the detector and the deductee. Our continuing failure to understand the origin of all forces and unify them is the

whose purpose has been limited to validate only the measurable data.

key bottlenecks behind the causal advancement in modern physics.

4.All forces, short or long range, have finite physical ranges. Therefore, all interactions are fundamentally "local" or physical range dependent.

Thus, we cannot generate observable (measurable) data without some physical transformation in an interactant (detector) whose intrinsic physical properties dictate its specific response characteristics to one or the other force to participate in any interaction, leading to a specified amount of energy exchange leading to the observable transformation. Obviously, recordable data generation is not possible if the interactants are physically beyond the range of their mutual force of interaction. Causal physics require that the interactants recognize each other through their mutually allowed force of interaction. Without a direct hit of a well-collimated laser beam within the active area of a detector of a power meter, we cannot ascertain the energy of the laser beam. Interaction-free data generation cannot take place in the causal world. "Spooky action at a distance" is an unfortunate cultural phenomenon that wants to mystify physics. Nature is systematically causal. That is why our "cause-effect" inter-relating causal mathematical equations, through centuries, have remained the key guiding tool to explore nature. Nature is not mystical.

Measured and analyzed "elliptical polarization" does not imply that the resultant electrical vector of the light beam is rotating circularly as the composite light beam (two collinear, phase-steady, and orthogonally polarized beams with 90° relative

*γ*-rays can stimulate the nuclear energy levels at the core of atoms.

detector element induced by a deductee-element.

(Ch. 12, in [5]):

**5**

Beams of structured light are generated by using classical optical components and the analytical tools of classical optics, which are Maxwell's wave equation and Huygens-Fresnel diffraction integral (or its advanced versions). Then, the concept of "indivisible light quanta" must have come from Einstein's paper on photoelectric effect [1]. However, Lamb et al. [2–4] have clearly shown that the use of semiclassical model, classical light, and quantized atoms yields a much more causal and selfconsistent model for light-matter interactions. This chapter, therefore, strengthens this concept behind the semiclassical model by underscoring some neglected but fundamental nature of light from two fundamental angles—"We never see light" (Section 2) and "Light does not see light" (Section 3). Then we discuss the consequences of assigning detector's intrinsic quantum properties to the energy donating classical and Maxwellian light waves as we do for the photoelectric effect (Section 4). Next, in Section 5, we discuss the consequences of ignoring interaction process visualization, which guides us to accept the necessity of introducing the concept of hybrid photon model. Hybrid photon model eliminates the need for accepting the postulate of "wave-particle duality." This duality postulate actually originated in late 1600s during debate between Newton ("corpuscular") and Huygens ("secondary wavelets"), and they agreed that their debate arose out of their ignorance about the deeper structure of light. Unfortunately, founders of mathematical formalism of quantum have promoted the "wave-particle duality" as the new confirmed knowledge. In reality, this postulate should energize us to keep exploring the deeper issues behind quantized emission and absorption of light and classical propagation properties of light. The last section on discussion underscores that we should always try to reevaluate working theories beyond its prevailing successes so we can advance our current understanding. Then discover new phenomena and then invent new tools and technologies.

## **2. We never "see" light**

We only perceive or measure the physical transformation induced by light energy in material bodies, which have their own unique response characteristics due to their unique response properties to light. Therefore, assigning any new physical property, to a physical entity under study, should be done carefully to ascertain that the observed (measured) property is not that of the detecting entity. This is especially important for light. We always infer the incidence (presence) of light after observing (or perceiving) some physical transformation in the detecting element. It could take place through a wide variety of already known phenomena like photoelectric effect, photochemical effect, photothermal effect, photo-acoustic effect, etc. In all such cases, a finite amount of energy from the EM wave is absorbed by the detecting element to undergo some quantitative physical

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light? DOI: http://dx.doi.org/10.5772/intechopen.88849*

transformation, which becomes our data. If the detecting element is inherently quantum mechanical in nature, then the amount of energy Δ*Emn* ¼ *hνmn* is absorbed by the detector out of the EM waves and will correspond to the specific quantum transition. However, all light-matter interactions are frequency dependent since all materials are individual dipolar atoms and molecules or their assemblies in solid or liquid states. X-rays and *γ*-rays do not interact with quantum mechanical Siphotoelectric detectors or classical photothermal detectors. We need appropriate materials where X-rays can stimulate the electrons in the inner shells of atoms and *γ*-rays can stimulate the nuclear energy levels at the core of atoms.

The strength of the *evidence-based science* lies with the corroboration of evidences with a suitable mathematical model. The model must help us to *visualize the interaction processes* that give rise to measurable data (evidence). This is the foundation of our causal approach to explore the laws of nature. This causal approach allows us to keep refining both the measurements and the modeling as we keep integrating diverse observations into a broader and well-validated theory. This is how our scientific advances have been continuing for centuries. Therefore, let us explore the physical process steps behind the generic detection/interaction processes (Ch. 12, in [5]):


Thus, we cannot generate observable (measurable) data without some physical transformation in an interactant (detector) whose intrinsic physical properties dictate its specific response characteristics to one or the other force to participate in any interaction, leading to a specified amount of energy exchange leading to the observable transformation. Obviously, recordable data generation is not possible if the interactants are physically beyond the range of their mutual force of interaction. Causal physics require that the interactants recognize each other through their mutually allowed force of interaction. Without a direct hit of a well-collimated laser beam within the active area of a detector of a power meter, we cannot ascertain the energy of the laser beam. Interaction-free data generation cannot take place in the causal world. "Spooky action at a distance" is an unfortunate cultural phenomenon that wants to mystify physics. Nature is systematically causal. That is why our "cause-effect" inter-relating causal mathematical equations, through centuries, have remained the key guiding tool to explore nature. Nature is not mystical.

Measured and analyzed "elliptical polarization" does not imply that the resultant electrical vector of the light beam is rotating circularly as the composite light beam (two collinear, phase-steady, and orthogonally polarized beams with 90° relative

The purpose of this chapter is to promote the development of out-of-box enquiring questions in physics leveraging the topic of structured light. It is a complex thinking and analytical process to describe a physical phenomenon simply based upon reproducible experimental data. This is because experimental data generation requires detector and deductee to undergo some physical transformation in their relevant parameters after exchanging some energy guided by some allowed force of interaction between them. Since we cannot directly observe the details of the physical interaction process, we cannot be certain from the properties of the measured data as to which property belongs to the detector and which belongs to the deductee. We have not been addressing this important enquiring question explicitly in physics. The field of structured light is a good test optical phenomenon to explore

Beams of structured light are generated by using classical optical components and the analytical tools of classical optics, which are Maxwell's wave equation and Huygens-Fresnel diffraction integral (or its advanced versions). Then, the concept of "indivisible light quanta" must have come from Einstein's paper on photoelectric effect [1]. However, Lamb et al. [2–4] have clearly shown that the use of semiclassical model, classical light, and quantized atoms yields a much more causal and selfconsistent model for light-matter interactions. This chapter, therefore, strengthens this concept behind the semiclassical model by underscoring some neglected but fundamental nature of light from two fundamental angles—"We never see light" (Section 2) and "Light does not see light" (Section 3). Then we discuss the consequences of assigning detector's intrinsic quantum properties to the energy donating classical and Maxwellian light waves as we do for the photoelectric effect (Section 4). Next, in Section 5, we discuss the consequences of ignoring interaction process visualization, which guides us to accept the necessity of introducing the concept of hybrid photon model. Hybrid photon model eliminates the need for accepting the postulate of "wave-particle duality." This duality postulate actually originated in late 1600s during debate between Newton ("corpuscular") and Huygens ("secondary wavelets"), and they agreed that their debate arose out of their ignorance about the deeper structure of light. Unfortunately, founders of mathematical formalism of quantum have promoted the "wave-particle duality" as the new confirmed knowledge. In reality, this postulate should energize us to keep exploring the deeper issues behind quantized emission and absorption of light and classical propagation properties of light. The last section on discussion underscores that we should always try to reevaluate working theories beyond its prevailing successes so we can advance our current understanding. Then discover new phenomena and then invent new

We only perceive or measure the physical transformation induced by light energy in material bodies, which have their own unique response characteristics due to their unique response properties to light. Therefore, assigning any new physical property, to a physical entity under study, should be done carefully to ascertain that the observed (measured) property is not that of the detecting entity. This is especially important for light. We always infer the incidence (presence) of light after observing (or perceiving) some physical transformation in the detecting element. It could take place through a wide variety of already known phenomena like photoelectric effect, photochemical effect, photothermal effect, photo-acoustic

effect, etc. In all such cases, a finite amount of energy from the EM wave is absorbed by the detecting element to undergo some quantitative physical

this enquiring question.

*Single Photon Manipulation*

tools and technologies.

**4**

**2. We never "see" light**

phase delay) is propagating with the resultant E-vector helically rotating around the Poynting vector. In this assumed and imaginary model, the energy of the composite light beam would have been also oscillating due to time-varying resultant amplitude of the E-vector. This would have implied that nature is violating the law of conservation of energy. Fortunately, in this case, our mathematics has been guiding us along the correct and causal path. Jones' matrix has been constructed to find the final energy of a composite light beam as the sum of the two separate energies contained in each of the two orthogonal polarization. The energy in each of the two orthogonal components is the square modulus of the sum of the X-component amplitudes and the Y-component amplitudes, carried out separately. Interested readers should consult Ch. 9 on polarization phenomenon in Ref. [5]. The chapter underscores, using elementary mathematics and the bulk dipolar polarizability *χ*, that without explicitly inserting this light-matter interaction parameter, the understanding of the ongoing physical process becomes difficult and confusing.

> *For I do not find that any one has yet given a probable explanation of the first and most notable phenomena of light, namely ……how visible rays, coming from an infinitude of diverse places, cross one another without hindering one another in any*

*Huygens' visualization of the wave propagation process through indefinite number of secondary wavelets, diffractively evolving through each other, "unperturbed by each other, cross one another without hindering one*

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?*

In p. 4 of his 1690 book, Huygens clearly anticipated the existence of a universal tension field, like pressure tension of air, but without any material particles, which facilitates the perpetual propagation of waves, as sound does in the air. In 1817, Fresnel gave the mathematical structure to the Huygens principle in

which NIW was automatically built in Huygens-Fresnel diffraction integral (HFDI). The integral literally propagates innumerable spherical wavelets that keep evolving through each other while diffractively evolving as co-propagating and cross propagating through each other [11]. *E P*ð Þ det*:* is the resultant amplitude:

*U P*ð Þ *source:*

the HFDI has automatically incorporated the NIW property of the waves.

Notice that the above HFDI propagates all spherical wave fronts out of every source point to the detector plane, irrespective of its distance *rs*!*<sup>d</sup>* from the source plane. These amplitude wave fronts evolve through each other completely independent of each other irrespective of how far they are propagating. In other words,

After Maxwell's wave equation was developed (1867), it was found that HFDI is a solution to Helmholtz equation, a time-independent form of Maxwell's generalized wave equation. Maxwell's wave equation accepts any linear superposition of wave amplitudes as its solution. The physical meaning, in the context of NIW, is that wave properties of the individual propagating wave remain unaltered as they cross propagate and/or co-propagate through each other. In other words, light does not interfere with light in the absence of interacting material, which we have logically derived in section on "We do not see light," except through the "eyes" of

It is then obvious that the generation and spatial superposition of multiple complex light wave fronts will continue to diffractively evolve and co-propagate as independent beams. However, when they finally interact with some materials, the energy transfer to any interacting material will be the square modulus of the sum of the finally evolved "local" wave front incident on the material. If the material is an anisotropic polarized detector, it will respond to the square modulus of the sum of all the amplitude components projected on to its polarization axis. If it is a very small-suspended anisotropic particle and the state of polarization is rotating slowly

exp ð Þ *ikrs*!*<sup>d</sup> rs*!*<sup>d</sup>*

cos *θ ds* (1)

*way. From p. 2 in [7].*

**Figure 2.**

interacting materials.

**7**

*E P*ð Þ¼ det*:*

*another in any way" [7]. This is non-interaction of waves (NIW) [5].*

*DOI: http://dx.doi.org/10.5772/intechopen.88849*

�*i λ* ðð Σ

## **3. Light does not "see" light**

The light wave amplitudes cross propagate and co-propagate through each other in the absence of interacting materials. This is why experimental astrophysics can image and analyze individual distant galaxies or stars even though the light selected by a telescope has crossed through the light beams of innumerable galaxies and/or stars. This is the same physical reason why we can see (recognize) each other from a distance, even though the scattered light beams from innumerable other faces and objects are crossing through each other. Alhazen experimentally validated this noninteraction of waves, or NIW, about a thousand years ago [6]. This brilliantly simple experiment is sketched in **Figure 1**.

Alhazen generated the inverted images of a set of candles through a pinhole camera. He found that blocking anyone or more candles does not create any changes in the images of the other candles. Inverted images clearly underscored that the light from all the candles were crossing through each other at the tiny pinhole. Alhazen underscored that he did not understand the deeper nature of light. He was humble.

Much later, Huygens formally postulated NIW in his 1690 book [7] when he presented his principle of wave propagation visualizing the process as the perpetual generation of innumerable secondary wavelets out of every point on the wave front. This also implied that the space is an energetic tension field to be able to support the perpetual wavelet generation and propagation (Ch. 11 in [5, 8–10]). Huygens explicitly articulated non-interaction of waves (**Figure 2**):

#### **Figure 1.**

*Alhazen's ancient experiment forced him to conclude that light does not interact with light. We still are struggling with the wave-particle duality [5].*

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light? DOI: http://dx.doi.org/10.5772/intechopen.88849*

**Figure 2.**

phase delay) is propagating with the resultant E-vector helically rotating around the Poynting vector. In this assumed and imaginary model, the energy of the composite light beam would have been also oscillating due to time-varying resultant amplitude of the E-vector. This would have implied that nature is violating the law of conservation of energy. Fortunately, in this case, our mathematics has been guiding us along the correct and causal path. Jones' matrix has been constructed to find the final energy of a composite light beam as the sum of the two separate energies contained in each of the two orthogonal polarization. The energy in each of the two orthogonal components is the square modulus of the sum of the X-component amplitudes and the Y-component amplitudes, carried out separately. Interested readers should consult Ch. 9 on polarization phenomenon in Ref. [5]. The chapter underscores, using elementary mathematics and the bulk dipolar polarizability *χ*, that without explicitly inserting this light-matter interaction parameter, the under-

standing of the ongoing physical process becomes difficult and confusing.

The light wave amplitudes cross propagate and co-propagate through each other in the absence of interacting materials. This is why experimental astrophysics can image and analyze individual distant galaxies or stars even though the light selected by a telescope has crossed through the light beams of innumerable galaxies and/or stars. This is the same physical reason why we can see (recognize) each other from a distance, even though the scattered light beams from innumerable other faces and objects are crossing through each other. Alhazen experimentally validated this noninteraction of waves, or NIW, about a thousand years ago [6]. This brilliantly

Alhazen generated the inverted images of a set of candles through a pinhole camera. He found that blocking anyone or more candles does not create any changes in the images of the other candles. Inverted images clearly underscored that the light from all the candles were crossing through each other at the tiny pinhole. Alhazen underscored that he did not understand the deeper nature of light. He was

Much later, Huygens formally postulated NIW in his 1690 book [7] when he presented his principle of wave propagation visualizing the process as the perpetual generation of innumerable secondary wavelets out of every point on the wave front. This also implied that the space is an energetic tension field to be able to support the perpetual wavelet generation and propagation (Ch. 11 in [5, 8–10]). Huygens

*Alhazen's ancient experiment forced him to conclude that light does not interact with light. We still are*

**3. Light does not "see" light**

*Single Photon Manipulation*

simple experiment is sketched in **Figure 1**.

explicitly articulated non-interaction of waves (**Figure 2**):

humble.

**Figure 1.**

**6**

*struggling with the wave-particle duality [5].*

*Huygens' visualization of the wave propagation process through indefinite number of secondary wavelets, diffractively evolving through each other, "unperturbed by each other, cross one another without hindering one another in any way" [7]. This is non-interaction of waves (NIW) [5].*

*For I do not find that any one has yet given a probable explanation of the first and most notable phenomena of light, namely ……how visible rays, coming from an infinitude of diverse places, cross one another without hindering one another in any way. From p. 2 in [7].*

In p. 4 of his 1690 book, Huygens clearly anticipated the existence of a universal tension field, like pressure tension of air, but without any material particles, which facilitates the perpetual propagation of waves, as sound does in the air.

In 1817, Fresnel gave the mathematical structure to the Huygens principle in which NIW was automatically built in Huygens-Fresnel diffraction integral (HFDI). The integral literally propagates innumerable spherical wavelets that keep evolving through each other while diffractively evolving as co-propagating and cross propagating through each other [11]. *E P*ð Þ det*:* is the resultant amplitude:

$$E(P\_{\text{det.}}) = \frac{-i}{\lambda} \iint\_{\Sigma} U(P\_{\text{source.}}) \frac{\exp\left(ikr\_{\text{s}\to d}\right)}{r\_{\text{s}\to d}} \cos\theta \,d\mathbf{s} \tag{1}$$

Notice that the above HFDI propagates all spherical wave fronts out of every source point to the detector plane, irrespective of its distance *rs*!*<sup>d</sup>* from the source plane. These amplitude wave fronts evolve through each other completely independent of each other irrespective of how far they are propagating. In other words, the HFDI has automatically incorporated the NIW property of the waves.

After Maxwell's wave equation was developed (1867), it was found that HFDI is a solution to Helmholtz equation, a time-independent form of Maxwell's generalized wave equation. Maxwell's wave equation accepts any linear superposition of wave amplitudes as its solution. The physical meaning, in the context of NIW, is that wave properties of the individual propagating wave remain unaltered as they cross propagate and/or co-propagate through each other. In other words, light does not interfere with light in the absence of interacting material, which we have logically derived in section on "We do not see light," except through the "eyes" of interacting materials.

It is then obvious that the generation and spatial superposition of multiple complex light wave fronts will continue to diffractively evolve and co-propagate as independent beams. However, when they finally interact with some materials, the energy transfer to any interacting material will be the square modulus of the sum of the finally evolved "local" wave front incident on the material. If the material is an anisotropic polarized detector, it will respond to the square modulus of the sum of all the amplitude components projected on to its polarization axis. If it is a very small-suspended anisotropic particle and the state of polarization is rotating slowly

in time and if the inertia permits, the anisotropic particle will rotate with the rotating polarizing field since most materials strongly respond to the resultant Evector. However, the original set of multiple co-propagating wave amplitudes does not reorganize themselves into a single composite wave front because of the overriding NIW property of waves.

## **4. Consequences of assigning detector's properties to the energy donating entity**

Can we logically confirm that the emission of a single photoelectron proves the existence of light as "indivisible light quanta"? We should recognize that the individual "clicks," which we register in photon counting electronics, are actually a brief current pulse, probably, consisting of billions of amplified electrons. It may not be difficult to validate that this highly amplified current pulse has been originally triggered by a single photoelectron. However, releasing a quantummechanically bound electron does not necessarily require the presence of a quantum photon of energy *hνmn* ¼ Δ*Emn*. The quantum cupful of energy Δ*Emn* can be acquired by a quantum entity from almost any source of energy under appropriate condition of interactions. Ancient humans used to create fire by using sparks generated by fast mechanical collisions between a pair of stones. They had no idea that they were inducing quantum transitions in the molecules of the stones while transferring the classical kinetic energy out of their moving hands! This is why the quantum formalism does not require any quantum postulate that energy providers to induce quantum transitions have to have energy-matching quantum states. In fact, Boltzmann's classical statistical thermal population density formula has been co-opted by the quantum mechanics. Un-quantized thermal energy can be absorbed during thermal collisions by quantum entities to fill up their quantum cups while accepting only that much of energy that fills up their quantum cups [12].

However, during very late 1800, emission of photoelectrons from photocathode showed some uniqueness. Below some optical frequency, there was no photoelectron emission in spite of increasing the radiation density. Young Einstein correctly surmised that there is some "quantumness" hidden behind this frequency dependence and no electron emission after a cutoff frequency. Unfortunately, Einstein assigned this quantumness to light, instead of to the electrons. However, we now know that electrons are always bound quantum mechanically in all materials, even when they are bound collectively in quantum energy bands in solid state [15]. Here, we must recognize that Bohr atom was formulated in 1913 and Quantum Mechanics was formulated in 1925, many years after the 1905 paper by Einstein on the photo-

*Transient quantum photon, immediately after emission from an atom, evolves into a diffractively spreading classical wave packet with quasi-exponential temporal envelope. All the quantum predictions are preserved. The total energy carried by the wave packet is ΔEmn* ¼ *hνmn with the unique carrier frequency νmn. The Fourier transform of the quasi-exponential envelope gives a spectral line width close to Lorentzian, which is the normal*

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?*

electric effect. However, had Einstein correctly assigned the quantumness,

**5. Consequences of ignoring interaction process visualization: the**

Let us first recognize that Einstein's photoelectric equation is an energybalancing equation to match the observed data. This is measurable data modeling epistemology (MDM-E). We need to incorporate interaction process mapping epistemology (IPM-E) over and above MDM-E. Einstein's formulation did not embody light-matter interaction process, as we have underscored in the section on "Light does not see light." For accurate semiclassical derivation of the photoelectric effect, the readers should consult the following references [2–4]. We will present here only a heuristic derivation of Einstein's energy-balancing photoelectric equation, but starting from light-matter amplitude-amplitude stimulation, the E-vector of light

the amplitude envelope of the light pulse (see **Figure 3**).

**necessity of the hybrid photon model**

**Figure 3.**

**9**

*spectral line width of spontaneous emissions.*

*DOI: http://dx.doi.org/10.5772/intechopen.88849*

observed in photoelectric effect, to electrons instead of to light, he would have been able to formulate quantum mechanics the Einsteinian way some 20 years earlier. We believe that Planck's view of light is correct. EM waves are classical waves, solutions of Maxwell's equation, and propagates following Huygens secondary wavelets. In fact, Refs. [2–4] have derived the equation for photoelectric emission using the semiclassical method. Here, we present a heuristic approach to present the hybrid photon model that is a quantum at the moment of emission but a classical wave packet with a quasi-exponential temporal envelope. The total energy under the envelope corresponds to the QM predictions: (i) total energy of the wave packet is Δ*Emn* ¼ *hνmn*, and (ii) the quasi-exponential envelope assures the observed spontaneous emission line width as very close to Lorentzian, the Fourier transform of

Toward the end of his life, Einstein, the originator of the concept of "indivisible light quanta," clearly stated that even after "50 years of brooding," he still did not understand "what are light quanta" [13]. Author was inspired by Einstein's doubt and delved into exploring the nature of light for many decades [14–16].

In the section on "We do not see light," we have underscored that detectors see light based on their internal physical properties. This is why visual observation of classical interferometry never pointed us to the quantumness of light. In fact, Newton was the first inventor of two-beam interferometry. He measured the radius of curvature of his plano-convex lens by putting it on a flat mirror and shining light from the top. Note that the debate over wave-particle duality started long time ago during late 1600 by Newton ("corpuscular") and Huygens ("wavelet"). Nevertheless, they recognized that their debate represented their deeper ignorance about the fundamental nature of light. We still have not fully understood the deeper nature of light. Therefore, we should not make the "wave-particle duality" as our confirmed new knowledge. We should humbly continue to explore the deeper nature of light. That is the purpose of this chapter.

The first solid reasoning behind theorizing emission and absorption of light from materials in discrete energy packets was presented by Planck in 1900 to match analytically the already measured blackbody radiation curve. However, Planck maintained his understanding that the quantum processes are real only during the instants of emission and absorption. After emission, the EM energy packet immediately evolves into Huygens' diffractive wavelets. This is how the diffractive radiation achieves the state of equilibrium energy density within the enclosed blackbody cavity [14].

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light? DOI: http://dx.doi.org/10.5772/intechopen.88849*

#### **Figure 3.**

in time and if the inertia permits, the anisotropic particle will rotate with the rotating polarizing field since most materials strongly respond to the resultant Evector. However, the original set of multiple co-propagating wave amplitudes does not reorganize themselves into a single composite wave front because of the over-

**4. Consequences of assigning detector's properties to the energy**

nally triggered by a single photoelectron. However, releasing a quantum-

accepting only that much of energy that fills up their quantum cups [12].

and delved into exploring the nature of light for many decades [14–16].

Toward the end of his life, Einstein, the originator of the concept of "indivisible light quanta," clearly stated that even after "50 years of brooding," he still did not understand "what are light quanta" [13]. Author was inspired by Einstein's doubt

In the section on "We do not see light," we have underscored that detectors see light based on their internal physical properties. This is why visual observation of classical interferometry never pointed us to the quantumness of light. In fact, Newton was the first inventor of two-beam interferometry. He measured the radius of curvature of his plano-convex lens by putting it on a flat mirror and shining light from the top. Note that the debate over wave-particle duality started long time ago during late 1600 by Newton ("corpuscular") and Huygens ("wavelet"). Nevertheless, they recognized that their debate represented their deeper ignorance about the fundamental nature of light. We still have not fully understood the deeper nature of light. Therefore, we should not make the "wave-particle duality" as our confirmed new knowledge. We should humbly continue to explore the deeper nature of light.

The first solid reasoning behind theorizing emission and absorption of light from

materials in discrete energy packets was presented by Planck in 1900 to match analytically the already measured blackbody radiation curve. However, Planck maintained his understanding that the quantum processes are real only during the instants of emission and absorption. After emission, the EM energy packet immediately evolves into Huygens' diffractive wavelets. This is how the diffractive radi-

ation achieves the state of equilibrium energy density within the enclosed

mechanically bound electron does not necessarily require the presence of a quantum photon of energy *hνmn* ¼ Δ*Emn*. The quantum cupful of energy Δ*Emn* can be acquired by a quantum entity from almost any source of energy under appropriate condition of interactions. Ancient humans used to create fire by using sparks generated by fast mechanical collisions between a pair of stones. They had no idea that they were inducing quantum transitions in the molecules of the stones while transferring the classical kinetic energy out of their moving hands! This is why the quantum formalism does not require any quantum postulate that energy providers to induce quantum transitions have to have energy-matching quantum states. In fact, Boltzmann's classical statistical thermal population density formula has been co-opted by the quantum mechanics. Un-quantized thermal energy can be absorbed during thermal collisions by quantum entities to fill up their quantum cups while

Can we logically confirm that the emission of a single photoelectron proves the existence of light as "indivisible light quanta"? We should recognize that the individual "clicks," which we register in photon counting electronics, are actually a brief current pulse, probably, consisting of billions of amplified electrons. It may not be difficult to validate that this highly amplified current pulse has been origi-

riding NIW property of waves.

That is the purpose of this chapter.

blackbody cavity [14].

**8**

**donating entity**

*Single Photon Manipulation*

*Transient quantum photon, immediately after emission from an atom, evolves into a diffractively spreading classical wave packet with quasi-exponential temporal envelope. All the quantum predictions are preserved. The total energy carried by the wave packet is ΔEmn* ¼ *hνmn with the unique carrier frequency νmn. The Fourier transform of the quasi-exponential envelope gives a spectral line width close to Lorentzian, which is the normal spectral line width of spontaneous emissions.*

However, during very late 1800, emission of photoelectrons from photocathode showed some uniqueness. Below some optical frequency, there was no photoelectron emission in spite of increasing the radiation density. Young Einstein correctly surmised that there is some "quantumness" hidden behind this frequency dependence and no electron emission after a cutoff frequency. Unfortunately, Einstein assigned this quantumness to light, instead of to the electrons. However, we now know that electrons are always bound quantum mechanically in all materials, even when they are bound collectively in quantum energy bands in solid state [15]. Here, we must recognize that Bohr atom was formulated in 1913 and Quantum Mechanics was formulated in 1925, many years after the 1905 paper by Einstein on the photoelectric effect. However, had Einstein correctly assigned the quantumness, observed in photoelectric effect, to electrons instead of to light, he would have been able to formulate quantum mechanics the Einsteinian way some 20 years earlier.

We believe that Planck's view of light is correct. EM waves are classical waves, solutions of Maxwell's equation, and propagates following Huygens secondary wavelets. In fact, Refs. [2–4] have derived the equation for photoelectric emission using the semiclassical method. Here, we present a heuristic approach to present the hybrid photon model that is a quantum at the moment of emission but a classical wave packet with a quasi-exponential temporal envelope. The total energy under the envelope corresponds to the QM predictions: (i) total energy of the wave packet is Δ*Emn* ¼ *hνmn*, and (ii) the quasi-exponential envelope assures the observed spontaneous emission line width as very close to Lorentzian, the Fourier transform of the amplitude envelope of the light pulse (see **Figure 3**).

## **5. Consequences of ignoring interaction process visualization: the necessity of the hybrid photon model**

Let us first recognize that Einstein's photoelectric equation is an energybalancing equation to match the observed data. This is measurable data modeling epistemology (MDM-E). We need to incorporate interaction process mapping epistemology (IPM-E) over and above MDM-E. Einstein's formulation did not embody light-matter interaction process, as we have underscored in the section on "Light does not see light." For accurate semiclassical derivation of the photoelectric effect, the readers should consult the following references [2–4]. We will present here only a heuristic derivation of Einstein's energy-balancing photoelectric equation, but starting from light-matter amplitude-amplitude stimulation, the E-vector of light

**Figure 4.**

*Left diagram: Emission of photoelectrons from a given material stops at a fixed specific frequency [16]. Middle diagram: Photoelectron emission from photocathode. Right diagram: Photoelectron transfer from valence to conduction band.*

stimulating the dipoles as *χ ν<sup>q</sup>* � �*E ν<sup>q</sup>* � �, containing the bound electrons, where *χ ν<sup>q</sup>* � � is the polarizability, or the light-matter *interaction parameter*. Since semiclassical thermal radiation consists of random wave groups emitted spontaneously with random phases, the total dipolar amplitude-amplitude stimulation of a bound electron can be expressed as

$$
\Psi = \sum\_{q} \mu\_{q} = \sum\_{q} \chi(\nu\_{q}) E(\nu\_{q}) \tag{2}
$$

are inserted into the structured beams and try to visualize the light-matter dipolar interaction processes. This would be a better system-engineering approach to

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?*

The key point of the author is that our unquestioned acceptance of the waveparticle duality has been hindering our deeper enquiry into the ultimate nature of light. The author has been attempting to inspire this process over decades by organizing special publications [13] and special conference series at SPIE from 2005 to 2015 [19], publishing experimental papers [20], and writing books [5, 21]. During this long arduous process, the author has recognized that Huygens principle (HP) of "secondary wavelets" has deeper enduring value for physics. HP requires space to be a physical tension field, a complex tension filed or CTF, to facilitate the perpetual and well-defined velocity of light. CTF possesses the necessary characteristic prop-

entire cosmic space. CTF must also possess other physical attributes that we have not been exploring actively. Thus, CTF could have serious implications in guiding us to reorganize our investigations to fulfill Einstein's dream of defining a unified field. CTF could be behind the emergence of both the EM waves and the particles as different kinds of oscillations of the same CTF, which holds 100% of the energy of

Optical physicists should note that the two major "successful theories" for grav-

ity, those of Newton and of Einstein, have been unable to explain the velocity distribution curves measured for a couple hundred galaxies. Therefore, theoretical physicists have proposed unnecessary postulates of the existence of Dark Matter and Dark Energy, neither of whose existence has been confirmed over the last

*Left-bottom curves [22]: Gravity theories of Newton and Einstein cannot match the measured velocity distribution of stars in one particular galaxy. This phenomenon turns out to be true for a couple of hundreds of galaxies. These two theories are very accurate for our solar system, but not good at the galactic scale. Bottom right: Huygens 1690 postulate of secondary wavelets [7], framed into Huygens-Fresnel diffraction integral, is still the core guiding analytical equation for the broad field of physical optics, including generating structured*

<sup>0</sup> *=μ*<sup>0</sup> <sup>1</sup>*=*<sup>2</sup>

, through the

understand different optical phenomena [17, 18].

*DOI: http://dx.doi.org/10.5772/intechopen.88849*

our cosmic system (Ch. 11 in [5, 8, 17]).

**Figure 5.**

*light.*

**11**

erties, which facilitates the perpetual velocity of light, *<sup>c</sup>* <sup>¼</sup> *<sup>ε</sup>*�<sup>1</sup>

The bound electron system must absorb the necessary amount of quantum-cupfilling energy, before the electron can be released to the conduction band or become a free-space electron. This energy exchange is a quadratic process:

$$\left|\Psi\right|^2 = \left|\sum\_{q} \chi(\nu\_q) E(\nu\_q)\right|^2 \mathfrak{so} h \nu\_q \tag{3}$$

For any quantum system, we must take the ensemble average. A single event (data point), as in Eq. (3), is never sufficient to verify a theory:

$$
\left\langle |\Psi|^2 \right\rangle = \left\langle \left| \sum\_{q} \chi(\nu\_q) E(\nu\_q) \right|^2 \right\rangle \Leftrightarrow \left\langle h\nu\_q \right\rangle = \left\langle \phi\_{\text{work }fn.} + (\mathbf{1}/2) m \mathbf{v}\_{el.}^2 \right\rangle \tag{4}
$$

In the right segment of Eq. (4), we have "recovered" Einstein's photoelectric energy-balancing equation out of dipole amplitude stimulations due to multitudes of waves. The left curve in **Figure 4** represents the photoelectric current [15]. Waves only fill up the quantum cups with the necessary energy if the dipoles are resonant to the frequency *ν* of the incident waves.

## **6. Discussions**

Visible light is always generated through orbital quantum electron transition processes in atoms. We have presented our *hybrid photon* model where light is released as a quantum energy packet, *hν*, as required by quantum formalism. Then we posit that immediately after the release of the *hν* packet, it evolves into a classical wave packet and follows Maxwell's wave equation and Huygens-Fresnel diffraction integral. We generate structured light using classical optical components and classical optical analytical tools. The possibility of introducing any quantumness in classical light during this process is difficult to imagine. The author is suggesting that we should explore the physical response properties of the material particles that *Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light? DOI: http://dx.doi.org/10.5772/intechopen.88849*

are inserted into the structured beams and try to visualize the light-matter dipolar interaction processes. This would be a better system-engineering approach to understand different optical phenomena [17, 18].

The key point of the author is that our unquestioned acceptance of the waveparticle duality has been hindering our deeper enquiry into the ultimate nature of light. The author has been attempting to inspire this process over decades by organizing special publications [13] and special conference series at SPIE from 2005 to 2015 [19], publishing experimental papers [20], and writing books [5, 21]. During this long arduous process, the author has recognized that Huygens principle (HP) of "secondary wavelets" has deeper enduring value for physics. HP requires space to be a physical tension field, a complex tension filed or CTF, to facilitate the perpetual and well-defined velocity of light. CTF possesses the necessary characteristic properties, which facilitates the perpetual velocity of light, *<sup>c</sup>* <sup>¼</sup> *<sup>ε</sup>*�<sup>1</sup> <sup>0</sup> *=μ*<sup>0</sup> <sup>1</sup>*=*<sup>2</sup> , through the entire cosmic space. CTF must also possess other physical attributes that we have not been exploring actively. Thus, CTF could have serious implications in guiding us to reorganize our investigations to fulfill Einstein's dream of defining a unified field. CTF could be behind the emergence of both the EM waves and the particles as different kinds of oscillations of the same CTF, which holds 100% of the energy of our cosmic system (Ch. 11 in [5, 8, 17]).

Optical physicists should note that the two major "successful theories" for gravity, those of Newton and of Einstein, have been unable to explain the velocity distribution curves measured for a couple hundred galaxies. Therefore, theoretical physicists have proposed unnecessary postulates of the existence of Dark Matter and Dark Energy, neither of whose existence has been confirmed over the last

#### **Figure 5.**

stimulating the dipoles as *χ ν<sup>q</sup>*

*Single Photon Manipulation*

**Figure 4.**

*conduction band.*

electron can be expressed as

j j <sup>Ψ</sup> <sup>2</sup> D E

**6. Discussions**

**10**

<sup>¼</sup> <sup>X</sup>

� �

*q χ ν<sup>q</sup>* � �*E ν<sup>q</sup>* � � �

resonant to the frequency *ν* of the incident waves.

<sup>2</sup> � �

� �*E ν<sup>q</sup>*

<sup>Ψ</sup> <sup>¼</sup> <sup>X</sup>

a free-space electron. This energy exchange is a quadratic process:

j j <sup>Ψ</sup> <sup>2</sup> <sup>¼</sup> <sup>X</sup>

(data point), as in Eq. (3), is never sufficient to verify a theory:

� �

> � � �

*q*

is the polarizability, or the light-matter *interaction parameter*. Since semiclassical thermal radiation consists of random wave groups emitted spontaneously with random phases, the total dipolar amplitude-amplitude stimulation of a bound

*Left diagram: Emission of photoelectrons from a given material stops at a fixed specific frequency [16]. Middle diagram: Photoelectron emission from photocathode. Right diagram: Photoelectron transfer from valence to*

*<sup>ψ</sup><sup>q</sup>* <sup>¼</sup> <sup>X</sup>

*q χ ν<sup>q</sup>* � �*E ν<sup>q</sup>* � � �

For any quantum system, we must take the ensemble average. A single event

⇔ *hν<sup>q</sup>*

In the right segment of Eq. (4), we have "recovered" Einstein's photoelectric energy-balancing equation out of dipole amplitude stimulations due to multitudes of waves. The left curve in **Figure 4** represents the photoelectric current [15]. Waves only fill up the quantum cups with the necessary energy if the dipoles are

Visible light is always generated through orbital quantum electron transition processes in atoms. We have presented our *hybrid photon* model where light is released as a quantum energy packet, *hν*, as required by quantum formalism. Then we posit that immediately after the release of the *hν* packet, it evolves into a classical wave packet and follows Maxwell's wave equation and Huygens-Fresnel diffraction integral. We generate structured light using classical optical components and classical optical analytical tools. The possibility of introducing any quantumness in classical light during this process is difficult to imagine. The author is suggesting that we should explore the physical response properties of the material particles that

*q χ ν<sup>q</sup>* � �*E ν<sup>q</sup>*

> � � � 2

� � <sup>¼</sup> *<sup>ϕ</sup>work fn:* <sup>þ</sup> ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>m</sup>*v2

D E

The bound electron system must absorb the necessary amount of quantum-cupfilling energy, before the electron can be released to the conduction band or become

� �, containing the bound electrons, where *χ ν<sup>q</sup>*

� � (2)

∝*hν<sup>q</sup>* (3)

*el:*

(4)

� �

*Left-bottom curves [22]: Gravity theories of Newton and Einstein cannot match the measured velocity distribution of stars in one particular galaxy. This phenomenon turns out to be true for a couple of hundreds of galaxies. These two theories are very accurate for our solar system, but not good at the galactic scale. Bottom right: Huygens 1690 postulate of secondary wavelets [7], framed into Huygens-Fresnel diffraction integral, is still the core guiding analytical equation for the broad field of physical optics, including generating structured light.*

several decades. These theories are definitely not wrong. Those who have been successfully launching and manipulating artificial satellites in our solar system rarely need to go beyond the mechanics of Newtonian gravity. Einstein's gravity correctly predicts the precession of the perihelion of Mercury. However, these two theories must be limited in capability to model gravity in the galactic scale (**Figure 5**).

**References**

Physik. 1905;**17**:132-148

honor of Alfred Kastler. In:

1969. pp. 363-369

Press; 2010

106-121

1970. p. 53

org/ebooks/14725

**13**

[1] Einstein A. Concerning a heuristic point of view about the creation and transformation of light. Annalen der

*DOI: http://dx.doi.org/10.5772/intechopen.88849*

[11] Born M, Wolf E. Principles of Optics. 6th ed. Cambridge University

[12] Roychoudhuri C. Hybrid photon model bridges classical and quantum optics.In: JW3A.32-1, OSA Annual

[13] Roychoudhuri C, Roy R, editors. The nature of light. What is a photon? Optics & Photonics News (OSA). 2003. Available from: https://www.osa-opn. org/opn/media/Images/PDFs/3185\_

[14] Planck M. The Theory of Heat Radiation. Blakistons Son & Co.; 1914. Free download from Gutenberg eBooks

Press; 1980

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light?*

Conference; 2017

6042\_30252.pdf?ext=.pdf

[15] Hufner S. Photoelectron Spectroscopy: Principles and Applications. Springer; 2010

[16] Bernstein J, Fishbane PM, Gasiorowicz S. Modern Physics.

[17] Roychoudhuri C. Urgency of evolution process congruent thinking in physics. In: An advocacy to elevate the prevailing abstract physics—Thinking towards a functionally useful reverseengineering thinking, Proc. SPIE. Vol.

[18] Roychoudhuri C. Consequences of repeated discovery and benign neglect of non-interaction of waves (NIW). In: Proc. SPIE. Vol. 10452 1045215; 2017. Available from: https://spie.org/Publications/ Proceedings/Paper/10.1117/12.2266216

[19] Roychoudhuri C. et al. Key organizer of the conference series: Nature of light: What (is) are photons? In: Proc. SPIE Vol. 5866, 2005; Proc. SPIE Vol. 6664, 2007; Proc. SPIE Vol. 7421, 2009; Proc. SPIE Vol. 8121, 2011; Proc. SPIE Vol. 8832, 2013; Proc. SPIE Vol. 9570, 2015

Prentice Hall; 2000

9570; 2015

[2] Lamb WE, Scully MO. Polarization, matter and radiation, Jubilee volume in

Photoelectric Effect without Photons. Paris: Presses Universitaires de France;

[3] Grynberg G, Aspect A, Fabre C. Introduction to Quantum Optics: From

Quantized Light. Cambridge University

[4] Stroud CR, Jaynes ET. Long-term solutions in semi-classical radiation theory. Physical Review A. 1970;**1**(1):

[5] Roychoudhuri C. Causal Physics: Photon by Non-Interaction of Waves. Boca Raton: Taylor & Francis; 2014

Cambridge: Harvard University Press;

[6] Ronchi V. Nature of Light.

[7] Huygens C. Treatise on Light, drafted in 1678. English translation. Chicago: University of Chicago Press; 2005 Project Gutenberg Edition: Available from: http://www.gutenberg.

[8] Roychoudhuri C. Next frontier in physics—Space as a complex tension field. Journal of Modern Physics. 2012

[9] Roychoudhuri C. Could space be considered as the inertial rest frame? In:

[10] Roychoudhuri C. The concept of "fundamental" must keep evolving, FQXi essay competition. 2015. Available from: https://fqxi.org/data/essay-contest-files/ Roychoudhuri\_180121\_Fundame.pdf

Proc. SPIE. Vol. 9570-30; 2015

the Semi-Classical Approach to

In contrast, in spite of subtle mathematical issues behind the Huygens postulate [23] of secondary wavelets, it remains the key foundational guide to propagate light through free space and non-interacting materials. To model light-matter interaction, Maxwell's equations have remained quite successful. In the history of physics, all theories eventually yield to new and better theories. Our attempts should be directed along these lines. We should not try to keep promoting the general validity of all working theories.

Optical physicists should explore the *deeper enduring values* behind the Huygens principle and find the limits of its application in different optical phenomena to advance further optical physics. Studies in optical phenomena have been guiding major advances in physics since ancient times. Starting from the 1600s to 1800s, advancements in physics were predominantly pioneered by scientists studying the broad field of optical sciences. However, starting from the early 1900s, this pioneering role has been shifted from guiding fundamental physics to finding only novel technical applications of optics. It is time for optical physicists to pick up more proactive roles in guiding the development of fundamental physics [24].

## **Author details**

ChandraSekhar Roychoudhuri Department of Physics, University of Connecticut, Storrs, CT, USA

\*Address all correspondence to: chandra.roychoudhuri@uconn.edu

© 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.

*Do We Manipulate Photons or Diffractive EM Waves to Generate Structured Light? DOI: http://dx.doi.org/10.5772/intechopen.88849*

## **References**

several decades. These theories are definitely not wrong. Those who have been successfully launching and manipulating artificial satellites in our solar system rarely need to go beyond the mechanics of Newtonian gravity. Einstein's gravity correctly predicts the precession of the perihelion of Mercury. However, these two theories must be limited in capability to model gravity in the galactic scale

In contrast, in spite of subtle mathematical issues behind the Huygens postulate [23] of secondary wavelets, it remains the key foundational guide to propagate light through free space and non-interacting materials. To model light-matter interaction, Maxwell's equations have remained quite successful. In the history of physics, all theories eventually yield to new and better theories. Our attempts should be directed along these lines. We should not try to keep promoting the general validity

Optical physicists should explore the *deeper enduring values* behind the Huygens principle and find the limits of its application in different optical phenomena to advance further optical physics. Studies in optical phenomena have been guiding major advances in physics since ancient times. Starting from the 1600s to 1800s, advancements in physics were predominantly pioneered by scientists studying the broad field of optical sciences. However, starting from the early 1900s, this pioneering role has been shifted from guiding fundamental physics to finding only novel technical applications of optics. It is time for optical physicists to pick up more proactive roles in guiding the development of fundamental physics [24].

(**Figure 5**).

of all working theories.

*Single Photon Manipulation*

**Author details**

**12**

ChandraSekhar Roychoudhuri

provided the original work is properly cited.

Department of Physics, University of Connecticut, Storrs, CT, USA

\*Address all correspondence to: chandra.roychoudhuri@uconn.edu

© 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,

[1] Einstein A. Concerning a heuristic point of view about the creation and transformation of light. Annalen der Physik. 1905;**17**:132-148

[2] Lamb WE, Scully MO. Polarization, matter and radiation, Jubilee volume in honor of Alfred Kastler. In: Photoelectric Effect without Photons. Paris: Presses Universitaires de France; 1969. pp. 363-369

[3] Grynberg G, Aspect A, Fabre C. Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light. Cambridge University Press; 2010

[4] Stroud CR, Jaynes ET. Long-term solutions in semi-classical radiation theory. Physical Review A. 1970;**1**(1): 106-121

[5] Roychoudhuri C. Causal Physics: Photon by Non-Interaction of Waves. Boca Raton: Taylor & Francis; 2014

[6] Ronchi V. Nature of Light. Cambridge: Harvard University Press; 1970. p. 53

[7] Huygens C. Treatise on Light, drafted in 1678. English translation. Chicago: University of Chicago Press; 2005 Project Gutenberg Edition: Available from: http://www.gutenberg. org/ebooks/14725

[8] Roychoudhuri C. Next frontier in physics—Space as a complex tension field. Journal of Modern Physics. 2012

[9] Roychoudhuri C. Could space be considered as the inertial rest frame? In: Proc. SPIE. Vol. 9570-30; 2015

[10] Roychoudhuri C. The concept of "fundamental" must keep evolving, FQXi essay competition. 2015. Available from: https://fqxi.org/data/essay-contest-files/ Roychoudhuri\_180121\_Fundame.pdf

[11] Born M, Wolf E. Principles of Optics. 6th ed. Cambridge University Press; 1980

[12] Roychoudhuri C. Hybrid photon model bridges classical and quantum optics.In: JW3A.32-1, OSA Annual Conference; 2017

[13] Roychoudhuri C, Roy R, editors. The nature of light. What is a photon? Optics & Photonics News (OSA). 2003. Available from: https://www.osa-opn. org/opn/media/Images/PDFs/3185\_ 6042\_30252.pdf?ext=.pdf

[14] Planck M. The Theory of Heat Radiation. Blakistons Son & Co.; 1914. Free download from Gutenberg eBooks

[15] Hufner S. Photoelectron Spectroscopy: Principles and Applications. Springer; 2010

[16] Bernstein J, Fishbane PM, Gasiorowicz S. Modern Physics. Prentice Hall; 2000

[17] Roychoudhuri C. Urgency of evolution process congruent thinking in physics. In: An advocacy to elevate the prevailing abstract physics—Thinking towards a functionally useful reverseengineering thinking, Proc. SPIE. Vol. 9570; 2015

[18] Roychoudhuri C. Consequences of repeated discovery and benign neglect of non-interaction of waves (NIW). In: Proc. SPIE. Vol. 10452 1045215; 2017. Available from: https://spie.org/Publications/ Proceedings/Paper/10.1117/12.2266216

[19] Roychoudhuri C. et al. Key organizer of the conference series: Nature of light: What (is) are photons? In: Proc. SPIE Vol. 5866, 2005; Proc. SPIE Vol. 6664, 2007; Proc. SPIE Vol. 7421, 2009; Proc. SPIE Vol. 8121, 2011; Proc. SPIE Vol. 8832, 2013; Proc. SPIE Vol. 9570, 2015

[20] Roychoudhuri C. Most of the author's papers and presentations for workshop, etc. Available from: http:// www.natureoflight.org/CP/

**Chapter 2**

**Abstract**

*Constantin Meis*

the same quantum vacuum field.

electron-positron charge

**1. Introduction**

**15**

Quantized Field of Single Photons

We present theoretical developments expressing the physical characteristics of a single photon in conformity with the experimental evidence. The quantization of the electromagnetic field vector potential amplitude is enhanced to a free of cavity photon state. Coupling the Schrödinger equation with the relativistic massless particle Hamiltonian to a symmetrical vector potential relation, we establish the *vector potential - energy* equation for the photon expressing the simultaneous *waveparticle* nature of a single photon, an intrinsic physical property. It is shown that the vector potential can be naturally considered as a real wave function for the photon entailing a coherent localization probability. We deduce directly the electric and magnetic field amplitudes of the cavity-free single photon, which are revealed to be proportional to the square of the angular frequency. The zero-energy electromagnetic field ground state (EFGS), a quantum vacuum real component, issues naturally from Maxwell's equations and the vector potential quantization procedure. The relation of the quantized amplitude of the photon vector potential to the electronpositron charge results directly showing the physical relationship between photons and electrons-positrons that might be at the origin of their mutual transformations. It becomes obvious that photons, as well as electrons-positrons, are issued from

**Keywords:** single photon, vector potential, photon wave-particle equation, photon wave function, photon electric field, electromagnetic field ground state,

During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. In this chapter, based on the present state of knowledge, we make a synthesis of the physical characteristics of a single photon put in evidence by the experiments, and we advance theoretical developments for its representation. Accordingly, the concept of the wave-particle nature of a single photon becomes physically compre-

However, before advancing in the theoretical developments, we consider that it is important starting with a brief historical review on the efforts carried out previously for understanding the nature of light while simultaneously making a synthesis

of the main experimental results which are of crucial importance for the

hensive and in agreement with the experimental evidence.

comprehension of the birth of the photon concept.

[21] Roychoudhuri C, Kracklauer AF, Creath K. The Nature of Light: What Is a Photon? Boca Raton: Taylor & Francis; 2008

[22] Mannheim PD, O'Brian JG. Fitting galactic rotation curves with conformal gravity and a global quadratic potential. Physical Review D. 2012;**85**:124020

[23] Goodman J. Fourier Optics. W. H. Freeman; 2017

[24] Roychoudhuri C, Tirfessa N. Bringing reality in physics: System engineering approach to optical phenomena following Huygens' Principle. In: Proc. SPIE 11143; 2019. DOI: 10.1117/12.2523602. Available from: https://www.spiedigitallibrary. org/conference-proceedings-of-spie on 08 Jul 2019

## **Chapter 2**

[20] Roychoudhuri C. Most of the author's papers and presentations for workshop, etc. Available from: http://

[21] Roychoudhuri C, Kracklauer AF, Creath K. The Nature of Light: What Is a Photon? Boca Raton: Taylor & Francis;

[22] Mannheim PD, O'Brian JG. Fitting galactic rotation curves with conformal gravity and a global quadratic potential. Physical Review D. 2012;**85**:124020

[23] Goodman J. Fourier Optics. W. H.

[24] Roychoudhuri C, Tirfessa N. Bringing reality in physics: System engineering approach to optical phenomena following Huygens' Principle. In: Proc. SPIE 11143; 2019. DOI: 10.1117/12.2523602. Available from: https://www.spiedigitallibrary. org/conference-proceedings-of-spie on

www.natureoflight.org/CP/

*Single Photon Manipulation*

2008

Freeman; 2017

08 Jul 2019

**14**

## Quantized Field of Single Photons

*Constantin Meis*

## **Abstract**

We present theoretical developments expressing the physical characteristics of a single photon in conformity with the experimental evidence. The quantization of the electromagnetic field vector potential amplitude is enhanced to a free of cavity photon state. Coupling the Schrödinger equation with the relativistic massless particle Hamiltonian to a symmetrical vector potential relation, we establish the *vector potential - energy* equation for the photon expressing the simultaneous *waveparticle* nature of a single photon, an intrinsic physical property. It is shown that the vector potential can be naturally considered as a real wave function for the photon entailing a coherent localization probability. We deduce directly the electric and magnetic field amplitudes of the cavity-free single photon, which are revealed to be proportional to the square of the angular frequency. The zero-energy electromagnetic field ground state (EFGS), a quantum vacuum real component, issues naturally from Maxwell's equations and the vector potential quantization procedure. The relation of the quantized amplitude of the photon vector potential to the electronpositron charge results directly showing the physical relationship between photons and electrons-positrons that might be at the origin of their mutual transformations. It becomes obvious that photons, as well as electrons-positrons, are issued from the same quantum vacuum field.

**Keywords:** single photon, vector potential, photon wave-particle equation, photon wave function, photon electric field, electromagnetic field ground state, electron-positron charge

## **1. Introduction**

During the last decades, an impressive technological development has been achieved permitting the manipulation of single photons with a high degree of statistical accuracy. However, despite the significant experimental advances, we still do not have a clear physical picture of a single photon state universally accepted by the scientific community, especially involved in quantum electrodynamics. In this chapter, based on the present state of knowledge, we make a synthesis of the physical characteristics of a single photon put in evidence by the experiments, and we advance theoretical developments for its representation. Accordingly, the concept of the wave-particle nature of a single photon becomes physically comprehensive and in agreement with the experimental evidence.

However, before advancing in the theoretical developments, we consider that it is important starting with a brief historical review on the efforts carried out previously for understanding the nature of light while simultaneously making a synthesis of the main experimental results which are of crucial importance for the comprehension of the birth of the photon concept.

The very first scientific publications on the nature of light are due to ancient Greeks who believed light is composed of corpuscles [1, 2]. Around 300 BC Euclid published the book *Optica* in which he developed the laws of reflection based on the rectilinear propagation of light. Two centuries later, Ptolemy of Alexandria published the book *Optics*, in which he included extensively all the previous knowledge on light. In this book, colours as well as refraction of the moonlight and sunlight by the earth's atmosphere were analysed. After Ptolemy of Alexandria, almost no progress has been reported until the seventeenth century.

are composed of discrete parts whose spots compose the diffraction patterns by gradual accumulation on the detection screen [14]. Compton published his studies on X-rays scattered by free electrons in 1923 advancing that the experimental results could only be interpreted based on the light quanta model [15].

the Greek word *phos* (Φωs, which means light) [1, 4].

sentation of light.

*Quantized Field of Single Photons*

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

**17**

*specific experimental condition* [1, 2, 19].

*natures in the same experimental conditions*.

Thus, the photoelectric effect and Compton scattering have been initially considered as the undoubtable demonstrations of the particle nature of light and historically were the strongest arguments in favour of the light quanta concept, which started to be universally accepted, and Lewis introduced the word "photon", from

Therein, it is extremely important to mention that Wentzel in 1926 [16] and Beck in 1927 [17], as well as much later Lamb and Scully in the 1960s [18], demonstrated that the photoelectric effect can be interpreted remarkably well by only considering the wave nature of light, without referring to photons at all [19]. Furthermore, the Compton scattering has been fully interpreted by Klein and Nishina in 1929 [20] also by considering the electromagnetic wave nature of light without invoking the photon concept. On the other hand, Young's experiment, initially presented as the most convincing argument for the wave nature of light, was applied by Taylor at very low intensities to demonstrate the particle concept of light [14]. Indeed, much later Jin et al. [21] published an excellent theoretical interpretation of Young's diffraction experiments based only on the particle repre-

Thus, the picture on the nature of light in the 1930s was rather confusing since both opposite sides defending the wave or the particle nature advanced equally strong arguments. Hence, Bohr, inspired by de Broglie's thesis on the simultaneous wave character of particles, announced the *complementarity principle* according to which *light has both wave and particle natures appearing mutually exclusively in each*

The development of lasers [22] in the 1960s and the revolutionary parametric down-convertion techniques [23, 24] in the 1970s, have made it possible to realise conditions in which, with a convenient statistical confidence, only a single photon may be present in the experimental apparatus. In this way, the double-prism experiment [25] realised in the 1990s contradicted for the first time Bohr's *mutual exclusiveness* demonstrating that a *single photon exhibits both the wave and particle*

According to the experimental investigations, it has been always stated that a photon has circular, left or right, polarisation with spin *h=*2*π* and cannot be conceived along the propagation axis within a length shorter than its wavelength [26]. Indeed, since Mandel's experiments in the 1960s [27, 28], all the efforts to localise precisely a single photon remained fruitless yielding the conclusion is that a photon cannot be better localised than within a volume of the order of the cube of its wavelength [29, 30]. Furthermore, Grangier et al. demonstrated experimentally the indivisibility of photons [19, 31], while in recent years the entangled state experiments [29, 30] have shown that the photon should be locally an integral entity during the detection procedure but with a real non-local wave function. The lateral expansion of a single photon, considered locally as an indivisible entity, was always an intriguing part of physics. With the purpose of studying the lateral expansion of the electromagnetic rays, Robinson in 1953 [32] and Hadlock in 1958 [33] carried out experiments using microwaves crossing small apertures and deduced that no energy is transmitted through apertures whose dimensions are smaller than roughly *λ*/4. In 1986, for the same purpose, Hunter and Wadlinger [34, 35] used X-band microwaves with *λ* = 28.5 mm and measured the

transmitted power through rectangular or circular apertures of different

dimensions. They concluded that no energy is transmitted when the apertures are

In the year of 1670, Newton revived the ideas of ancient Greeks and advanced the theory following that light is composed of corpuscles that travel rectilinearly [3]. Ten years later, Huygens developed the principles of the wave theory of light [1, 4, 5]. Huygens' wave theory was a hard opponent to Newton's corpuscle concept. In the beginning of the nineteenth century, Young obtained experimentally interference patterns using different sources of light and explained some polarisation observations by assuming that light oscillations are perpendicular to the propagation axis [1, 6]. Euler and Fresnel explained the diffraction patterns observed experimentally by applying the wave theory [6]. In 1865, Maxwell published his theory on the electromagnetic waves establishing the relations between the electric and magnetic fields and showing that light is composed of electromagnetic waves [7]. A few years later, Hertz confirmed Maxwell's theory by discovering the longwavelength electromagnetic radiation [1, 7]. Thus, at the end of the nineteenth century, the scientific community started to accept officially the wave nature of light replacing Newton's theory.

Nevertheless, new events supporting the particle nature of light occurred in the beginning of the twentieth century. Stefan and Wien discovered the direct relationship between the thermal radiation energy and the temperature of a black body [8, 9]. However, the emitted radiation energy density as a function of the temperature calculated by Rayleigh failed to describe the experimental results at short wavelengths. Scientists had given the name "UV catastrophe" to this problem revealing the necessity of a new theoretical approach. Planck managed to establish the correct energy density expression for the radiation emitted by a black body with respect to temperature, in excellent agreement with the experiment [8]. For that purpose, he assumed that the bodies are composed of "oscillators" which have the particularity of emitting the electromagnetic energy in "packets" of *hν*, where *ν* is the frequency and *h* is a constant that was later called Planck's constant. During the same period, the experiments carried out by Michelson et al. [10] demonstrated that the speed of light in vacuum is a universal physical constant corresponding to the product of the frequency *ν* times the wavelength *λ*, that is, *c = λ ν*.

In 1902, Lenard pointed out that the photoelectric effect, discovered by Hertz 15 years earlier [11], occurs beyond a threshold frequency of light and the kinetic energy of the emitted electrons does not depend on the incident light intensity. Based on Planck's works, Einstein proposed a simple interpretation of the photoelectric effect assuming that the electromagnetic radiation is composed of quanta with energy *hν* [12]. He advanced that *the energy of a light ray when spreading from a point consists of a finite number of energy quanta localised in points in space, which move without dividing and are only absorbed and emitted as a whole.* Although that was a decisive step towards the particle theory of light, the concept of the light quanta was still not generally accepted, and Bohr, who was strongly opposed to the particle concept of light [13], announced in his Nobel lecture (1922) that *the light quantum hypothesis is not compatible with the interference phenomena and consequently it cannot throw light in the nature of radiation.* Bohr's statement was rather surprising because Taylor's experiments, consisting of repeating Young's double slit diffraction at extremely low light intensities, had already demonstrated since 1909 that light rays

### *Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

The very first scientific publications on the nature of light are due to ancient Greeks who believed light is composed of corpuscles [1, 2]. Around 300 BC Euclid published the book *Optica* in which he developed the laws of reflection based on the

published the book *Optics*, in which he included extensively all the previous knowledge on light. In this book, colours as well as refraction of the moonlight and sunlight by the earth's atmosphere were analysed. After Ptolemy of Alexandria,

In the year of 1670, Newton revived the ideas of ancient Greeks and advanced the theory following that light is composed of corpuscles that travel rectilinearly [3]. Ten years later, Huygens developed the principles of the wave theory of light [1, 4, 5]. Huygens' wave theory was a hard opponent to Newton's corpuscle concept. In the beginning of the nineteenth century, Young obtained experimentally interference patterns using different sources of light and explained some polarisation observations by assuming that light oscillations are perpendicular to the propagation axis [1, 6]. Euler and Fresnel explained the diffraction patterns observed experimentally by applying the wave theory [6]. In 1865, Maxwell published his theory on the electromagnetic waves establishing the relations between the electric and magnetic fields and showing that light is composed of electromagnetic waves [7]. A few years later, Hertz confirmed Maxwell's theory by discovering the longwavelength electromagnetic radiation [1, 7]. Thus, at the end of the nineteenth century, the scientific community started to accept officially the wave nature of

Nevertheless, new events supporting the particle nature of light occurred in the beginning of the twentieth century. Stefan and Wien discovered the direct relationship between the thermal radiation energy and the temperature of a black body [8, 9]. However, the emitted radiation energy density as a function of the temperature calculated by Rayleigh failed to describe the experimental results at short wavelengths. Scientists had given the name "UV catastrophe" to this problem revealing the necessity of a new theoretical approach. Planck managed to establish the correct energy density expression for the radiation emitted by a black body with respect to temperature, in excellent agreement with the experiment [8]. For that purpose, he assumed that the bodies are composed of "oscillators" which have the particularity of emitting the electromagnetic energy in "packets" of *hν*, where *ν* is the frequency and *h* is a constant that was later called Planck's constant. During the same period, the experiments carried out by Michelson et al. [10] demonstrated that the speed of light in vacuum is a universal physical constant corresponding to the

In 1902, Lenard pointed out that the photoelectric effect, discovered by Hertz 15

years earlier [11], occurs beyond a threshold frequency of light and the kinetic energy of the emitted electrons does not depend on the incident light intensity. Based on Planck's works, Einstein proposed a simple interpretation of the photoelectric effect assuming that the electromagnetic radiation is composed of quanta with energy *hν* [12]. He advanced that *the energy of a light ray when spreading from a point consists of a finite number of energy quanta localised in points in space, which move without dividing and are only absorbed and emitted as a whole.* Although that was a decisive step towards the particle theory of light, the concept of the light quanta was still not generally accepted, and Bohr, who was strongly opposed to the particle concept of light [13], announced in his Nobel lecture (1922) that *the light quantum hypothesis is not compatible with the interference phenomena and consequently it cannot throw light in the nature of radiation.* Bohr's statement was rather surprising because Taylor's experiments, consisting of repeating Young's double slit diffraction at extremely low light intensities, had already demonstrated since 1909 that light rays

rectilinear propagation of light. Two centuries later, Ptolemy of Alexandria

almost no progress has been reported until the seventeenth century.

product of the frequency *ν* times the wavelength *λ*, that is, *c = λ ν*.

light replacing Newton's theory.

*Single Photon Manipulation*

**16**

are composed of discrete parts whose spots compose the diffraction patterns by gradual accumulation on the detection screen [14]. Compton published his studies on X-rays scattered by free electrons in 1923 advancing that the experimental results could only be interpreted based on the light quanta model [15].

Thus, the photoelectric effect and Compton scattering have been initially considered as the undoubtable demonstrations of the particle nature of light and historically were the strongest arguments in favour of the light quanta concept, which started to be universally accepted, and Lewis introduced the word "photon", from the Greek word *phos* (Φωs, which means light) [1, 4].

Therein, it is extremely important to mention that Wentzel in 1926 [16] and Beck in 1927 [17], as well as much later Lamb and Scully in the 1960s [18], demonstrated that the photoelectric effect can be interpreted remarkably well by only considering the wave nature of light, without referring to photons at all [19]. Furthermore, the Compton scattering has been fully interpreted by Klein and Nishina in 1929 [20] also by considering the electromagnetic wave nature of light without invoking the photon concept. On the other hand, Young's experiment, initially presented as the most convincing argument for the wave nature of light, was applied by Taylor at very low intensities to demonstrate the particle concept of light [14]. Indeed, much later Jin et al. [21] published an excellent theoretical interpretation of Young's diffraction experiments based only on the particle representation of light.

Thus, the picture on the nature of light in the 1930s was rather confusing since both opposite sides defending the wave or the particle nature advanced equally strong arguments. Hence, Bohr, inspired by de Broglie's thesis on the simultaneous wave character of particles, announced the *complementarity principle* according to which *light has both wave and particle natures appearing mutually exclusively in each specific experimental condition* [1, 2, 19].

The development of lasers [22] in the 1960s and the revolutionary parametric down-convertion techniques [23, 24] in the 1970s, have made it possible to realise conditions in which, with a convenient statistical confidence, only a single photon may be present in the experimental apparatus. In this way, the double-prism experiment [25] realised in the 1990s contradicted for the first time Bohr's *mutual exclusiveness* demonstrating that a *single photon exhibits both the wave and particle natures in the same experimental conditions*.

According to the experimental investigations, it has been always stated that a photon has circular, left or right, polarisation with spin *h=*2*π* and cannot be conceived along the propagation axis within a length shorter than its wavelength [26]. Indeed, since Mandel's experiments in the 1960s [27, 28], all the efforts to localise precisely a single photon remained fruitless yielding the conclusion is that a photon cannot be better localised than within a volume of the order of the cube of its wavelength [29, 30]. Furthermore, Grangier et al. demonstrated experimentally the indivisibility of photons [19, 31], while in recent years the entangled state experiments [29, 30] have shown that the photon should be locally an integral entity during the detection procedure but with a real non-local wave function.

The lateral expansion of a single photon, considered locally as an indivisible entity, was always an intriguing part of physics. With the purpose of studying the lateral expansion of the electromagnetic rays, Robinson in 1953 [32] and Hadlock in 1958 [33] carried out experiments using microwaves crossing small apertures and deduced that no energy is transmitted through apertures whose dimensions are smaller than roughly *λ*/4. In 1986, for the same purpose, Hunter and Wadlinger [34, 35] used X-band microwaves with *λ* = 28.5 mm and measured the transmitted power through rectangular or circular apertures of different dimensions. They concluded that no energy is transmitted when the apertures are

smaller than �*λ/π* confirming that the lateral expansion of the photons is a fraction of the wavelength.

*<sup>H</sup>* <sup>¼</sup> <sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

*Quantized Field of Single Photons*

with ℏ ¼ *h=*2*π* as Planck's reduced constant.

depends on the magnetic field flux in the solenoid:

*i*ℏ *∂ ∂t* Ψ *r* !*; t* � � <sup>¼</sup> <sup>1</sup>

> Ψ *r* !*; t* � � <sup>¼</sup> <sup>Ψ</sup>*<sup>p</sup> <sup>r</sup>*

*i*ℏ *∂ ∂t* Ψ*<sup>p</sup> r* !*; t* � � ¼ � <sup>ℏ</sup><sup>2</sup>

and electric fields and of any other potential.

cylindrical coordinates.

potential:

with *A* !

where Ψ*<sup>p</sup> r*

potential:

**19**

!*; t*

tions:

<sup>2</sup>*<sup>m</sup>* �*i*<sup>ℏ</sup> <sup>∇</sup>

*A* ! <sup>¼</sup> <sup>1</sup> 2*π r* ð *s B* ! � *dS* !

! �*qA* ! *r* !*; t*

� � � � <sup>2</sup>

If the solenoid is extremely long along the *z* axis, then the magnetic field is uniform in the region inside and zero outside. The scalar potential Φ can be put to zero by assuming that the solenoid is not charged. In this case, in the outside region, the electric and magnetic fields are zero, but the vector potential is not zero and

where *r* is the radial distance from the *z* axis of the solenoid, *S* is the surface of

The Schrödinger equation for a charged particle outside the solenoid, where the

*i*ℏ ∇ ! <sup>þ</sup>*qA* � � ! <sup>2</sup>

!*; t* � �*<sup>e</sup> iq* ℏ Ð *r* ! <sup>0</sup> *<sup>A</sup>* ! *r* !0 ð Þ�*d r*!0

given by Eq. (4). The solutions of the last equation are the wave func-

� � is the solution of Schrödinger's equation in absence of the vector

2*m* ∇ !<sup>2</sup> Ψ*<sup>p</sup> r* !*; t*

The exponential part of the wave function of Eq. (6) entails that two particles have equal charge and mass moving both outside the solenoid at the same distance

Interference patterns for electrons in analogue conditions have been observed experimentally [44–46] demonstrating that the vector potential is a real physical field and interacts directly with charged particles in complete absence of magnetic

The vector potential, being a real field, is considered as the fundamental link between the electromagnetic wave theory issued from Maxwell's equations and the

from the axis, but the first in the same direction with the vector potential *A*

*<sup>δ</sup>*<sup>Θ</sup> <sup>¼</sup> *<sup>q</sup>* ℏ ð *s B* ! � *dS* !

second in the opposite direction will suffer a phase difference:

**2.2 The radiation vector potential: classical to quantum link**

Ψ *r* !*; t*

the circle with radius *r* perpendicular to *z*, and ^*e<sup>θ</sup>* is the angular unit vector in

vector potential is not zero, writes in complete absence of any other external

2*m*

þ *q*Φ *r* !*; t*

� � (3)

^*e<sup>θ</sup>* (4)

� � (5)

� � (7)

!

and the

(8)

(6)

Thus, the experiments have shown that the single photon is not a point and cannot be localised at a coordinate, as stated by Einstein, while it can exhibit both the wave and particle natures in the same experimental conditions contradicting Bohr's mutual exclusiveness. However, quantum electrodynamics (QED) has been developed during the 1930s to 1960s based upon the point particle model for the photon [36–39]. In fact, the point photon concept has permitted to establish an efficient mathematical approach for describing states before and after an interaction processes [19, 39–41], but it is naturally inappropriate for the description of the real nature of a single photon.

Finally, what we can essentially draw out by summing up the experimental evidence is that a single photon is a minimum, local, indivisible part of the electromagnetic field with precise energy *hν* and momentum *hν/c*, having circular left or right polarisation with spin �*h=*2*π*, respectively. It is not a point particle since it expands over a wavelength *λ* along the propagation axis and is detectable within a volume of the order of *λ* <sup>3</sup> entailing that its lateral expansion is a fraction of its wavelength. Hence, it appears to be a local "wave-corpuscle" guided by a non-local wave function, absorbed and emitted as a whole and capable of interacting with charges increasing or decreasing its frequency and consequently its energy.

In what follows, we present first the standard theoretical representation of the electromagnetic field quantization resulting in photons, and next we proceed to recent advances based on the vector potential quantization enhanced to a single photon state.

## **2. The electromagnetic field vector potential**

#### **2.1 Reality of the vector potential**

Since the formulation of Maxwell's equations, the vector potential *A* ! *r* !*; t* was considered as a pure mathematical artefact [4, 5, 7] used to calculate the electric field:

$$\overrightarrow{E}\left(\overrightarrow{r},t\right) = -\overrightarrow{\text{d}A}\left(\overrightarrow{r},t\right)/\partial t - \overrightarrow{\nabla}\Phi\left(\overrightarrow{r},t\right) \tag{1}$$

where Φ *r* !*; t* is the scalar potential, as well as the magnetic field:

$$
\overrightarrow{B}\left(\overrightarrow{r},t\right) = \overrightarrow{\nabla} \times \overrightarrow{A}\left(\overrightarrow{r},t\right) \tag{2}
$$

In 1949, Ehrenberg and Siday were the first to put in evidence the influence of the vector potential on charged particles [42] deducing that it is a real physical field. Ten years later, Aharonov and Bohm re-infirmed the influence of the vector potential on electrons in complete absence of electric and magnetic fields [43]. That was confirmed experimentally by Chambers [44], Tonomura et al. [45], and Osakabe et al. [46] demonstrating without any doubt the reality of the vector potential field end its direct influence on charges.

From a theoretical point of view [43], the behaviour of a particle with charge *q* and mass *m* in the vicinity of a solenoid where the vector potential is present is described by the Hamiltonian:

*Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

smaller than �*λ/π* confirming that the lateral expansion of the photons is a fraction

Thus, the experiments have shown that the single photon is not a point and cannot be localised at a coordinate, as stated by Einstein, while it can exhibit both the wave and particle natures in the same experimental conditions contradicting Bohr's mutual exclusiveness. However, quantum electrodynamics (QED) has been developed during the 1930s to 1960s based upon the point particle model for the photon [36–39]. In fact, the point photon concept has permitted to establish an efficient mathematical approach for describing states before and after an interaction processes [19, 39–41], but it is naturally inappropriate for the description of

Finally, what we can essentially draw out by summing up the experimental evidence is that a single photon is a minimum, local, indivisible part of the electromagnetic field with precise energy *hν* and momentum *hν/c*, having circular left or right polarisation with spin �*h=*2*π*, respectively. It is not a point particle since it expands over a wavelength *λ* along the propagation axis and is detectable within a

In what follows, we present first the standard theoretical representation of the electromagnetic field quantization resulting in photons, and next we proceed to recent advances based on the vector potential quantization enhanced to a single

wavelength. Hence, it appears to be a local "wave-corpuscle" guided by a non-local wave function, absorbed and emitted as a whole and capable of interacting with charges increasing or decreasing its frequency and consequently

Since the formulation of Maxwell's equations, the vector potential *A*

¼ �∂*A* ! *r* !*; t* 

*B* ! *r* !*; t* 

considered as a pure mathematical artefact [4, 5, 7] used to calculate the electric field:

is the scalar potential, as well as the magnetic field:

¼∇ ! � *A* ! *r* !*; t* 

In 1949, Ehrenberg and Siday were the first to put in evidence the influence of the vector potential on charged particles [42] deducing that it is a real physical field. Ten years later, Aharonov and Bohm re-infirmed the influence of the vector potential on electrons in complete absence of electric and magnetic fields [43]. That was confirmed experimentally by Chambers [44], Tonomura et al. [45], and Osakabe et al. [46] demonstrating without any doubt the reality of the vector potential field

From a theoretical point of view [43], the behaviour of a particle with charge *q* and mass *m* in the vicinity of a solenoid where the vector potential is present is

*=*∂*t*� ∇ ! Φ *r* !*; t* 

**2. The electromagnetic field vector potential**

*E* ! *r* !*; t* 

**2.1 Reality of the vector potential**

<sup>3</sup> entailing that its lateral expansion is a fraction of its

! *r* !*; t* 

was

(1)

(2)

of the wavelength.

*Single Photon Manipulation*

the real nature of a single photon.

volume of the order of *λ*

its energy.

photon state.

where Φ *r*

!*; t* 

end its direct influence on charges.

described by the Hamiltonian:

**18**

$$H = \frac{1}{2m} \left( -i\hbar \vec{\nabla} \cdot \vec{qA} \left( \vec{r}, t \right) \right)^2 + q\Phi \left( \vec{r}, t \right) \tag{3}$$

with ℏ ¼ *h=*2*π* as Planck's reduced constant.

If the solenoid is extremely long along the *z* axis, then the magnetic field is uniform in the region inside and zero outside. The scalar potential Φ can be put to zero by assuming that the solenoid is not charged. In this case, in the outside region, the electric and magnetic fields are zero, but the vector potential is not zero and depends on the magnetic field flux in the solenoid:

$$\overrightarrow{A} = \frac{1}{2\pi r} \int\_{s} \overrightarrow{B} \cdot d\overrightarrow{S} \,\hat{e}\_{\theta} \tag{4}$$

where *r* is the radial distance from the *z* axis of the solenoid, *S* is the surface of the circle with radius *r* perpendicular to *z*, and ^*e<sup>θ</sup>* is the angular unit vector in cylindrical coordinates.

The Schrödinger equation for a charged particle outside the solenoid, where the vector potential is not zero, writes in complete absence of any other external potential:

$$i\hbar\frac{\partial}{\partial t}\Psi\left(\overrightarrow{r},t\right) = \frac{1}{2m}\left(i\hbar\overrightarrow{\nabla} + q\overrightarrow{A}\right)^2\Psi\left(\overrightarrow{r},t\right) \tag{5}$$

with *A* ! given by Eq. (4). The solutions of the last equation are the wave functions:

$$
\Psi\left(\overrightarrow{r},t\right) = \Psi\_p\left(\overrightarrow{r},t\right)e^{\frac{i}{\hbar}\int\_0^{\overrightarrow{r}}\overrightarrow{A}\left(\overrightarrow{r}'\right)\cdot d\overrightarrow{r}'}\tag{6}
$$

where Ψ*<sup>p</sup> r* !*; t* � � is the solution of Schrödinger's equation in absence of the vector potential:

$$i\hbar\frac{\partial}{\partial t}\Psi\_p\left(\overrightarrow{r},t\right) = -\frac{\hbar^2}{2m}\overrightarrow{\nabla}^2\Psi\_p\left(\overrightarrow{r},t\right) \tag{7}$$

The exponential part of the wave function of Eq. (6) entails that two particles have equal charge and mass moving both outside the solenoid at the same distance from the axis, but the first in the same direction with the vector potential *A* ! and the second in the opposite direction will suffer a phase difference:

$$
\delta\Theta = \frac{q}{\hbar} \int\_s \overrightarrow{B} \cdot d\overrightarrow{S} \tag{8}
$$

Interference patterns for electrons in analogue conditions have been observed experimentally [44–46] demonstrating that the vector potential is a real physical field and interacts directly with charged particles in complete absence of magnetic and electric fields and of any other potential.

#### **2.2 The radiation vector potential: classical to quantum link**

The vector potential, being a real field, is considered as the fundamental link between the electromagnetic wave theory issued from Maxwell's equations and the particle concept in quantum electrodynamics (QED) [19, 36, 39]. We will show analytically how this link is established.

In the classical theory [5, 7], the energy density of a mode *k* of the electromagnetic wave writes:

$$\mathcal{W}\_k\left(\overrightarrow{r},t\right) = \frac{1}{2}\left(\varepsilon\_0 \left|\overrightarrow{E}\_k\left(\overrightarrow{r},t\right)\right|^2 + \frac{1}{\mu\_0}\left|\overrightarrow{B}\_k\left(\overrightarrow{r},t\right)\right|^2\right) \tag{9}$$

The last relation is the fundamental link between the classical and quantum theory of light which is used to define in QED the vector potential amplitude

*ak<sup>λ</sup> <sup>A</sup>*<sup>~</sup> <sup>∗</sup>

Therein, it is worth noting that an external arbitrary volume parameter *V* appears in the vector potential amplitude of the single photon, expressed by Eq. (15), which is supposed to be an intrinsic physical property. This could entail the unphysical interpretation that a single photon in an infinite cavity has zero vector potential, thus zero electric and magnetic fields and consequently zero energy. This ambiguity, which is scarcely quoted in the literature, is lifted by considering that, in the case of a single photon, the volume *V* in Eq. (15) is equivalent to that defined by the boundary conditions in a cavity for the single radiation

**3. Electromagnetic field quantization and the photon description**

The energy of the electromagnetic field in a volume *V* considered as a superposition of different *k*-modes and *λ*-polarisations is obtained directly from

where the summation over the *λ*-polarisations takes only two values

"normal ordering" radiation Hamiltonian corresponding to the order *a*<sup>þ</sup>

*<sup>H</sup>*<sup>~</sup> *NO* <sup>¼</sup> <sup>X</sup> *k, <sup>λ</sup> a*<sup>þ</sup>

*k, <sup>λ</sup>*

where we have used the fundamental commutation relation in quantum

*akλ; a*<sup>þ</sup> *kλ*

*<sup>H</sup>*<sup>~</sup> *ANO* <sup>¼</sup> <sup>X</sup>

X *k, λ ω*2

Replacing in Eq. (17) the vector potential amplitude and its conjugate by the relations of the vector potential amplitude operators defined in Eq. (16), we get the

and the "anti-normal ordering" Hamiltonian corresponding to the order *akλa*<sup>þ</sup>

*a*<sup>þ</sup>

*<sup>k</sup>*j j *Akλ*ð Þ *ω<sup>k</sup>*

**3.1 Harmonic oscillator representation of the electromagnetic field**

*EEM* ¼ 2*ε*0*V*

corresponding to circular left and right [19, 36–41].

creation and annihilation operators:

electrodynamics:

**21**

*k<sup>λ</sup>* ¼

*<sup>k</sup><sup>λ</sup>* are, respectively, the annihilation and creation non-Hermitian

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ωkV*

*a*þ

<sup>2</sup> (17)

*<sup>k</sup><sup>λ</sup>akλ*ℏ*ω<sup>k</sup>* (18)

*<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* <sup>þ</sup> <sup>1</sup> � �ℏ*ω<sup>k</sup>* (19)

� � <sup>¼</sup> <sup>1</sup> (20)

*<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* of the

*kλ*

*<sup>k</sup><sup>λ</sup>* (16)

operators for a single photon [19, 26, 29, 36–41]:

s

operators for a *k*-mode and *λ*-polarisation photon.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ωkV*

*<sup>A</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>* <sup>¼</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

where *ak<sup>λ</sup>* and *a*<sup>þ</sup>

*Quantized Field of Single Photons*

mode *k*.

Eq. (13):

where *ε<sup>0</sup>* and *μ<sup>0</sup>* are the electric permittivity and magnetic permeability of the vacuum, respectively, related to the speed of light in vacuum *<sup>c</sup>* by *<sup>ε</sup>*0*μ*0*c*<sup>2</sup> <sup>¼</sup> 1.

In the case of a monochromatic plane wave with angular frequency *ωk*, the electric *Ek* ! *r* !*; t* � � and magnetic *Bk* ! *r* !*; t* � � fields are proportional to the vector potential amplitude *A*0*k*ð Þ *ω<sup>k</sup>* :

$$\overrightarrow{E}\_{k}\left(\overrightarrow{r},t\right) = -2\alpha\rho\_{k}A\_{0k}\left(\alpha\rho\_{k}\right)\hat{\varepsilon}\sin\left(\overrightarrow{k}\cdot\overrightarrow{r}\right.\tag{10}$$

$$\overrightarrow{B}\_{k}\left(\overrightarrow{r},t\right) = -\frac{1}{c}\mathcal{Q}\rho\_{k}A\_{0k}(\rho\_{k})\left(\hat{k}\times\hat{\varepsilon}\right)\sin\left(\overrightarrow{k}\cdot\overrightarrow{r}\right) - \alpha\_{k}t\right\,\tag{11}$$

where ^*ε* is a unit vector perpendicular to the propagation axis, *k* � �! � � � � <sup>¼</sup> <sup>2</sup>*π=λ<sup>k</sup>* is the wave vector along the propagation axis, and *λ<sup>k</sup>* is the wavelength of the mode *k*.

Introducing Eqs. (10) and (11) in Eq. (9), the energy density now depends on the square of the vector potential amplitude:

$$\mathcal{W}\_k\left(\overrightarrow{r},t\right) = 4\epsilon\_0 a\_k^2 |A\_{0k}(\boldsymbol{\alpha}\_k)|^2 \sin^2\left(\overrightarrow{k}\cdot\overrightarrow{r}\cdot\overrightarrow{\boldsymbol{\sigma}} - \boldsymbol{\alpha}\_k t\right) \tag{12}$$

The mean value over a period, thus over a wavelength, is time independent:

$$\mathcal{W}\_k = 2\varepsilon\_0 a\_k^2 \left| A\_{0k}(a\_k) \right|^2 \tag{13}$$

Note that the last equation expressing the mean energy density over a period of the mode *k* of the electromagnetic wave is independent on any external volume yielding that in the classical description, a free of cavity electromagnetic radiation mode expands naturally within a minimum volume. In a given cavity, this volume corresponds roughly to that imposed by the boundary conditions and the cut-off wave vectors [4, 5, 7].

On the other hand, in the quantum description, the energy density for a number *N*ð Þ *ω<sup>k</sup>* of *k*-mode photons with angular frequency *ω<sup>k</sup>* and energy ℏ*ω<sup>k</sup>* in a volume *V* writes simply:

$$W\_k = \frac{N(\alpha\_k)\hbar\alpha\_k}{V} \tag{14}$$

In order to link the classical to the quantum description [4, 9, 19], the classical mean energy density over a period, expressed by Eq. (13), is imposed to be equivalent to the quantum mechanics expression of Eq. (14) for *N*ð Þ¼ *ω<sup>k</sup>* 1 getting the vector potential amplitude for a single *k*-mode photon:

$$|A\_{0k}(a\_k)| = \sqrt{\frac{\hbar}{2\varepsilon\_0 a\_k V}}\tag{15}$$

particle concept in quantum electrodynamics (QED) [19, 36, 39]. We will show

In the classical theory [5, 7], the energy density of a mode *k* of the electromag-

where *ε<sup>0</sup>* and *μ<sup>0</sup>* are the electric permittivity and magnetic permeability of the

¼ �2*ω<sup>k</sup> A*0*<sup>k</sup>*ð Þ *ω<sup>k</sup>* ^*ε* sin *k*

<sup>2</sup>*ω<sup>k</sup> <sup>A</sup>*0*<sup>k</sup>*ð Þ *<sup>ω</sup><sup>k</sup>* ^

wave vector along the propagation axis, and *λ<sup>k</sup>* is the wavelength of the mode *k*. Introducing Eqs. (10) and (11) in Eq. (9), the energy density now depends on

*<sup>k</sup>*j j *A*0*<sup>k</sup>*ð Þ *ω<sup>k</sup>*

The mean value over a period, thus over a wavelength, is time independent:

Note that the last equation expressing the mean energy density over a period of the mode *k* of the electromagnetic wave is independent on any external volume yielding that in the classical description, a free of cavity electromagnetic radiation mode expands naturally within a minimum volume. In a given cavity, this volume corresponds roughly to that imposed by the boundary conditions and the cut-off

On the other hand, in the quantum description, the energy density for a number *N*ð Þ *ω<sup>k</sup>* of *k*-mode photons with angular frequency *ω<sup>k</sup>* and energy ℏ*ω<sup>k</sup>* in a volume *V*

*Wk* <sup>¼</sup> *<sup>N</sup>*ð Þ *<sup>ω</sup><sup>k</sup>* <sup>ℏ</sup>*ω<sup>k</sup>*

In order to link the classical to the quantum description [4, 9, 19], the classical mean energy density over a period, expressed by Eq. (13), is imposed to be equivalent to the quantum mechanics expression of Eq. (14) for *N*ð Þ¼ *ω<sup>k</sup>* 1 getting the

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ωkV*

j j *A*0*<sup>k</sup>*ð Þ *ω<sup>k</sup>* ¼

vector potential amplitude for a single *k*-mode photon:

*Wk* <sup>¼</sup> <sup>2</sup>*ε*0*ω*<sup>2</sup>

where ^*ε* is a unit vector perpendicular to the propagation axis, *k*

<sup>¼</sup> <sup>4</sup>*ε*0*ω*<sup>2</sup>

� � � 2 þ 1 *μ*0 *B* ! *<sup>k</sup> r* !*; t* � � � �

2 � �

�

! � *r* ! �*ω<sup>k</sup> t* � �

sin *k* ! � *r* ! �*ωkt* � �

*k* � ^*ε* � �

> <sup>2</sup> sin <sup>2</sup> *k* ! � *r* ! �*ω<sup>k</sup> t* � �

*<sup>k</sup>*j j *A*0*<sup>k</sup>*ð Þ *ω<sup>k</sup>*

� � �

fields are proportional to the vector

� �! � � �

<sup>2</sup> (13)

*<sup>V</sup>* (14)

(9)

(10)

(11)

(12)

(15)

� <sup>¼</sup> <sup>2</sup>*π=λ<sup>k</sup>* is the

analytically how this link is established.

*Wk r* !*; t* � �

> *Ek* ! *r* !*; t* � �

the square of the vector potential amplitude:

*Wk r* !*; t* � � ¼ 1 <sup>2</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>E</sup>* ! *<sup>k</sup> r* !*; t* � � � �

> ! *r* !*; t* � �

and magnetic *Bk*

¼ � <sup>1</sup> *c*

�

vacuum, respectively, related to the speed of light in vacuum *<sup>c</sup>* by *<sup>ε</sup>*0*μ*0*c*<sup>2</sup> <sup>¼</sup> 1. In the case of a monochromatic plane wave with angular frequency *ωk*, the

netic wave writes:

*Single Photon Manipulation*

electric *Ek* ! *r* !*; t* � �

potential amplitude *A*0*k*ð Þ *ω<sup>k</sup>* :

*Bk* ! *r* !*; t* � �

wave vectors [4, 5, 7].

writes simply:

**20**

The last relation is the fundamental link between the classical and quantum theory of light which is used to define in QED the vector potential amplitude operators for a single photon [19, 26, 29, 36–41]:

$$
\tilde{A}\_{k\lambda} = \sqrt{\frac{\hbar}{2\varepsilon\_0 a \nu\_k V}} a\_{k\lambda} \qquad \qquad \tilde{A}\_{k\lambda}^{\*} = \sqrt{\frac{\hbar}{2\varepsilon\_0 a \nu\_k V}} a\_{k\lambda}^{+} \tag{16}
$$

where *ak<sup>λ</sup>* and *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>* are, respectively, the annihilation and creation non-Hermitian operators for a *k*-mode and *λ*-polarisation photon.

Therein, it is worth noting that an external arbitrary volume parameter *V* appears in the vector potential amplitude of the single photon, expressed by Eq. (15), which is supposed to be an intrinsic physical property. This could entail the unphysical interpretation that a single photon in an infinite cavity has zero vector potential, thus zero electric and magnetic fields and consequently zero energy. This ambiguity, which is scarcely quoted in the literature, is lifted by considering that, in the case of a single photon, the volume *V* in Eq. (15) is equivalent to that defined by the boundary conditions in a cavity for the single radiation mode *k*.

## **3. Electromagnetic field quantization and the photon description**

#### **3.1 Harmonic oscillator representation of the electromagnetic field**

The energy of the electromagnetic field in a volume *V* considered as a superposition of different *k*-modes and *λ*-polarisations is obtained directly from Eq. (13):

$$E\_{\rm EM} = 2\varepsilon\_0 V \sum\_{k\_2, \lambda} \alpha\_k^2 |A\_{k\lambda}(a\eta\_k)|^2 \tag{17}$$

where the summation over the *λ*-polarisations takes only two values corresponding to circular left and right [19, 36–41].

Replacing in Eq. (17) the vector potential amplitude and its conjugate by the relations of the vector potential amplitude operators defined in Eq. (16), we get the "normal ordering" radiation Hamiltonian corresponding to the order *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* of the creation and annihilation operators:

$$
\tilde{H}\_{\rm NO} = \sum\_{k\_\lambda \lambda} a\_{k\lambda}^+ a\_{k\lambda} \hbar o\_k \tag{18}
$$

and the "anti-normal ordering" Hamiltonian corresponding to the order *akλa*<sup>þ</sup> *kλ*

$$\tilde{H}\_{\rm ANO} = \sum\_{k\_2,k} \left( a\_{k\bar{\imath}}^+ a\_{k\bar{\imath}} + \mathbf{1} \right) \hbar o\_k \tag{19}$$

where we have used the fundamental commutation relation in quantum electrodynamics:

$$\begin{bmatrix} \mathbf{a}\_{kl}, \mathbf{a}\_{kl}^{+} \end{bmatrix} = \mathbf{1} \tag{20}$$

In Dirac's representation the eigenfunctions take the simple expression *nk<sup>λ</sup>* j i, and the action of the creation and annihilation operators of a single *k-*mode and *λ*-polarisation photon writes:

$$a\_{k\dot{i}}^{+}|n\_{k\dot{i}}\rangle = \sqrt{n\_{k\dot{i}}+\mathbf{1}}|n\_{k\dot{i}}+\mathbf{1}\rangle; \quad a\_{k\dot{i}}|n\_{k\dot{i}}\rangle = \sqrt{n\_{k\dot{i}}}|n\_{k\dot{i}}-\mathbf{1}\rangle \tag{21}$$

The successive action of both operators in the normal order corresponds to the photon number Hermitian operator *n*~*k<sup>λ</sup>* ¼ *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* having the eigenvalue *nk<sup>λ</sup>* representing the number of *k-*mode and *λ*-polarisation photons:

$$
\langle a\_{k\dot{\lambda}}^{+} a\_{k\dot{\lambda}} | n\_{k\dot{\lambda}} \rangle = \tilde{n}\_{k\dot{\lambda}} | n\_{k\dot{\lambda}} \rangle = n\_{k\dot{\lambda}} | n\_{k\dot{\lambda}} \rangle \tag{22}
$$

because *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>* are simply canonical variables, getting the energy of an

*P*2 *<sup>k</sup><sup>λ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *kQ*<sup>2</sup> *kλ*

Replacing in the last equation the classical canonical variables of position and

� � (27)

ffiffiffiffiffiffiffiffi ℏ 2*ω<sup>k</sup>*

!¼ *mdq*!*=dt*, with canonical variables of position

*a*þ *<sup>k</sup><sup>λ</sup>* þ *ak<sup>λ</sup>*

� �ℏ*ω<sup>k</sup>* (29)

*m*p , the transition from the classical expres-

*Q*<sup>2</sup> � � (30)

1 2

*a*ð Þ <sup>þ</sup> þ *a* , and *n*~ ¼ *a*þ*a* is direct and needs no

� �ℏ*<sup>ω</sup>* (31)

� � (28)

s

*<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* ¼ *n*~*k<sup>λ</sup>*, one gets the harmonic oscillator Hamiltonian for the

*n*~*k, <sup>λ</sup>* þ 1 2

At that level it is important to note that, for a harmonic oscillator of a particle

*<sup>P</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

Conversely, this is not the case for the electromagnetic field [19, 29, 39] because commutations between the canonical variables *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>* occur during the mathematical transition from Eq. (17) to Eq. (26). It is then considered that *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 in order to obtain Eq. (27) just before replacing the canonical variables by the corresponding quantum mechanics operators. Therein, it is important to remark

*k*0

*<sup>λ</sup>*<sup>0</sup> � ¼ *i*ℏ*δkk*<sup>0</sup>*δλλ*<sup>0</sup>

� is a fundamental

1 2 � �ℏ*<sup>ω</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>~</sup> <sup>þ</sup>

*EEM* <sup>¼</sup> <sup>1</sup> 2 X *k, <sup>λ</sup>*

ffiffiffiffiffiffiffiffi ℏ*ω<sup>k</sup>* 2

r

momentum with the corresponding Hermitian operators [19, 29, 41]:

*a*þ *<sup>k</sup><sup>λ</sup>* � *ak<sup>λ</sup>* � �*; <sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>* <sup>¼</sup>

*<sup>H</sup>*<sup>~</sup> *EM* <sup>¼</sup> <sup>X</sup> *k, λ*

> ! � � � � � �*<sup>=</sup>* ffiffiffiffi

*<sup>E</sup>* <sup>¼</sup> <sup>1</sup> 2

*<sup>Q</sup>*<sup>~</sup> <sup>2</sup> � � <sup>¼</sup> *<sup>a</sup>*þ*<sup>a</sup>* <sup>þ</sup>

ffiffiffiffi ℏ 2*ω* q

commutation operations between the canonical variables *P* and *Q* [19, 39]. Consequently, the harmonic oscillator Hamiltonian for a particle of mass *m* expressed by Eq. (31) is a quite physical result (e.g., phonons in solid-state physics) obtained with a perfect correspondence between the classical canonical variables of momentum and position *P* and *Q*, respectively, and the corresponding Hermitian

concept of quantum mechanics, which should not be ignored when replacing classical variables by the corresponding quantum mechanics operators [19]. In fact, without dropping *Qkλ; Pk<sup>λ</sup>* ½ � in Eq. (26) and replacing the canonical variables by the corresponding quantum operators of Eq. (28), we get naturally the same normal

ordering and anti-normal ordering radiation Hamiltonians as in Eq. (23):

ensemble of harmonic oscillators:

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

*Quantized Field of Single Photons*

*<sup>P</sup>*~*k<sup>λ</sup>* <sup>¼</sup> *<sup>i</sup>*

and putting *a*<sup>þ</sup>

with mass *m* and momentum *p*

*<sup>m</sup>*<sup>p</sup> and momentum *<sup>P</sup>* <sup>¼</sup> *<sup>p</sup>*

to the quantum mechanics Hamiltonian:

that Heisenberg's commutation relation *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup>

*<sup>a</sup>*ð Þ <sup>þ</sup> � *<sup>a</sup>* , *<sup>Q</sup>*<sup>~</sup> <sup>¼</sup>

*<sup>H</sup>*<sup>~</sup> <sup>¼</sup> <sup>1</sup> 2 *P*~2 <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

ffiffiffiffi ℏ*ω* 2 q

radiation field:

*Q* ¼ *q* ! � � � � � � ffiffiffiffi

sion of energy:

where *<sup>P</sup>*<sup>~</sup> <sup>¼</sup> *<sup>i</sup>*

operators *P*~ and *Q*~ .

**23**

In this representation the normal and anti-normal ordering radiation Hamiltonians write, respectively:

$$
\tilde{H}\_{\rm NO} = \sum\_{k\_2 \lambda} \tilde{n}\_{k\lambda} \hbar o\_{k\lambda}; \quad \tilde{H}\_{\rm ANO} = \sum\_{k\_2 \lambda} (\tilde{n}\_{k\lambda} + 1)\hbar o\_k \tag{23}
$$

We obtain a harmonic oscillator Hamiltonian for the electromagnetic field by considering the mean value of the normal ordering and anti-normal ordering Hamiltonians:

$$
\tilde{H}\_{EM} = \frac{1}{2} \left( \tilde{H}\_{NO} + \tilde{H}\_{ANO} \right) = \sum\_{k,\lambda} \left( \tilde{n}\_{k,\lambda} + \frac{1}{2} \right) \hbar \nu\_k \tag{24}
$$

Thus, in QED the electromagnetic field is considered to be an ensemble of harmonic oscillators each represented by a point particle, the photon, whose eigenfunction is denoted simply by 1*k<sup>λ</sup>* j i [19, 39, 41].

Although we have no experimental facts showing the harmonic oscillator nature of a single photon, this representation has been adopted since the 1930s [37].

In a different way, a harmonic oscillator representation for the electromagnetic field can be obtained by the intermediate of the canonical variables of position *Qk<sup>λ</sup>* and momentum *Pkλ*. For that purpose we introduce the definitions expressing the vector potential amplitude and its complex conjugate with respect to *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>* [19, 29, 41]:

$$A\_{k\lambda} = \frac{(o\_k Q\_{k\lambda} + iP\_{k\lambda})}{2o\_k \sqrt{\varepsilon\_0 V}}; \qquad A\_{k\lambda}^\* = \frac{(o\_k Q\_{k\lambda} - iP\_{k\lambda})}{2o\_k \sqrt{\varepsilon\_0 V}}\tag{25}$$

Introducing the last expressions in Eq. (17), we get the electromagnetic field energy:

$$E\_{\rm EM} = \frac{1}{2} \sum\_{k\_2, \lambda} \left( P\_{k\lambda}^2 + o\_k^2 Q\_{k\lambda}^2 \right) \pm i o\_k [Q\_{k\lambda}, P\_{k\lambda}] \tag{26}$$

where the (+) sign is obtained when Eq. (17) is considered initially to be in the "normal order", *A*<sup>∗</sup> *<sup>k</sup><sup>λ</sup>Akλ*, and the (�) one when in the "anti-normal order" *AkλA*<sup>∗</sup> *kλ*.

With the purpose of establishing a harmonic oscillator representation for the electromagnetic field, it is generally considered that *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 in Eq. (26),

In Dirac's representation the eigenfunctions take the simple expression *nk<sup>λ</sup>* j i, and the action of the creation and annihilation operators of a single *k-*mode and

*nk<sup>λ</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> j i *nk<sup>λ</sup>* <sup>þ</sup> <sup>1</sup> *; ak<sup>λ</sup> nk<sup>λ</sup>* j i <sup>¼</sup> ffiffiffiffiffiffi

The successive action of both operators in the normal order corresponds to the

*nk<sup>λ</sup>*

*<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* having the eigenvalue *nk<sup>λ</sup>*

*n*~*k<sup>λ</sup>* þ 1Þℏ*ω<sup>k</sup>*

� (23)

ℏ*ω<sup>k</sup>* (24)

*<sup>k</sup><sup>λ</sup>ak<sup>λ</sup> nk<sup>λ</sup>* j i ¼ *n*~*k<sup>λ</sup> nk<sup>λ</sup>* j i ¼ *nk<sup>λ</sup> nk<sup>λ</sup>* j i (22)

*k, λ*

*k, λ*

*n*~*k, <sup>λ</sup>* þ 1 2

� �

*<sup>k</sup><sup>λ</sup>* <sup>¼</sup> ð Þ *<sup>ω</sup>kQk<sup>λ</sup>* � *iPk<sup>λ</sup>* 2*ω<sup>k</sup>*

� � � *<sup>i</sup>ω<sup>k</sup> Qkλ; Pk<sup>λ</sup>* ½ � (26)

ffiffiffiffiffiffiffiffi

*<sup>ε</sup>*0*<sup>V</sup>* <sup>p</sup> (25)

*kλ*.

<sup>p</sup> j i *nk<sup>λ</sup>* � <sup>1</sup> (21)

*λ*-polarisation photon writes:

*Single Photon Manipulation*

*a*þ

Hamiltonians write, respectively:

iltonians:

[19, 29, 41]:

energy:

**22**

"normal order", *A*<sup>∗</sup>

*<sup>k</sup><sup>λ</sup> nk<sup>λ</sup>* j i <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

representing the number of *k-*mode and *λ*-polarisation photons:

In this representation the normal and anti-normal ordering radiation

*<sup>H</sup>*<sup>~</sup> *NO* <sup>þ</sup> *<sup>H</sup>*<sup>~</sup> *ANO* � � <sup>¼</sup> <sup>X</sup>

*<sup>n</sup>*~*k<sup>λ</sup>*ℏ*ωk; <sup>H</sup>*<sup>~</sup> *ANO* <sup>¼</sup> <sup>X</sup>

We obtain a harmonic oscillator Hamiltonian for the electromagnetic field by considering the mean value of the normal ordering and anti-normal ordering Ham-

Thus, in QED the electromagnetic field is considered to be an ensemble of harmonic oscillators each represented by a point particle, the photon, whose

of a single photon, this representation has been adopted since the 1930s [37].

Although we have no experimental facts showing the harmonic oscillator nature

In a different way, a harmonic oscillator representation for the electromagnetic field can be obtained by the intermediate of the canonical variables of position *Qk<sup>λ</sup>* and momentum *Pkλ*. For that purpose we introduce the definitions expressing the vector potential amplitude and its complex conjugate with respect to *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>*

Introducing the last expressions in Eq. (17), we get the electromagnetic field

where the (+) sign is obtained when Eq. (17) is considered initially to be in the

With the purpose of establishing a harmonic oscillator representation for the electromagnetic field, it is generally considered that *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 in Eq. (26),

*<sup>k</sup><sup>λ</sup>Akλ*, and the (�) one when in the "anti-normal order" *AkλA*<sup>∗</sup>

*a*þ

photon number Hermitian operator *n*~*k<sup>λ</sup>* ¼ *a*<sup>þ</sup>

*<sup>H</sup>*<sup>~</sup> *NO* <sup>¼</sup> <sup>X</sup> *k, λ*

*<sup>H</sup>*<sup>~</sup> *EM* <sup>¼</sup> <sup>1</sup> 2

eigenfunction is denoted simply by 1*k<sup>λ</sup>* j i [19, 39, 41].

*Ak<sup>λ</sup>* <sup>¼</sup> ð Þ *<sup>ω</sup>kQk<sup>λ</sup>* <sup>þ</sup> *iPk<sup>λ</sup>* 2*ω<sup>k</sup>*

*EEM* <sup>¼</sup> <sup>1</sup> 2 X *k, <sup>λ</sup>*

ffiffiffiffiffiffiffiffi *<sup>ε</sup>*0*<sup>V</sup>* <sup>p</sup> *; A*<sup>∗</sup>

*P*2 *<sup>k</sup><sup>λ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *kQ*<sup>2</sup> *kλ* because *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>* are simply canonical variables, getting the energy of an ensemble of harmonic oscillators:

$$E\_{\rm EM} = \frac{1}{2} \sum\_{k,\lambda} \left( P\_{k\lambda}^2 + a\_k^2 Q\_{k\lambda}^2 \right) \tag{27}$$

Replacing in the last equation the classical canonical variables of position and momentum with the corresponding Hermitian operators [19, 29, 41]:

$$\tilde{P}\_{k\dot{\lambda}} = i\sqrt{\frac{\hbar o\_k}{2}}(a\_{k\dot{\lambda}}^+ - a\_{k\dot{\lambda}}); \quad \tilde{Q}\_{k\dot{\lambda}} = \sqrt{\frac{\hbar}{2o\_k}}(a\_{k\dot{\lambda}}^+ + a\_{k\dot{\lambda}}) \tag{28}$$

and putting *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>ak<sup>λ</sup>* ¼ *n*~*k<sup>λ</sup>*, one gets the harmonic oscillator Hamiltonian for the radiation field:

$$\tilde{H}\_{EM} = \sum\_{k,\lambda} \left(\tilde{n}\_{k,\lambda} + \frac{1}{2}\right) \hbar o\_{\mathbb{k}} \tag{29}$$

At that level it is important to note that, for a harmonic oscillator of a particle with mass *m* and momentum *p* !¼ *mdq*!*=dt*, with canonical variables of position *Q* ¼ *q* ! � � � � � � ffiffiffiffi *<sup>m</sup>*<sup>p</sup> and momentum *<sup>P</sup>* <sup>¼</sup> *<sup>p</sup>* ! � � � � � �*<sup>=</sup>* ffiffiffiffi *m*p , the transition from the classical expression of energy:

$$E = \frac{1}{2} \left( P^2 + a o^2 Q^2 \right) \tag{30}$$

to the quantum mechanics Hamiltonian:

$$\tilde{H} = \frac{1}{2} \left( \tilde{P}^2 + \alpha^2 \tilde{\mathbf{Q}}^2 \right) = \left( a^+ a + \frac{1}{2} \right) \hbar \omega = \left( \tilde{n} + \frac{1}{2} \right) \hbar \omega \tag{31}$$

where *<sup>P</sup>*<sup>~</sup> <sup>¼</sup> *<sup>i</sup>* ffiffiffiffi ℏ*ω* 2 q *<sup>a</sup>*ð Þ <sup>þ</sup> � *<sup>a</sup>* , *<sup>Q</sup>*<sup>~</sup> <sup>¼</sup> ffiffiffiffi ℏ 2*ω* q *a*ð Þ <sup>þ</sup> þ *a* , and *n*~ ¼ *a*þ*a* is direct and needs no commutation operations between the canonical variables *P* and *Q* [19, 39].

Consequently, the harmonic oscillator Hamiltonian for a particle of mass *m* expressed by Eq. (31) is a quite physical result (e.g., phonons in solid-state physics) obtained with a perfect correspondence between the classical canonical variables of momentum and position *P* and *Q*, respectively, and the corresponding Hermitian operators *P*~ and *Q*~ .

Conversely, this is not the case for the electromagnetic field [19, 29, 39] because commutations between the canonical variables *Qk<sup>λ</sup>* and *Pk<sup>λ</sup>* occur during the mathematical transition from Eq. (17) to Eq. (26). It is then considered that *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 in order to obtain Eq. (27) just before replacing the canonical variables by the corresponding quantum mechanics operators. Therein, it is important to remark that Heisenberg's commutation relation *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup> *k*0 *<sup>λ</sup>*<sup>0</sup> � ¼ *i*ℏ*δkk*<sup>0</sup>*δλλ*<sup>0</sup> � is a fundamental concept of quantum mechanics, which should not be ignored when replacing classical variables by the corresponding quantum mechanics operators [19]. In fact, without dropping *Qkλ; Pk<sup>λ</sup>* ½ � in Eq. (26) and replacing the canonical variables by the corresponding quantum operators of Eq. (28), we get naturally the same normal ordering and anti-normal ordering radiation Hamiltonians as in Eq. (23):

$$\begin{pmatrix} \bar{H}\_{EM}^{(+)} = \frac{1}{2} \sum\_{k\_l \lambda} \left( \bar{\tilde{P}}\_{k\lambda}^2 + \alpha\_k^2 \bar{\tilde{Q}}\_{k\lambda}^2 \right) + i\alpha\_k \left[ \bar{Q}\_{k\lambda}, \hat{P}\_{k'\lambda'} \right] \\\\ \bar{H}\_{EM}^{(-)} = \frac{1}{2} \sum\_{k\_l \lambda} \left( \bar{\tilde{P}}\_{k\lambda}^2 + \alpha\_k^2 \bar{\tilde{Q}}\_{k\lambda}^2 \right) - i\alpha\_k \left[ \bar{Q}\_{k\lambda}, \bar{P}\_{k'\lambda'} \right] \end{pmatrix} = \begin{pmatrix} \bar{H}\_{NO} = \sum\_{k\_l \lambda} \bar{n}\_{k\lambda} \hbar \nu\_k \\\\ \bar{H}\_{NO} = \sum\_{k\_l \lambda} \left( \bar{n}\_{k\lambda} + 1 \right) \hbar \nu\_k \end{pmatrix} \tag{32}$$

Obviously, as frequently quoted [2, 19, 39], the fundamental mathematical ambiguity consisting of cancelling the commuting classical variable term *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 before the substitution by non-commuting quantum mechanics operators *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup> *k*0 *<sup>λ</sup>*<sup>0</sup> � ¼ *i*ℏ*δkk*<sup>0</sup>*δλλ*<sup>0</sup> � leads to the harmonic oscillator Hamiltonian for the electromagnetic field.

In fact, since no experiment has yet demonstrated that a single photon is a harmonic oscillator, the main reason for considering the electromagnetic field as an ensemble of harmonic oscillators lies in the importance of the zero-point energy (ZPE) issued in absence of photons from the eigenvalue *nk<sup>λ</sup>* ¼ 0 of Eq. (29) corresponding to the vacuum energy:

$$E\_{\rm ZPE} = \sum\_{k,\lambda} \frac{1}{2} \hbar o\_k \tag{33}$$

Considering the natural units ð Þ ℏ ¼ *c* ¼ 1 , the radiation vector potential writes

^*εk<sup>λ</sup> <sup>α</sup>*ð Þ*<sup>λ</sup>* ð Þ*<sup>k</sup> <sup>e</sup>*

! � *k* !0

*<sup>α</sup>k<sup>λ</sup>; <sup>α</sup>*ð Þþ*<sup>λ</sup>* ð Þ¼ *<sup>k</sup>* ð Þ <sup>2</sup>*k*<sup>0</sup>

*λ*¼1

� <sup>¼</sup> *<sup>ω</sup><sup>k</sup>* and Eq. (37) writes:

*λ*¼1

*λ*¼1

On the basis of the density of state theory, the quantization of a field in a cavity of volume *V* permits to transform the continuous summation over the modes to a

<sup>0</sup> � *k* � �! <sup>2</sup>

ffiffiffiffiffiffiffiffi 1 2*ω<sup>k</sup>* <sup>r</sup> <sup>X</sup> 2

Suppressing the natural units (i.e., introducing *c* and ℏ) and transforming the last equation in the SI system, which is generally used in QED, we get

> ffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ω<sup>k</sup>* <sup>s</sup> <sup>X</sup> 2

ð *d*<sup>3</sup> *k* ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup> <sup>¼</sup>

The last transformation is only valid for an ensemble of modes *k* whose wavelengths *λ<sup>k</sup>* are much shorter than the characteristic dimensions of the volume *V*

�*iωkt ; α*<sup>þ</sup>

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

¼ *l* 2 .

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

ffiffiffiffi 1 *V* <sup>r</sup> <sup>X</sup>

*k*

*<sup>k</sup><sup>λ</sup>*ðÞ¼ *t α*<sup>þ</sup> *kλe*

�*ikx* <sup>þ</sup> *<sup>α</sup>*ð Þþ*<sup>λ</sup>* ð Þ*<sup>k</sup> <sup>e</sup> ikx* h i (34)

� � is the Dirac delta function, and

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup> *α*þ

1*=*2

�*ikx* <sup>þ</sup> *<sup>α</sup>*<sup>þ</sup> *kλe ikx* � � (37)

�*ikx* <sup>þ</sup> *<sup>α</sup>*<sup>þ</sup> *kλe ikx* � � (38)

�*ikx* <sup>þ</sup> *<sup>α</sup>*<sup>þ</sup>

*<sup>k</sup><sup>λ</sup>*ð Þ*k e ikx* � � (39)

(40)

*<sup>i</sup>ωkt* (41)

� � (35)

*<sup>k</sup><sup>λ</sup>* (36)

*δλλ*0*δ*<sup>3</sup> *k* ! � *k* !0

within the frame of the quantum field theory (QFT) [26, 38]:

X 2

*λ*¼1

*<sup>α</sup>*ð Þ*<sup>λ</sup>* ð Þ*<sup>k</sup> ; <sup>α</sup> <sup>λ</sup>*<sup>0</sup> ð Þþ *<sup>k</sup>*<sup>0</sup> � � h i <sup>¼</sup> <sup>2</sup>*k*0ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>

ð *d*<sup>3</sup> *k* ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup> 2*k*<sup>0</sup>

where *δλλ*<sup>0</sup> is the Kronecker delta, *δ*<sup>3</sup> *k*

1*=*2

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup>

Using Eq. (36) in Eq. (34), the vector potential becomes:

*k*

ð Þ 2*k*<sup>0</sup> 1*=*2 X 2

ð *d*<sup>3</sup>

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup>

� �! � � �

ð *d*<sup>3</sup> *k* ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup>

ð *d*<sup>3</sup> *k* ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup>

Switching now to Heisenberg's representation:

*αk<sup>λ</sup>*ðÞ¼ *t αk<sup>λ</sup>e*

and *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

*A* ! ð Þ¼ *x*

*Quantized Field of Single Photons*

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

*<sup>α</sup>*ð Þ*<sup>λ</sup>* ð Þ¼ *<sup>k</sup>* ð Þ <sup>2</sup>*k*<sup>0</sup>

*A* ! ð Þ¼ *x*

!2 þ *l* 2 � �<sup>1</sup>*=*<sup>2</sup>

For *k* ¼ *l* ¼ 0 then *k*<sup>0</sup> ¼ *k*

*A* ! ð Þ¼ *x*

discrete one [19, 51]:

[19, 29, 39, 41].

**25**

*A* ! ð Þ¼ *x*

with *ω<sup>k</sup>* ¼ *k*

[38, 41, 51].

with

The summation of the last expression over all modes and polarisations is infinite and represents the principal singularity in the QED formalism [19, 26, 29, 36, 39].

Nevertheless, the zero-point energy is very important because it is considered to be the basis for the explanation of the vacuum effects such as the spontaneous emission, the Lamb shift and the Casimir effect. However, as pointed out by many authors [19, 26, 39, 41], it is important to underline that the explanation of the spontaneous emission and the Lamb shift in QED is not due to Eq. (33) but precisely to the commutation properties of the photon creation and annihilation operators, *a*<sup>þ</sup> *kλ* and *akλ*, respectively. It has to be emphasized that in quantum mechanics theory Eq. (33), being a constant, commutes with all Hermitian operators corresponding to physical observables and consequently has absolutely no influence to any quantum process.

Conversely, the zero-point energy expressed by Eq. (33) is useful for the explanation of the spontaneous emission and the Lamb shift in the classical description of radiation [2, 39, 47].

Regarding the Casimir effect, it is often commented that caution has to be taken concerning the interpretation of its physical origin because it has been demonstrated by different methods [48–50] that it can be easily explained using classical electrodynamics without invoking at all the zero-point energy.

Hence, in view of the above, the normal ordering Hamiltonian is the one mainly used in QED, casting aside the vacuum singularity issued from the harmonic oscillator formalism, while the zero-point energy issued from the harmonic oscillator Hamiltonian is principally useful in the classical formalism for the interpretation of the vacuum effects [2, 19, 39, 47].

#### **3.2 Electromagnetic field vector potential quantization in QED**

We have analysed in Section 3.1 the electromagnetic field energy quantization according to the harmonic oscillator representation. Now, we will analyse the vector potential field quantization following the second quantisation process.

Considering the natural units ð Þ ℏ ¼ *c* ¼ 1 , the radiation vector potential writes within the frame of the quantum field theory (QFT) [26, 38]:

$$\overrightarrow{A}\ (\infty) = \int \frac{d^3k}{(2\pi)^3 2k\_0} \sum\_{\lambda=1}^2 \hat{e}\_{k\lambda} \left[ a^{(\lambda)}(k) e^{-ikx} + a^{(\lambda)+}(k) e^{ikx} \right] \tag{34}$$

with

*H*~ ð Þ <sup>þ</sup> *EM* <sup>¼</sup> <sup>1</sup> 2 X *k, <sup>λ</sup>*

0

BBBB@

*H*~ ð Þ � *EM* <sup>¼</sup> <sup>1</sup> 2 X *k, <sup>λ</sup>*

operators *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup>

electromagnetic field.

[19, 26, 29, 36, 39].

quantum process.

radiation [2, 39, 47].

**24**

the vacuum effects [2, 19, 39, 47].

*P*~2 *<sup>k</sup><sup>λ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *kQ*~ 2 *kλ*

*Single Photon Manipulation*

*P*~2 *<sup>k</sup><sup>λ</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *kQ*~ 2 *kλ*

*k*0

corresponding to the vacuum energy:

� �

� �

*<sup>λ</sup>*<sup>0</sup> � ¼ *i*ℏ*δkk*<sup>0</sup>*δλλ*<sup>0</sup>

<sup>þ</sup> *<sup>i</sup>ω<sup>k</sup> <sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*^*k*<sup>0</sup>

� *<sup>i</sup>ω<sup>k</sup> <sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup> *<sup>k</sup>*<sup>0</sup>

ambiguity consisting of cancelling the commuting classical variable term *Qkλ; Pk<sup>λ</sup>* ½ �¼ 0 before the substitution by non-commuting quantum mechanics

Obviously, as frequently quoted [2, 19, 39], the fundamental mathematical

In fact, since no experiment has yet demonstrated that a single photon is a harmonic oscillator, the main reason for considering the electromagnetic field as an ensemble of harmonic oscillators lies in the importance of the zero-point energy (ZPE) issued in absence of photons from the eigenvalue *nk<sup>λ</sup>* ¼ 0 of Eq. (29)

> *EZPE* <sup>¼</sup> <sup>X</sup> *k, λ*

The summation of the last expression over all modes and polarisations is

be the basis for the explanation of the vacuum effects such as the spontaneous emission, the Lamb shift and the Casimir effect. However, as pointed out by many authors [19, 26, 39, 41], it is important to underline that the explanation of the spontaneous emission and the Lamb shift in QED is not due to Eq. (33) but precisely to the commutation properties of the photon creation and annihilation operators, *a*<sup>þ</sup>

and *akλ*, respectively. It has to be emphasized that in quantum mechanics theory Eq. (33), being a constant, commutes with all Hermitian operators corresponding to physical observables and consequently has absolutely no influence to any

Conversely, the zero-point energy expressed by Eq. (33) is useful for the explanation of the spontaneous emission and the Lamb shift in the classical description of

Regarding the Casimir effect, it is often commented that caution has to be taken

Hence, in view of the above, the normal ordering Hamiltonian is the one mainly used in QED, casting aside the vacuum singularity issued from the harmonic oscillator formalism, while the zero-point energy issued from the harmonic oscillator Hamiltonian is principally useful in the classical formalism for the interpretation of

We have analysed in Section 3.1 the electromagnetic field energy quantization according to the harmonic oscillator representation. Now, we will analyse the vector

concerning the interpretation of its physical origin because it has been demonstrated by different methods [48–50] that it can be easily explained using classical

electrodynamics without invoking at all the zero-point energy.

**3.2 Electromagnetic field vector potential quantization in QED**

potential field quantization following the second quantisation process.

infinite and represents the principal singularity in the QED formalism

*λ*0 � � 1

CCCCA ¼

0

BB@

*<sup>H</sup>*<sup>~</sup> *NO* <sup>¼</sup> <sup>X</sup> *k, <sup>λ</sup>*

*<sup>H</sup>*<sup>~</sup> *ANO* <sup>¼</sup> <sup>X</sup>

ℏ*ω<sup>k</sup>* (33)

*k, <sup>λ</sup>*

*n*~*k<sup>λ</sup>* ℏ*ω<sup>k</sup>*

ð Þ *n*~*k<sup>λ</sup>* þ 1 ℏ*ω<sup>k</sup>*

(32)

1

CCA

*kλ*

*λ*0 � �

� leads to the harmonic oscillator Hamiltonian for the

1 2

Nevertheless, the zero-point energy is very important because it is considered to

$$\left[a^{(\boldsymbol{\lambda})}(\boldsymbol{k}), a^{(\boldsymbol{\lambda'})+}\left(\boldsymbol{k'}\right)\right] = 2\boldsymbol{k}\_0(2\pi)^3 \delta\_{\boldsymbol{\lambda}\boldsymbol{\lambda'}} \delta^3\left(\overrightarrow{\boldsymbol{k}} - \overrightarrow{\boldsymbol{k'}}\right) \tag{35}$$

where *δλλ*<sup>0</sup> is the Kronecker delta, *δ*<sup>3</sup> *k* ! � *k* !0 � � is the Dirac delta function, and

$$a^{(\lambda)}(k) = (2k\_0)^{1/2} (2\pi)^{3/2} a\_{k\lambda}; \quad a^{(\lambda)+}(k) = (2k\_0)^{1/2} (2\pi)^{3/2} a\_{k\lambda}^+ \tag{36}$$

Using Eq. (36) in Eq. (34), the vector potential becomes:

$$\overrightarrow{A}\ (\infty) = \int \frac{d^3k}{\left(2\pi\right)^{3/2} \left(2k\_0\right)^{1/2}} \sum\_{k=1}^2 \hat{\varepsilon}\_{k\lambda} \left[a\_{k\lambda}e^{-ikx} + a\_{k\lambda}^+e^{ikx}\right] \tag{37}$$

with *ω<sup>k</sup>* ¼ *k* !2 þ *l* 2 � �<sup>1</sup>*=*<sup>2</sup> and *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> <sup>0</sup> � *k* � �! <sup>2</sup> ¼ *l* 2 . For *k* ¼ *l* ¼ 0 then *k*<sup>0</sup> ¼ *k* � �! � � � � <sup>¼</sup> *<sup>ω</sup><sup>k</sup>* and Eq. (37) writes:

$$\overrightarrow{A}\ (\mathbf{x}) = \int \frac{d^3k}{(2\pi)^{3/2}} \sqrt{\frac{1}{2\alpha\_k}} \sum\_{\lambda=1}^2 \hat{e}\_{k\lambda} \left[a\_{k\lambda}e^{-ik\mathbf{x}} + a\_{k\lambda}^+e^{ik\mathbf{x}}\right] \tag{38}$$

Suppressing the natural units (i.e., introducing *c* and ℏ) and transforming the last equation in the SI system, which is generally used in QED, we get [38, 41, 51].

$$\overrightarrow{A}\ (\varkappa) = \int \frac{d^3k}{(2\pi)^{3/2}} \sqrt{\frac{\hbar}{2\varepsilon\_0 a \nu\_k}} \sum\_{l=1}^2 \hat{e}\_{k\lambda} \left[a\_{k\ell}e^{-ik\chi} + a\_{k\ell}^+(k)e^{ik\chi}\right] \tag{39}$$

On the basis of the density of state theory, the quantization of a field in a cavity of volume *V* permits to transform the continuous summation over the modes to a discrete one [19, 51]:

$$\int \frac{d^3k}{(2\pi)^{3/2}} = \sqrt{\frac{1}{V}} \sum\_{k} \tag{40}$$

The last transformation is only valid for an ensemble of modes *k* whose wavelengths *λ<sup>k</sup>* are much shorter than the characteristic dimensions of the volume *V* [19, 29, 39, 41].

Switching now to Heisenberg's representation:

$$a\_{kl}(t) = a\_{kl}e^{-i\nu\_k t} \quad ; \quad a\_{kl}^+(t) = a\_{kl}^+ e^{i\nu\_k t} \tag{41}$$

Generalizing the coordinate system, adapting the phase and using Eq. (40), the vector potential of the electromagnetic field writes in QED [19, 29, 39, 41, 51]:

$$\overrightarrow{A}\left(\overrightarrow{r},t\right) = \sum\_{k} \sqrt{\frac{\hbar}{2\varepsilon\_{0}a\nu\_{k}V}} \sum\_{\lambda=1}^{2} \hat{e}\_{k\lambda} \left[a\_{k\lambda}\hat{e}^{i\left(\overrightarrow{k\cdot r}-a\mu t\right)} + a\_{k\lambda}^{+}\hat{e}^{-i\left(\overrightarrow{k\cdot r}-a\mu t\right)}\right] \tag{42}$$

Considering the scalar potential to be constant, the electric field is:

$$\overrightarrow{E}\left(\overrightarrow{r},t\right) = i\sum\_{k}\sqrt{\frac{\hbar o\eta\_{k}}{2\varepsilon\_{0}V}}\sum\_{k=1}^{2}\hat{e}\_{k\boldsymbol{k}}\left[a\_{k}e^{i\left(\overrightarrow{k\cdot\overrightarrow{r}-a\eta\_{k}}\right)}-a\_{k}^{+}e^{-i\left(\overrightarrow{k\cdot\overrightarrow{r}-a\eta\_{k}}\right)}\right] \tag{43}$$

The last expressions represent in a given volume *V* the quantized vector potential and the electric field of the electromagnetic radiation composed of a large number of modes *k* each with angular frequency *ω<sup>k</sup>* and wavelength *λ<sup>k</sup>* ¼ 2*πc=ω<sup>k</sup>* much smaller than *V*1/3:

$$
\lambda\_k \ll V^{1/3} \quad (\forall k) \tag{44}
$$

under the condition of Eq. (44), concern a large number of modes in a considerably big volume compared to their wavelengths. Thus, with the aim of obtaining a clearer picture of the single photon, we will now complement the previous descriptions by enhancing the vector potential amplitude quantization to the photon level.

As mentioned in Section 2.2, the classical expression of the mean energy density over a period for a single electromagnetic mode *k*, represented by Eq. (13), can be considered equivalent to that for a single photon in the quantum representation, given by Eq. (14) for *N*ð Þ¼ *ω<sup>k</sup>* 1, on the condition that the volume *V* is not arbitrary but corresponds to that defined by the boundary conditions in a cavity for the considered electromagnetic mode. In fact, the physical properties of a free photon are independent on any surrounding volume unless the characteristic dimensions of the last one are of the order of the photon wavelength [52].We recall again that the experimental evidence has shown [19, 27, 29, 51] that a single photon with angular frequency *ω<sup>k</sup>* and wavelength *λ<sup>k</sup>* ¼ 2*πc=ω<sup>k</sup>* can only be localised within a volume *Vk*

*<sup>k</sup>* ) *Vk* <sup>∝</sup>*ω*�<sup>3</sup>

From a theoretical point of view, this is also compatible with the density of state theory according to which the spatial volume corresponding to a single state of the

*<sup>k</sup>* [19, 29, 39, 41]. On the other hand, the dimension analysis of the vector potential issued from the general solution of Maxwell's equations yields that it is proportional to an

> !� *<sup>r</sup>* !'j *c* � �

*d*3

*<sup>k</sup>* (47)

*r*' ∝ *ω* (48)

(49)

) and *μ* the magnetic permeability.

**4.1 Photon vector potential amplitude and quantization volume**

whose dimensions are roughly the cube of its wavelength:

quantized field is proportional to *ω*�<sup>3</sup>

*A* ! *r* !*; t* � � <sup>¼</sup> *<sup>μ</sup>*

where *J* is the current density (*C m*�*<sup>2</sup> s*

amplitude is normally proportional to *ω* [4, 5, 7].

� � <sup>¼</sup> *<sup>α</sup>*0*<sup>k</sup>*ð Þ *<sup>ω</sup><sup>k</sup>* ^*εkλαk<sup>λ</sup><sup>e</sup>*

� � <sup>¼</sup> *<sup>α</sup>*0*<sup>k</sup>*ð Þ *<sup>ω</sup><sup>k</sup>* ^*εk<sup>λ</sup><sup>e</sup>*

where, following to the above analysis, the amplitude writes:

angular frequency [5, 7, 9]:

*Quantized Field of Single Photons*

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

potential.

**27**

and classical formalism:

*α*~*k<sup>λ</sup> r* !*; t*

> *αk<sup>λ</sup>* ! *r* !*; t*

*Vk* ∝*λ*<sup>3</sup>

4*π*

ð *J* ! *r* ! '*, t* � <sup>j</sup>*<sup>r</sup>*

> j *r* ! � *r* ! 'j

a dipole is proportional to *ω*<sup>4</sup> entailing from Eq. (12) that the vector potential

*i k*! �*r* !�*<sup>ω</sup>kt* � �

> *i k*! �*r* !�*<sup>ω</sup>kt* � �

�1

Indeed, it is well established experimentally that the energy density radiated by

This result is gauge independent since it concerns the natural units of the vector

According to the previous considerations, for a free single *k*-mode photon with *λ*-polarisation (left or right circular), the vector potential can be written in quantum

> <sup>þ</sup> ^*<sup>ε</sup>* <sup>∗</sup> *kλα*<sup>þ</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*

> > <sup>þ</sup> ^*<sup>ε</sup>* <sup>∗</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*

*α*0*<sup>k</sup>*ð Þ¼ *ω<sup>k</sup>* j j *ξ ω<sup>k</sup>* (50)

� � � �

� � � �

The amplitudes in Eqs. (42) and (43) have been obtained using the density of state theory and are valid only on the condition of Eq. (44). Furthermore, the boundary conditions of the electromagnetic waves considered in cavities and waveguides impose the wave vectors *k* of the modes to be higher than a characteristic cut-off value *<sup>k</sup>*>*kcut*�*off <sup>λ</sup><sup>k</sup>* <sup>&</sup>lt; *<sup>λ</sup>cut*�*off* � � depending on the dimensions and the shape of the volume containing the radiation field [4–7]. Consequently, for a volume *V* with finite dimensions, the summation in Eqs. (42) and (43) runs only over the modes *k* with wave vectors higher than the minimum cut-off value *kcut*�*off*ð Þ *V* imposed by the shape and dimensions of *V* so that we can write more precisely:

$$\overrightarrow{A}\left(\overrightarrow{r},t\right) = \sum\_{k>k\_{\text{cat}}-\text{eff}} \sqrt{\frac{\hbar}{2\varepsilon\_{0}\alpha\_{k}V}} \sum\_{k=1}^{2} \hat{e}\_{k\boldsymbol{l}} \left[a\_{k\boldsymbol{l}}e^{i\left(\overrightarrow{k\cdot r}-\alpha\_{k}t\right)} + a\_{k\boldsymbol{l}}^{+}e^{-i\left(\overrightarrow{k\cdot r}-\alpha\_{k}t\right)}\right] \tag{45}$$

$$\overrightarrow{E}\left(\overrightarrow{r},t\right) = i \sum\_{k>k\_{\text{tot}}-\text{eff}} \sqrt{\frac{\hbar o\_k}{2\varepsilon\_0 V}} \sum\_{k=1}^{2} \hat{\varepsilon}\_{k\lambda} \left[a\_{k\hat{\varepsilon}} \mathbf{e}^{i\left(\overrightarrow{k\cdot r}-a\_k t\right)} - a\_{k\hat{\varepsilon}}^{+} \mathbf{e}^{-i\left(\overrightarrow{k\cdot r}-a\_k t\right)}\right] \tag{46}$$

The last equations represent the vector potential and the electric field of a large number of modes *k* of the quantized electromagnetic field in a finite volume *V* with *<sup>λ</sup><sup>k</sup>* <sup>≪</sup> *<sup>V</sup>*<sup>1</sup>*=*<sup>3</sup> ð Þ <sup>∀</sup>*<sup>k</sup>* .

### **4. Quantized vector potential of a single photon**

We have seen in Section 3.1 that according to the energy quantization procedure, a *k*-mode and *λ*-polarisation photon is considered to be a point harmonic oscillator represented by the simplified eigenfunction 1*k<sup>λ</sup>* j i. On the other hand, following the field quantization procedure in Section 3.2, it appears clearly that the established vector potential and electric field expressions in Eqs. (42) and (43),

### *Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

Generalizing the coordinate system, adapting the phase and using Eq. (40), the vector potential of the electromagnetic field writes in QED [19, 29, 39, 41, 51]:

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

*i k*! �*r* !�*<sup>ω</sup>kt* � �

*i k*! �*r* !�*<sup>ω</sup>kt* � �

The last expressions represent in a given volume *V* the quantized vector poten-

The amplitudes in Eqs. (42) and (43) have been obtained using the density of state theory and are valid only on the condition of Eq. (44). Furthermore, the boundary conditions of the electromagnetic waves considered in cavities and waveguides impose the wave vectors *k* of the modes to be higher than a character-

þ *α*<sup>þ</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*

(42)

(43)

(45)

(46)

� � � �

� *α*<sup>þ</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*

*<sup>λ</sup><sup>k</sup>* <sup>≪</sup> *<sup>V</sup>*<sup>1</sup>*=*<sup>3</sup> ð Þ <sup>∀</sup>*<sup>k</sup>* (44)

þ *α*<sup>þ</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*

� � � �

� *α*<sup>þ</sup> *kλe* �*i k*! �*r* !�*ωkt*

� � � �

� � depending on the dimensions and the

� � � �

*A* ! *r* !*; t* � �

*Single Photon Manipulation*

*E* ! *r* !*; t* � �

precisely:

*A* ! *r* !*; t* � �

*E* ! *r* !*; t* � �

**26**

with *<sup>λ</sup><sup>k</sup>* <sup>≪</sup> *<sup>V</sup>*<sup>1</sup>*=*<sup>3</sup> ð Þ <sup>∀</sup>*<sup>k</sup>* .

much smaller than *V*1/3:

<sup>¼</sup> <sup>X</sup> *k*

> ¼ *i* X *k*

istic cut-off value *k*>*kcut*�*off λ<sup>k</sup>* < *λcut*�*off*

<sup>¼</sup> <sup>X</sup> *k*>*kcut*�*off*ð Þ *V*

<sup>¼</sup> *<sup>i</sup>* <sup>X</sup> *k*>*kcut*�*off*ð Þ *V*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ωkV* <sup>s</sup> <sup>X</sup>

> ffiffiffiffiffiffiffiffiffiffi ℏ*ω<sup>k</sup>* 2*ε*0*V* <sup>r</sup> <sup>X</sup> 2

2

*λ*¼1

*λ*¼1

Considering the scalar potential to be constant, the electric field is:

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

tial and the electric field of the electromagnetic radiation composed of a large number of modes *k* each with angular frequency *ω<sup>k</sup>* and wavelength *λ<sup>k</sup>* ¼ 2*πc=ω<sup>k</sup>*

shape of the volume containing the radiation field [4–7]. Consequently, for a volume *V* with finite dimensions, the summation in Eqs. (42) and (43) runs only over the modes *k* with wave vectors higher than the minimum cut-off value *kcut*�*off*ð Þ *V* imposed by the shape and dimensions of *V* so that we can write more

2

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

^*εk<sup>λ</sup> αk<sup>λ</sup>e*

The last equations represent the vector potential and the electric field of a large number of modes *k* of the quantized electromagnetic field in a finite volume *V*

We have seen in Section 3.1 that according to the energy quantization procedure, a *k*-mode and *λ*-polarisation photon is considered to be a point harmonic oscillator represented by the simplified eigenfunction 1*k<sup>λ</sup>* j i. On the other hand, following the field quantization procedure in Section 3.2, it appears clearly that the established vector potential and electric field expressions in Eqs. (42) and (43),

*i k*! �*r* !�*<sup>ω</sup>kt* � �

*i k*! �*r* !�*ωkt* � �

*λ*¼1

*λ*¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 2*ε*0*ωkV* <sup>s</sup> <sup>X</sup>

> ffiffiffiffiffiffiffiffiffiffi ℏ*ω<sup>k</sup>* 2*ε*0*V* <sup>r</sup> <sup>X</sup> 2

**4. Quantized vector potential of a single photon**

under the condition of Eq. (44), concern a large number of modes in a considerably big volume compared to their wavelengths. Thus, with the aim of obtaining a clearer picture of the single photon, we will now complement the previous descriptions by enhancing the vector potential amplitude quantization to the photon level.

## **4.1 Photon vector potential amplitude and quantization volume**

As mentioned in Section 2.2, the classical expression of the mean energy density over a period for a single electromagnetic mode *k*, represented by Eq. (13), can be considered equivalent to that for a single photon in the quantum representation, given by Eq. (14) for *N*ð Þ¼ *ω<sup>k</sup>* 1, on the condition that the volume *V* is not arbitrary but corresponds to that defined by the boundary conditions in a cavity for the considered electromagnetic mode. In fact, the physical properties of a free photon are independent on any surrounding volume unless the characteristic dimensions of the last one are of the order of the photon wavelength [52].We recall again that the experimental evidence has shown [19, 27, 29, 51] that a single photon with angular frequency *ω<sup>k</sup>* and wavelength *λ<sup>k</sup>* ¼ 2*πc=ω<sup>k</sup>* can only be localised within a volume *Vk* whose dimensions are roughly the cube of its wavelength:

$$V\_k \propto \lambda\_k^3 \qquad \Rightarrow \qquad V\_k \propto o\_k^{-\mathfrak{J}} \tag{47}$$

From a theoretical point of view, this is also compatible with the density of state theory according to which the spatial volume corresponding to a single state of the quantized field is proportional to *ω*�<sup>3</sup> *<sup>k</sup>* [19, 29, 39, 41].

On the other hand, the dimension analysis of the vector potential issued from the general solution of Maxwell's equations yields that it is proportional to an angular frequency [5, 7, 9]:

$$\overrightarrow{A}\left(\overrightarrow{r},t\right) = \frac{\mu}{4\pi} \left[\frac{\overrightarrow{J}\left(\overrightarrow{r}\,',t-\frac{|\overrightarrow{r}-\overrightarrow{r}\,'|}{c}\right)}{|\overrightarrow{r}-\overrightarrow{r}\,'|}d^3r' \quad \text{or} \quad \rho o\right] \tag{48}$$

where *J* is the current density (*C m*�*<sup>2</sup> s* �1 ) and *μ* the magnetic permeability.

Indeed, it is well established experimentally that the energy density radiated by a dipole is proportional to *ω*<sup>4</sup> entailing from Eq. (12) that the vector potential amplitude is normally proportional to *ω* [4, 5, 7].

This result is gauge independent since it concerns the natural units of the vector potential.

According to the previous considerations, for a free single *k*-mode photon with *λ*-polarisation (left or right circular), the vector potential can be written in quantum and classical formalism:

$$\begin{aligned} \ddot{a}\_{k\boldsymbol{\lambda}}\left(\overrightarrow{r},t\right) &= a\_{0k}(\boldsymbol{\omega}\_{k}) \left[ \hat{\boldsymbol{e}}\_{k\boldsymbol{\lambda}} \boldsymbol{a}\_{k\boldsymbol{\lambda}} \boldsymbol{e}^{\boldsymbol{\left(\hat{k}\cdot\overrightarrow{r}-\boldsymbol{\omega}\_{k}t\right)}} + \hat{\boldsymbol{e}}\_{k\boldsymbol{\lambda}}^{\*} \boldsymbol{a}\_{k\boldsymbol{\lambda}}^{+} \boldsymbol{e}^{-\boldsymbol{i}\left(\overrightarrow{k\cdot r}-\boldsymbol{\omega}\_{k}t\right)} \right] \\\\ \boldsymbol{a}\_{k\boldsymbol{\lambda}}^{\cdot}\left(\overrightarrow{r},t\right) &= a\_{0k}(\boldsymbol{\omega}\_{k}) \left[ \hat{\boldsymbol{e}}\_{k\boldsymbol{\lambda}} \boldsymbol{e}^{\boldsymbol{i}\left(\overrightarrow{k\cdot r}-\boldsymbol{\omega}\_{k}t\right)} + \hat{\boldsymbol{e}}\_{k\boldsymbol{\lambda}}^{\*} \boldsymbol{e}^{-\boldsymbol{i}\left(\overrightarrow{k\cdot r}-\boldsymbol{\omega}\_{k}t\right)} \right] \end{aligned} \tag{49}$$

where, following to the above analysis, the amplitude writes:

$$a\_{0k}(a\_k) = |\xi| a\_k \tag{50}$$

with *ξ* being a constant [2, 53–55].

We can evaluate *ξ* [2, 53] by using Eqs. (49) and (50) in Eq. (13) and normalising the energy to that of a single photon, ℏ*ωk*, by integrating over a wavelength along the propagation direction while taking into account the experimental results on the lateral expansion of the photon [32–35, 56]. We get:

$$|\xi| = \left| \frac{1}{(2\pi)^{3/2}} \sqrt{\frac{\hbar}{8\alpha\varepsilon\_0 c^3}} \right| = \left| \frac{\hbar}{4\pi\varepsilon c} \right| = 1.747 \quad \text{10}^{-25} \text{ Volt } m^{-1}\text{s}^2 \tag{51}$$

where *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*2*=*4*πε*0ℏ*c*≈1*=*137*:*036 is the fine structure constant and *<sup>e</sup>* is the electron charge. Obviously, when introducing Eq. (50) in Eq. (17) for a single *k*-mode photon, an appropriate volume *V*<sup>k</sup> has to be considered for the equation to hold:

$$E\_k = \hbar o\_k = 2\varepsilon\_0 V\_k \xi^2 o\_k^4 \tag{52}$$

With the same token considering circular polarisation [4, 5, 7, 9] for the ampli-

According to the classical electromagnetic theory, the spin can be written through the electric and magnetic field components; hence, using again the circular

*r* ¼ *ε*0ð Þ *c=ω<sup>k</sup>*

2 <sup>p</sup> *<sup>ω</sup>kα*0*k=<sup>c</sup>* � � *Vk <sup>k</sup>*

ffiffi 2 <sup>p</sup> *<sup>ω</sup>kα*0*<sup>k</sup>* � � ffiffi

� (58)

! � � � � �

The fact that the quantum properties, energy, momentum, and spin, of the photon can be expressed through the classical electromagnetic fields integrated over the volume *Vk* signifies that the photon has naturally a spatial extension, and consequently when employing the term "wave-particle", one must have in mind

We can now obtain Heisenberg's uncertainty relation for position and momen-

<sup>0</sup>*k<sup>λ</sup>* ¼ j j *ξ ωkα*<sup>þ</sup>

ffiffiffiffiffiffiffiffiffiffi *ε*0*Vk* <sup>p</sup> *<sup>α</sup>*~0*k<sup>λ</sup>* � *<sup>α</sup>*<sup>~</sup> <sup>∗</sup>

*<sup>λ</sup>*<sup>0</sup> � *ξa*<sup>þ</sup> *k*0 *λ*0 h i � � <sup>¼</sup> *<sup>i</sup>*ℏ*δkk*<sup>0</sup>*δλλ*<sup>0</sup> (61)

*c* ¼ *α*0*<sup>k</sup>=*j j *ξ* ¼ *ω<sup>k</sup>*

� (62)

*δEk δt*≥ ℏ (63)

*δα*0*<sup>k</sup>δt*≥ j j *ξ* (64)

tum using *Vk*. Indeed, replacing *V* in Eq. (16) by *Vk*, we get the photon vector

The corresponding position *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>* and momentum *<sup>P</sup>*~*k<sup>λ</sup>* Hermitian operators

Thus, introducing Eq. (59) in Eq. (60) and using Eq. (20) with Eq. (53), Heisenberg's commutation relation, a fundamental concept in quantum theory,

> *kλ* � �*; <sup>ξ</sup>ak*<sup>0</sup>

, are complemented by the vector potential amplitude *α*0*<sup>k</sup>* expressing its

*<sup>α</sup>*~0*k<sup>λ</sup>* <sup>¼</sup> j j *<sup>ξ</sup> <sup>ω</sup>kαk<sup>λ</sup> ; <sup>α</sup>*<sup>~</sup> <sup>∗</sup>

0*kλ* � � *; <sup>P</sup>*~*k<sup>λ</sup>* ¼ �*iω<sup>k</sup>*

> ffiffiffiffiffiffiffiffiffiffiffiffi *VkVk*<sup>0</sup>

*Ek=*ℏ ¼ *pk*

� �

Considering Heisenberg's *energy-time* uncertainty principle:

<sup>p</sup> *<sup>ξ</sup>ak<sup>λ</sup>* <sup>þ</sup> *<sup>ξ</sup>a*<sup>þ</sup>

The fundamental properties of the photon, energy *Ek*, momentum *pk*

! j*c=*ℏ ¼ *k* � �! � � � �

we directly deduce from Eq. (62) the *vector potential-time* uncertainty:

! *= k* � �! � � � � <sup>¼</sup> <sup>ℏ</sup> *<sup>k</sup>* !

2 <sup>p</sup> *<sup>ω</sup>kα*0*k=<sup>c</sup>* � �*Vk* <sup>¼</sup> <sup>ℏ</sup>

�>*<sup>k</sup>* <sup>¼</sup> *<sup>c</sup>=ω<sup>k</sup>* obtained for a single

*<sup>k</sup><sup>λ</sup>* (59)

!

, and wave

0*kλ* � � (60)

(57)

tudes of the electric and magnetic fields in Eq. (55), the momentum is:

ffiffi 2 <sup>p</sup> *<sup>ω</sup>kα*0*<sup>k</sup>* � � ffiffi

*pk* ! ¼ ð *Vk ε*<sup>0</sup> *εk<sup>λ</sup>* ! �*βk<sup>λ</sup>* ! *d*3 *r* ¼ *ε*<sup>0</sup>

*S* ! � � � � � � ¼ ð *Vk ε*<sup>0</sup> *r*

polarisation, we get:

photon state [57].

[19, 29, 51] write:

results directly [2]:

electromagnetic nature:

*<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>; <sup>P</sup>*<sup>~</sup> *<sup>k</sup>*<sup>0</sup> *λ*0 � � ¼ �*iε*0*ω*<sup>2</sup>

vector *k* !

**29**

� �

*Quantized Field of Single Photons*

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

potential amplitude operators:

*<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi *ε*0*Vk* <sup>p</sup> *<sup>α</sup>*~0*k<sup>λ</sup>* <sup>þ</sup> *<sup>α</sup>*<sup>~</sup> <sup>∗</sup>

*<sup>k</sup>*<sup>0</sup>*ω<sup>k</sup>*

! � *εk<sup>λ</sup>*

! �*βk<sup>λ</sup>* ! Þ

where we have taken the mean value < *r*

that a single photon is a "three-dimensional particle".

� � *d*3

� �

Thus, the characteristic volume of a free single photon writes in agreement with Eq. (47):

$$V\_k = \left(\frac{\hbar}{2\varepsilon\_0 \xi^2}\right) a\_k^{-\frac{\pi}{3}}\tag{53}$$

Replacing *ξ* expressed by Eq. (51) in Eq. (53), we obtain the photon quantization volume:

$$V\_k \approx 4a\lambda\_k^3\tag{54}$$

Equations (50) and (53) express the quantized vector potential amplitude and the spatial extension of a single photon with the constant *ξ* evaluated to be j j *ξ* ¼ ℏ*=*4*π*j j *e c*.

#### **4.2 Photon classical-quantum (wave-particle) physical properties**

For a free *k*-mode photon, the volume *Vk* corresponds to the space in which the quantized vector potential oscillates at the angular frequency *ω<sup>k</sup>* over a period along the propagation axis generating orthogonal electric and magnetic fields whose amplitudes are, respectively:

$$\left|\overrightarrow{\boldsymbol{e}\_{k}}\right| = \left|-\partial\overrightarrow{\boldsymbol{a}\_{k}}\right| \left(\overrightarrow{\boldsymbol{r}},\boldsymbol{t}\right) / \partial\boldsymbol{t} \middle| \infty \left|\xi\right| \boldsymbol{o}\_{k}^{2}; \qquad \left|\overrightarrow{\boldsymbol{\beta}}\_{k}\right| \propto \sqrt{\varepsilon\_{0}\mu\_{0}} \left|\xi\right| \boldsymbol{o}\_{k}^{2} \tag{55}$$

which are independent on any external arbitrary volume parameter and are directly proportional to the square of the angular frequency [2, 54, 55].

We can now express the quantum properties of the photon, energy, momentum, and spin by integrating the classical electromagnetic expressions over the volume *Vk* and by using the vector potential amplitude obtained in Eq. (50), linking in this way the classical (wave) to the quantum (particle) representations [2]. The energy writes:

$$E\_k = \int\_{V\_k} 2\varepsilon\_0 a\_{0k}^2 a\_k^2 d^3 r = \int\_{V\_k} 2\varepsilon\_0 \xi^2 a\_k^4 d^3 r = 2\varepsilon\_0 \xi^2 a\_k^4 V\_k = \hbar o\_k \tag{56}$$

*Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

with *ξ* being a constant [2, 53–55].

We get:

Eq. (47):

volume:

j j *ξ* ¼ ℏ*=*4*π*j j *e c*.

energy writes:

**28**

*Ek* ¼ ð *Vk* 2*ε*0*α*<sup>2</sup> 0*kω*<sup>2</sup> *kd*3 *r* ¼ ð *Vk* 2*ε*0*ξ*<sup>2</sup> *ω*4 *kd*3

amplitudes are, respectively:

*εk*

� �

!j¼ �*∂αk<sup>λ</sup>*

� � � ! *r* !*; t* � �

j j *<sup>ξ</sup>* <sup>¼</sup> <sup>1</sup>

*Single Photon Manipulation*

�

� �

ð Þ <sup>2</sup>*<sup>π</sup>* <sup>3</sup>*=*<sup>2</sup>

� <sup>r</sup> �

We can evaluate *ξ* [2, 53] by using Eqs. (49) and (50) in Eq. (13) and normalising the energy to that of a single photon, ℏ*ωk*, by integrating over a

account the experimental results on the lateral expansion of the photon [32–35, 56].

� � �

*Ek* <sup>¼</sup> <sup>ℏ</sup>*ω<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*ε*0*Vkξ*<sup>2</sup>

*Vk* <sup>¼</sup> <sup>ℏ</sup>

where *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*2*=*4*πε*0ℏ*c*≈1*=*137*:*036 is the fine structure constant and *<sup>e</sup>* is the electron charge. Obviously, when introducing Eq. (50) in Eq. (17) for a single *k*-mode photon, an appropriate volume *V*<sup>k</sup> has to be considered for the equation to hold:

Thus, the characteristic volume of a free single photon writes in agreement with

2*ε*0*ξ*<sup>2</sup> � �

Replacing *ξ* expressed by Eq. (51) in Eq. (53), we obtain the photon quantization

*Vk* ≈ 4*αλ*<sup>3</sup>

Equations (50) and (53) express the quantized vector potential amplitude and

For a free *k*-mode photon, the volume *Vk* corresponds to the space in which the quantized vector potential oscillates at the angular frequency *ω<sup>k</sup>* over a period along the propagation axis generating orthogonal electric and magnetic fields whose

> *<sup>k</sup> ; β<sup>k</sup>* !

> > *<sup>r</sup>* <sup>¼</sup> <sup>2</sup>*ε*0*ξ*<sup>2</sup>

*ω*4

� (55)

� � �

j∝ ffiffiffiffiffiffiffiffiffi *ε*0*μ*<sup>0</sup> <sup>p</sup> j j *<sup>ξ</sup> <sup>ω</sup>*<sup>2</sup>

*k*

*kVk* ¼ ℏ*ω<sup>k</sup>* (56)

*<sup>=</sup>∂t*j<sup>∝</sup> j j *<sup>ξ</sup> <sup>ω</sup>*<sup>2</sup>

which are independent on any external arbitrary volume parameter and are

We can now express the quantum properties of the photon, energy, momentum, and spin by integrating the classical electromagnetic expressions over the volume *Vk* and by using the vector potential amplitude obtained in Eq. (50), linking in this way the classical (wave) to the quantum (particle) representations [2]. The

directly proportional to the square of the angular frequency [2, 54, 55].

the spatial extension of a single photon with the constant *ξ* evaluated to be

**4.2 Photon classical-quantum (wave-particle) physical properties**

� <sup>¼</sup> <sup>1</sup>*:*747 10�<sup>25</sup> *Volt m*�<sup>1</sup>

*ω*4

*ω*�<sup>3</sup>

*s*

*<sup>k</sup>* (52)

*<sup>k</sup>* (53)

*<sup>k</sup>* (54)

<sup>2</sup> (51)

wavelength along the propagation direction while taking into

� � � � �

<sup>¼</sup> <sup>ℏ</sup> 4*π e c* � � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ 8*αε*0*c*<sup>3</sup>

With the same token considering circular polarisation [4, 5, 7, 9] for the amplitudes of the electric and magnetic fields in Eq. (55), the momentum is:

$$\overrightarrow{p\_k} = \left\{ \begin{array}{c} \varepsilon\_0 \varepsilon\_{k\lambda} \times \overrightarrow{\boldsymbol{\rho}}\_{k\lambda} \ d^3 \boldsymbol{r} = \varepsilon\_0 \left( \sqrt{2} a \boldsymbol{\rho}\_k \boldsymbol{a}\_{0k} \right) \left( \sqrt{2} a \boldsymbol{\rho}\_k \boldsymbol{a}\_{0k} / \boldsymbol{\varepsilon} \right) \boldsymbol{V}\_k \ \overrightarrow{\boldsymbol{k}} \ / \left| \overrightarrow{\boldsymbol{k}} \right| = \hbar \ \overrightarrow{\boldsymbol{k}} \right\} \end{array} \tag{57}$$

According to the classical electromagnetic theory, the spin can be written through the electric and magnetic field components; hence, using again the circular polarisation, we get:

$$\left|\overrightarrow{\mathbf{S}}\right| = \int\_{V\_{k}} \varepsilon\_{0} \left|\overrightarrow{r} \times \left(\overrightarrow{\boldsymbol{\varepsilon}\_{k\boldsymbol{l}}} \times \overrightarrow{\boldsymbol{\rho}\_{k\boldsymbol{l}}}\right)\right| d^{3}r = \varepsilon\_{0} (\boldsymbol{\varepsilon}/a\boldsymbol{\rho}\_{k}) \left(\sqrt{2}a\rho\_{k}a\_{0\boldsymbol{k}}\right) \left(\sqrt{2}a\rho\_{k}a\_{0\boldsymbol{k}}/\boldsymbol{\varepsilon}\right) \mathbf{V}\_{k} = \hbar \tag{58}$$

where we have taken the mean value < *r* ! � � � � � �>*<sup>k</sup>* <sup>¼</sup> *<sup>c</sup>=ω<sup>k</sup>* obtained for a single photon state [57].

The fact that the quantum properties, energy, momentum, and spin, of the photon can be expressed through the classical electromagnetic fields integrated over the volume *Vk* signifies that the photon has naturally a spatial extension, and consequently when employing the term "wave-particle", one must have in mind that a single photon is a "three-dimensional particle".

We can now obtain Heisenberg's uncertainty relation for position and momentum using *Vk*. Indeed, replacing *V* in Eq. (16) by *Vk*, we get the photon vector potential amplitude operators:

$$
\tilde{a}\_{0k\dot{\lambda}} = |\xi| \, o\nu\_k a\_{k\dot{\lambda}}; \qquad \tilde{a}^\*\_{0k\dot{\lambda}} = |\xi| \, o\nu\_k a^+\_{k\dot{\lambda}} \tag{59}
$$

The corresponding position *<sup>Q</sup>*<sup>~</sup> *<sup>k</sup><sup>λ</sup>* and momentum *<sup>P</sup>*~*k<sup>λ</sup>* Hermitian operators [19, 29, 51] write:

$$
\tilde{Q}\_{k\dot{\lambda}} = \sqrt{\varepsilon\_0 V\_k} \left( \tilde{a}\_{0k\dot{\lambda}} + \tilde{a}\_{0k\dot{\lambda}}^\* \right); \quad \tilde{P}\_{k\dot{\lambda}} = -i a \nu\_k \sqrt{\varepsilon\_0 V\_k} \left( \tilde{a}\_{0k\dot{\lambda}} - \tilde{a}\_{0k\dot{\lambda}}^\* \right) \tag{60}
$$

Thus, introducing Eq. (59) in Eq. (60) and using Eq. (20) with Eq. (53), Heisenberg's commutation relation, a fundamental concept in quantum theory, results directly [2]:

$$\left[\tilde{Q}\_{k\dot{\lambda}}, \tilde{P}\_{k'\dot{\lambda}'}\right] = -i\varepsilon\_0 a\_{k'}^2 a\_{k} \sqrt{V\_k V\_{k'}} \left[\left(\xi a\_{k\dot{\lambda}} + \xi a\_{k\dot{\lambda}}^+\right), \left(\xi a\_{k'\dot{\lambda}'} - \xi a\_{k'\dot{\lambda}'}^+\right)\right] = i\hbar \delta\_{kk'} \delta\_{\lambda k'} \tag{61}$$

The fundamental properties of the photon, energy *Ek*, momentum *pk* ! , and wave vector *k* ! , are complemented by the vector potential amplitude *α*0*<sup>k</sup>* expressing its electromagnetic nature:

$$E\_k/\hbar = \left| \overrightarrow{p\_k} \right| c/\hbar = \left| \overrightarrow{k} \right| c = a\_{0k}/|\xi| = o\_k \tag{62}$$

Considering Heisenberg's *energy-time* uncertainty principle:

$$
\delta E\_k \,\delta t \ge \hbar \,\tag{63}
$$

we directly deduce from Eq. (62) the *vector potential-time* uncertainty:

$$
\delta \alpha\_{0k} \delta t \geq |\xi| \tag{64}$$

The energy and vector potential uncertainties with respect to time are intrinsic physical properties of the wave-particle nature of the photon.

## **4.3 Photon wave-particle equation and wave function**

Obviously, the photon vector potential function *αk<sup>λ</sup>* ! *r* !*; t* expressed in Eq. (49) satisfies the wave propagation equation in vacuum issued from Maxwell's equations:

$$\vec{\nabla}^2 \vec{a\_{k\bar{\imath}}} \left( \vec{r}, t \right) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \vec{a\_{k\bar{\imath}}} \left( \vec{r}, t \right) = \mathbf{0} \tag{65}$$

Obviously, the shorter the wavelength of the photon, the higher the localization probability in agreement with Heisenberg's uncertainty and the

**4.4 Electromagnetic field ground state, photons, and electrons-positrons**

<sup>þ</sup> ^*<sup>ε</sup>* <sup>∗</sup> *kλe* �*i k*! �*r* !�*ωkt*þ*ϕ*

þ *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>*^*ε* <sup>∗</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>*

� � � �

wave considered as a superposition of plane wave modes writes:

*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>* � �

*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>* � �

*<sup>i</sup><sup>ϕ</sup>* <sup>þ</sup> ^*<sup>ε</sup>* <sup>∗</sup> *kλe* �*i<sup>ϕ</sup>* � � *;* <sup>Ξ</sup><sup>~</sup> <sup>0</sup>

� � � �

The photon vector potential is composed of a fundamental function Ξ*k<sup>λ</sup>* times the angular frequency *ω<sup>k</sup>* and writes in the classical (wave) and quantum (particle)

In this way, the general equation for the vector potential of the electromagnetic

<sup>þ</sup> ^*<sup>ε</sup>* <sup>∗</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>*

and that of a large number of cavity-free photons in quantum electrodynamics is:

� � � �

þ *a*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>*^*ε* <sup>∗</sup> *kλe* �*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>*

According to Eqs. (55) and (62), for *ω<sup>k</sup>* ! 0 all the physical properties of the photon vanish entailing that the photon exists only for a non-zero frequency of the vector potential oscillation. However, the zero-frequency level does not

correspond to perfect inexistence because the fundamental field Ξ*k<sup>λ</sup>* does not vanish

*<sup>k</sup><sup>λ</sup>* ¼ j j *ξ akλ*^*εk<sup>λ</sup>e*

*<sup>k</sup><sup>λ</sup>* is the *electromagnetic field ground state* (EFGS) permeating all the

of the polarisation vectors ^*εk<sup>λ</sup>* [63, 64] and writes in the classical and quantum

space (*λ<sup>k</sup>* ! ∞) and having zero energy and zero vector potential as well as zero electric and magnetic fields. This physical state lies beyond the Bohm-Aharonov situation in which the energy and the electric and magnetic fields are zero but a vector potential is present in space [43]. Thus, in complete absence of energy and

constituting the main "skeleton" of any photon which now clearly appears to be a

� � � �

¼ *ωk*Ξ !

*<sup>k</sup><sup>λ</sup> ωk; r* !*; t* � � (69)

<sup>¼</sup> *<sup>ω</sup>k*Ξ~*k<sup>λ</sup> akλ; <sup>a</sup>*<sup>þ</sup>

<sup>¼</sup> <sup>X</sup> *k, λ*

<sup>¼</sup> <sup>X</sup> *k, λ*

*<sup>k</sup><sup>λ</sup>* involving the amplitude *ξ* and the general expression

*<sup>k</sup><sup>λ</sup>* can be assimilated to a quantum vacuum component

*<sup>i</sup><sup>ϕ</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>þ</sup> *<sup>k</sup><sup>λ</sup>*^*ε* <sup>∗</sup> *kλe* �*i<sup>ϕ</sup>* � � (73)

*ωk*Ξ !

*<sup>k</sup><sup>λ</sup> ωk; r* !*; t* � �

*ωk*Ξ~*k<sup>λ</sup> akλ; a*<sup>þ</sup>

(71)

*kλ* � �

(72)

*kλ* � � (70)

experimental evidence.

*Quantized Field of Single Photons*

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

formalisms:

*αkλ*

*A* ! *r* !*; t* � � <sup>¼</sup> <sup>X</sup>

> *<sup>A</sup>*<sup>~</sup> <sup>¼</sup> <sup>X</sup> *k, λ*

representations:

Ξ !<sup>0</sup>

The field Ξ<sup>0</sup>

**31**

! ¼ j j *ξ ω<sup>k</sup>* ^*εk<sup>λ</sup>e*

*α*~*k<sup>λ</sup>* ¼ j j *ξ ω<sup>k</sup> akλ*^*εkλe*

*k, λ*

for *<sup>ω</sup><sup>k</sup>* <sup>¼</sup> 0 but reduces to <sup>Ξ</sup><sup>0</sup>

*<sup>k</sup><sup>λ</sup>* ¼ j j *ξ* ^*εk<sup>λ</sup>e*

vector potential, the field Ξ<sup>0</sup>

vacuum oscillation [2, 63, 64].

j j *ξ ω<sup>k</sup> akλ*^*εkλe*

*i k*! �*r* !�*ωkt*þ*ϕ* � �

*i k*! �*r* !�*<sup>ω</sup>kt*þ*<sup>ϕ</sup>* � �

j j *ξ ω<sup>k</sup>* ^*εkλe*

as well as the *vector potential energy* (*wave-particle*) equation for the photon [2, 54]:

$$i\begin{pmatrix} \xi \\ \hbar \end{pmatrix} \frac{\partial}{\partial t} \overrightarrow{a\_{k\iota}} \left( \overrightarrow{r}, t \right) = \begin{pmatrix} \tilde{a}\_{0k} \\ \tilde{H} \end{pmatrix} \overrightarrow{a\_{k\iota}} \left( \overrightarrow{r}, t \right) \tag{66}$$

where the vector potential operator *α*~0*<sup>k</sup>* ¼ �*iξc* ∇ ! and the relativistic

Hamiltonian for a massless particle *<sup>H</sup>*<sup>~</sup> ¼ �*i*ℏ*<sup>c</sup>* <sup>∇</sup> ! have the eigenvalues *ξω<sup>k</sup>* and ℏ*ωk*, respectively [2, 53].

It is worth remarking the symmetry between the pairs f g *Ek;* ℏ and f g *α*0*<sup>k</sup>; ξ* for a single photon characterising, respectively, the particle (energy) and electromagnetic wave (vector potential) natures, having in mind that the energy corresponds to the integration of the single-mode electromagnetic field energy density over the volume *Vk*.

Now, when considering the propagation of a *k*-mode photon with wavelength *λk*, the difficulties for defining a position operator are widely commented in the literature [27, 29, 30, 39–41, 56]. At the same time, the efforts for defining a wave function for the photon based on the electric and magnetic fields were rather fruitless [58–62]. It has been emphasized several times [19, 26, 27, 30, 56] that a photon cannot be localised along the propagation axis in a length shorter than the wavelength *λ<sup>k</sup>* and within a volume smaller than roughly *λ*<sup>3</sup> *k*.

In fact, from a theoretical point of view, for a photon propagating in the *z* direction, Heisenberg's uncertainty for the position *z* and momentum *Pz* ¼ ℏ*kz* ¼ *h=λ<sup>k</sup>* writes:

$$\delta \mathbf{z} \; \delta P\_z \ge h \quad \rightarrow \quad \delta \mathbf{z} \; \delta (\mathbf{1}/\lambda\_k) \ge \mathbf{1} \tag{67}$$

Notice that the momentum uncertainty along the propagation axis is expressed through the uncertainty over the inverse of the wavelength.

Considering now the vector potential function with the quantized amplitude *α*0*<sup>k</sup>* ¼ j j *ξ ω<sup>k</sup>* as a real wave function for the photon, then when a *k*-mode photon is emitted at a coordinate *z*<sup>e</sup> at an instant *t*<sup>e</sup> and propagates in vacuum along the *z* axis, the probability Π*k*ð Þ*z* to be localised at time *t* and at the coordinate *z* = *z*<sup>e</sup> + *c* (*t-t*e) corresponds to the square of the modulus of the vector potential and is consequently proportional to the square of the angular frequency [54, 55]:

$$\left|\Pi\_{\mathbf{k}}(\mathbf{z})\right|\mathfrak{s}\left|\overrightarrow{a\_{\mathbf{k}}}\right|^{2} = \xi^{2}a\_{\mathbf{k}}^{2}\mathfrak{s}\left\lambda\_{\mathbf{k}}^{-2}\tag{68}$$

*Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

The energy and vector potential uncertainties with respect to time are intrinsic

satisfies the wave propagation equation in vacuum issued from Maxwell's equations:

<sup>¼</sup> *<sup>α</sup>*~0*<sup>k</sup> H*~ 

!

It is worth remarking the symmetry between the pairs f g *Ek;* ℏ and f g *α*0*<sup>k</sup>; ξ* for a single photon characterising, respectively, the particle (energy) and electromagnetic wave (vector potential) natures, having in mind that the energy corresponds to the integration of the single-mode electromagnetic field energy density over the

Now, when considering the propagation of a *k*-mode photon with wavelength *λk*, the difficulties for defining a position operator are widely commented in the literature [27, 29, 30, 39–41, 56]. At the same time, the efforts for defining a wave function for the photon based on the electric and magnetic fields were rather fruitless [58–62]. It has been emphasized several times [19, 26, 27, 30, 56] that a photon cannot be localised along the propagation axis in a length shorter than the

In fact, from a theoretical point of view, for a photon propagating in the *z*

Notice that the momentum uncertainty along the propagation axis is expressed

Considering now the vector potential function with the quantized amplitude *α*0*<sup>k</sup>* ¼ j j *ξ ω<sup>k</sup>* as a real wave function for the photon, then when a *k*-mode photon is emitted at a coordinate *z*<sup>e</sup> at an instant *t*<sup>e</sup> and propagates in vacuum along the *z* axis, the probability Π*k*ð Þ*z* to be localised at time *t* and at the coordinate *z* = *z*<sup>e</sup> + *c* (*t-t*e) corresponds to the square of the modulus of the vector potential and is consequently

! ð Þj *z; t*

2 <sup>¼</sup> *<sup>ξ</sup>*<sup>2</sup> *ω*2 *<sup>k</sup>* ∝ *λ*�<sup>2</sup> *k*

direction, Heisenberg's uncertainty for the position *z* and momentum

� 1 *c*2 *∂*2 *<sup>∂</sup>t*<sup>2</sup> *<sup>α</sup>k<sup>λ</sup>* ! *r* !*; t* 

as well as the *vector potential energy* (*wave-particle*) equation for the

! *r* !*; t* 

*αkλ* ! *r* !*; t* 

!

expressed in Eq. (49)

(66)

¼ 0 (65)

and the relativistic

*k*.

*δz δPz* ≥*h* ! *δz δ*ð Þ 1*=λ<sup>k</sup>* ≥ 1 (67)

(68)

have the eigenvalues *ξω<sup>k</sup>* and ℏ*ωk*,

physical properties of the wave-particle nature of the photon.

**4.3 Photon wave-particle equation and wave function**

Obviously, the photon vector potential function *αk<sup>λ</sup>*

∇ !2 *αkλ* ! *r* !*; t* 

*<sup>i</sup> <sup>ξ</sup>* ℏ *∂ ∂t αkλ* ! *r* !*; t* 

Hamiltonian for a massless particle *<sup>H</sup>*<sup>~</sup> ¼ �*i*ℏ*<sup>c</sup>* <sup>∇</sup>

where the vector potential operator *α*~0*<sup>k</sup>* ¼ �*iξc* ∇

wavelength *λ<sup>k</sup>* and within a volume smaller than roughly *λ*<sup>3</sup>

through the uncertainty over the inverse of the wavelength.

proportional to the square of the angular frequency [54, 55]:

Π*k*ð Þ*z* ∝ *αk<sup>λ</sup>*

 

photon [2, 54]:

*Single Photon Manipulation*

respectively [2, 53].

*Pz* ¼ ℏ*kz* ¼ *h=λ<sup>k</sup>* writes:

volume *Vk*.

**30**

Obviously, the shorter the wavelength of the photon, the higher the localization probability in agreement with Heisenberg's uncertainty and the experimental evidence.

### **4.4 Electromagnetic field ground state, photons, and electrons-positrons**

The photon vector potential is composed of a fundamental function Ξ*k<sup>λ</sup>* times the angular frequency *ω<sup>k</sup>* and writes in the classical (wave) and quantum (particle) formalisms:

$$\overrightarrow{a}\_{k\boldsymbol{\ell}} = |\xi| \, o\boldsymbol{\mu}\_{k} \left[ \hat{e}\_{k\boldsymbol{\ell}} \boldsymbol{e}^{i\left(\overrightarrow{k}\cdot\overrightarrow{r} - \alpha\_{k}t + \phi\right)} + \hat{e}\_{k\boldsymbol{\ell}} \boldsymbol{e}^{-i\left(\overrightarrow{k}\cdot\overrightarrow{r} - \alpha\_{k}t + \phi\right)} \right] = o\boldsymbol{\mu}\_{k} \overrightarrow{\Xi}\_{k\boldsymbol{\ell}} \left( o\boldsymbol{\eta}\_{k}, \overrightarrow{r}, t\right) \tag{69}$$

$$\tilde{a}\_{k\boldsymbol{\lambda}} = |\xi| a\_k \left[ a\_{k\boldsymbol{\lambda}} \hat{e}\_{k\boldsymbol{\lambda}} e^{i \left( \vec{k} \cdot \vec{r} - a\boldsymbol{u}t + \phi \right)} + a\_{k\boldsymbol{\lambda}}^{+} \hat{e}\_{k\boldsymbol{\lambda}} e^{-i \left( \vec{k} \cdot \vec{r} - a\boldsymbol{u}t + \phi \right)} \right] = a\nu\_{k\boldsymbol{\lambda}} \tilde{\boldsymbol{\Delta}}\_{k\boldsymbol{\lambda}} \left( a\_{k\boldsymbol{\lambda}}, a\_{k\boldsymbol{\lambda}}^{+} \right) \tag{70}$$

In this way, the general equation for the vector potential of the electromagnetic wave considered as a superposition of plane wave modes writes:

$$\overrightarrow{A}\left(\overrightarrow{r},t\right) = \sum\_{k\_2\lambda} |\xi| \alpha y\_k \left[ \hat{e}\_{k\lambda} e^{i\left(\overrightarrow{k}\cdot\overrightarrow{r} - \alpha\_k t + \phi\right)} + \hat{e}\_{k\lambda}^\* e^{-i\left(\overrightarrow{k}\cdot\overrightarrow{r} - \alpha\_k t + \phi\right)} \right] = \sum\_{k\_2\lambda} \alpha y\_k \overrightarrow{\Xi}\_{k\lambda} \left(\alpha y\_k, \overrightarrow{r}, t\right) \tag{71}$$

and that of a large number of cavity-free photons in quantum electrodynamics is:

$$\tilde{A} = \sum\_{k\_{\mathbf{k}}\lambda} |\xi| \, o\_{\mathbf{k}} \left[ a\_{k\vec{\iota}} \hat{e}\_{k\vec{\ell}} \epsilon^{i\left(\vec{k}\cdot\vec{r} - o\_{\vec{k}}t + \phi\right)} + a\_{k\vec{\iota}} \hat{e}\_{k\vec{\ell}} \epsilon^{-i\left(\vec{k}\cdot\vec{r} - o\_{\vec{k}}t + \phi\right)} \right] = \sum\_{k\_{\mathbf{k}}\lambda} o\_{\vec{\kappa}} \tilde{\Xi}\_{k\vec{\ell}} \left( a\_{k\vec{\iota}}, a\_{k\vec{\iota}}^{+} \right) \tag{72}$$

According to Eqs. (55) and (62), for *ω<sup>k</sup>* ! 0 all the physical properties of the photon vanish entailing that the photon exists only for a non-zero frequency of the vector potential oscillation. However, the zero-frequency level does not correspond to perfect inexistence because the fundamental field Ξ*k<sup>λ</sup>* does not vanish for *<sup>ω</sup><sup>k</sup>* <sup>¼</sup> 0 but reduces to <sup>Ξ</sup><sup>0</sup> *<sup>k</sup><sup>λ</sup>* involving the amplitude *ξ* and the general expression of the polarisation vectors ^*εk<sup>λ</sup>* [63, 64] and writes in the classical and quantum representations:

$$
\stackrel{\rightarrow}{\Xi}\_{k\dot{\lambda}}^{0} = |\xi| \left[ \hat{e}\_{k\dot{\lambda}} e^{i\phi} + \hat{e}\_{k\dot{\lambda}}^{\*} e^{-i\phi} \right] \quad ; \quad \bar{\Xi}\_{k\dot{\lambda}}^{0} = |\xi| \left[ a\_{k\dot{\lambda}} \hat{e}\_{k\dot{\lambda}} e^{i\phi} + a\_{k\dot{\lambda}}^{+} \hat{e}\_{k\dot{\lambda}}^{\*} e^{-i\phi} \right] \tag{73}
$$

The field Ξ<sup>0</sup> *<sup>k</sup><sup>λ</sup>* is the *electromagnetic field ground state* (EFGS) permeating all the space (*λ<sup>k</sup>* ! ∞) and having zero energy and zero vector potential as well as zero electric and magnetic fields. This physical state lies beyond the Bohm-Aharonov situation in which the energy and the electric and magnetic fields are zero but a vector potential is present in space [43]. Thus, in complete absence of energy and vector potential, the field Ξ<sup>0</sup> *<sup>k</sup><sup>λ</sup>* can be assimilated to a quantum vacuum component constituting the main "skeleton" of any photon which now clearly appears to be a vacuum oscillation [2, 63, 64].

Combination of the expression j j *ξ* ¼ ℏ*=*4*π*j j*e c* to the fine structure constant definition *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*2*=*4*πε*0ℏ*<sup>c</sup>* permits to draw directly the electron-positron elementary charge *<sup>e</sup>* ¼ �1*:*602 10�19*C*, a fundamental physical constant, which now is expressed exactly through the EFGS amplitude *ξ* [64, 65]:

$$e = \pm (4\pi)^2 a \frac{|\xi|}{\mu\_0} \tag{74}$$

a given point on the propagation axis is obtained by the square modulus of the vector potential and is proportional to the square of the angular frequency *ξ*<sup>2</sup>

quantum vacuum component, issues naturally from the vector potential wave function putting in evidence that photons are oscillations of the vacuum field. Furthermore, the electron-positron charge and mass are directly proportional to the vector potential amplitude quantization constant showing the strong physical relationship with the photons. Obviously, the origin of the mechanisms governing the transformations of photons to electrons-positrons and inversely lies in the nature of

1 National Institute for Nuclear Science and Technology, CEA - Saclay, France

© 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,

2 Department of Physics, University of Versailles, UVSQ, France

\*Address all correspondence to: constantin.meis@cea.fr

provided the original work is properly cited.

in agreement with the experiments.

*DOI: http://dx.doi.org/10.5772/intechopen.88378*

*Quantized Field of Single Photons*

the electromagnetic field ground state.

**Author details**

Constantin Meis1,2

**33**

(Eq. (68)) which signifies that the higher the frequency, the better the localization,

Finally, the electromagnetic field ground state (EFGS) at zero frequency, a real

*ω*2 *k*

Using again Eq. (51) and recalling that the electron mass may be written as *me* ¼ *e*ℏ*=*2*μB*, where *μ<sup>B</sup>* is the Bohr magneton, we deduce that the electron mass is also expressed as a function of the EFGS amplitude *ξ* [64]:

$$m\_{\epsilon} = 2\pi c e^2 \frac{|\xi|}{\mu\_B} \tag{75}$$

entailing that the mass derives also from the EFGS and is proportional to the charge square.

Equations (50), (74), and (75) show the strong physical relationship between photons and electrons-positrons which are all related directly to the EFGS through the amplitude *ξ*. Obviously, photons and electrons-positrons, also probably leptonsantileptons, are issued from the same quantum vacuum field. This may be at the origin of the physical mechanism governing the photon generation during the electron-positron (and probably lepton-antilepton) annihilation and that of the electron-positron (lepton-antilepton) pair creation during the annihilation of highenergy gamma photons in the vicinity of very heavy nucleus.

### **5. Conclusion**

In this chapter we have presented recent theoretical developments complementing the standard formalism with the purpose of describing a single photon state in conformity with the experiments. We resume below the principal features.

The quantization of the vector potential amplitude *α*0*<sup>k</sup>*, a real physical entity, for a single free of cavity *k*-mode photon with angular frequency *ω<sup>k</sup>* is expressed by *α*0*<sup>k</sup>* ¼ j j *ξ ωk*, where j j *ξ* ¼ ℏ*=*4*π*j j*e c*, and leads to the establishment of a *vector potential - energy* (electromagnetic wave-particle) formalism (Eq. (66)) expressing the simultaneous wave-particle nature of the photon. A single photon state is a local indivisible entity of the electromagnetic field extending over a wavelength *λ<sup>k</sup>* and consisting of the quantized vector potential oscillating at the angular frequency *ωk*, with circular polarisation, giving birth to orthogonally oscillating electric and magnetic fields whose amplitudes are proportional to the square of the angular frequency j j *<sup>ξ</sup> <sup>ω</sup>*<sup>2</sup> *<sup>k</sup>* (Eq. (55)). Its lateral expansion, confirmed experimentally, yields a minimum photon volume *Vk* which is proportional to *λ*<sup>3</sup> *<sup>k</sup>*. The quantum properties of the photon, energy, momentum, and spin are obtained directly from the classical electromagnetic expressions integrated over the volume *Vk* (Eqs. (56)–(58)). It is also shown (Eq. (61)) that the Heisenberg uncertainty can be readily obtained through the use of the volume *Vk*.

A single photon, as a local three-dimensional entity of the electromagnetic field, is absorbed and emitted as a whole and propagates guided by the non-local vector potential function (Eq. (49)), which appears to be a natural wave function for the photon satisfying both the propagation equation (Eq. (65)) and the *vector potential - energy* equation (Eq. (66)). The probability for detecting a photon around

## *Quantized Field of Single Photons DOI: http://dx.doi.org/10.5772/intechopen.88378*

Combination of the expression j j *ξ* ¼ ℏ*=*4*π*j j*e c* to the fine structure constant definition *<sup>α</sup>* <sup>¼</sup> *<sup>e</sup>*2*=*4*πε*0ℏ*<sup>c</sup>* permits to draw directly the electron-positron elementary charge *<sup>e</sup>* ¼ �1*:*602 10�19*C*, a fundamental physical constant, which now is

*<sup>e</sup>* ¼ �ð Þ <sup>4</sup>*<sup>π</sup>* <sup>2</sup>

Using again Eq. (51) and recalling that the electron mass may be written as *me* ¼ *e*ℏ*=*2*μB*, where *μ<sup>B</sup>* is the Bohr magneton, we deduce that the electron mass is

*me* <sup>¼</sup> <sup>2</sup>*πc e*<sup>2</sup> j j *<sup>ξ</sup>*

entailing that the mass derives also from the EFGS and is proportional to the

Equations (50), (74), and (75) show the strong physical relationship between photons and electrons-positrons which are all related directly to the EFGS through the amplitude *ξ*. Obviously, photons and electrons-positrons, also probably leptonsantileptons, are issued from the same quantum vacuum field. This may be at the origin of the physical mechanism governing the photon generation during the electron-positron (and probably lepton-antilepton) annihilation and that of the electron-positron (lepton-antilepton) pair creation during the annihilation of high-

In this chapter we have presented recent theoretical developments complementing the standard formalism with the purpose of describing a single photon state in con-

The quantization of the vector potential amplitude *α*0*<sup>k</sup>*, a real physical entity, for a single free of cavity *k*-mode photon with angular frequency *ω<sup>k</sup>* is expressed by *α*0*<sup>k</sup>* ¼ j j *ξ ωk*, where j j *ξ* ¼ ℏ*=*4*π*j j*e c*, and leads to the establishment of a *vector potential - energy* (electromagnetic wave-particle) formalism (Eq. (66)) expressing the simultaneous wave-particle nature of the photon. A single photon state is a local indivisible entity of the electromagnetic field extending over a wavelength *λ<sup>k</sup>* and consisting of the quantized vector potential oscillating at the angular frequency *ωk*, with circular polarisation, giving birth to orthogonally oscillating electric and magnetic fields whose amplitudes are proportional to the square of the angular fre-

*<sup>k</sup>* (Eq. (55)). Its lateral expansion, confirmed experimentally, yields a

the photon, energy, momentum, and spin are obtained directly from the classical electromagnetic expressions integrated over the volume *Vk* (Eqs. (56)–(58)). It is also shown (Eq. (61)) that the Heisenberg uncertainty can be readily obtained

is absorbed and emitted as a whole and propagates guided by the non-local vector potential function (Eq. (49)), which appears to be a natural wave function for the photon satisfying both the propagation equation (Eq. (65)) and the *vector potential - energy* equation (Eq. (66)). The probability for detecting a photon around

A single photon, as a local three-dimensional entity of the electromagnetic field,

*<sup>k</sup>*. The quantum properties of

*<sup>α</sup>* j j *<sup>ξ</sup> μ*0

*μB*

(74)

(75)

expressed exactly through the EFGS amplitude *ξ* [64, 65]:

also expressed as a function of the EFGS amplitude *ξ* [64]:

energy gamma photons in the vicinity of very heavy nucleus.

formity with the experiments. We resume below the principal features.

minimum photon volume *Vk* which is proportional to *λ*<sup>3</sup>

through the use of the volume *Vk*.

charge square.

*Single Photon Manipulation*

**5. Conclusion**

quency j j *<sup>ξ</sup> <sup>ω</sup>*<sup>2</sup>

**32**

a given point on the propagation axis is obtained by the square modulus of the vector potential and is proportional to the square of the angular frequency *ξ*<sup>2</sup> *ω*2 *k* (Eq. (68)) which signifies that the higher the frequency, the better the localization, in agreement with the experiments.

Finally, the electromagnetic field ground state (EFGS) at zero frequency, a real quantum vacuum component, issues naturally from the vector potential wave function putting in evidence that photons are oscillations of the vacuum field. Furthermore, the electron-positron charge and mass are directly proportional to the vector potential amplitude quantization constant showing the strong physical relationship with the photons. Obviously, the origin of the mechanisms governing the transformations of photons to electrons-positrons and inversely lies in the nature of the electromagnetic field ground state.

## **Author details**

## Constantin Meis1,2

1 National Institute for Nuclear Science and Technology, CEA - Saclay, France

2 Department of Physics, University of Versailles, UVSQ, France

\*Address all correspondence to: constantin.meis@cea.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.

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**37**

Section 2

Generation
