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THEORY OF COHERENCE FOR THE QUANTIZED ELECTROMAGNETIC FIELD,

Solimene,Nicholas

Published Feb 1, 1965• 0 citations
OpenAlex

Abstract

A preliminary account of the quantum theory of coherence for electromagnetic radiation is presented using the method of Schwinger's variational action principle. The generating functional for the quantum expectation values of arbitrary products of field operators for the free electromagnetic field subject to any homogeneous boundary conditions is explicitly given in terms of the dyadic Green's function for the corresponding classical problem. Although thermal expectation values are generated directly by the process of functional differentiation in this formulation, it is possible to generate expectation values for any free field state by employing projection operations. In particular Glauber's coherent state expectation values may be obtained. Results for the radiation field coupled only to c-number sources may trivially be extracted from the free field formulation. (Author)

Related papers

EFFECTS OF COHERENCE IN ELECTROMAGNETIC RADIATION.

E. Wolf

Abstract : The report summarizes researches concerned with theoretical studies of coherence effects in electromagnetic fields, with special reference to the electromangetic range of the spectrum. Various properties of fluctuating fields were investigated and second and higher order coherence effects were analyzed. Relation between the classical and quantum theory of coherence was also considered. A degree of coherence of arbitrary order was introduced which has some advantages over that introduced earlier by other authors. Further, diagonal coherent state representation of quantum operators was studied. Some related investigations in statistical mechanics, and in diffraction theory of the electromagnetic field were also carried out. (Author)

•Sep 1, 1967•0 citations

Coherent and Incoherent States of the Radiation Field

Roy J. Glauber

Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.

Physical Review•Sep 15, 1963•6,794 citations

Functional Formulation of the Coherence Theory

Andrzej Zardecki

A functional formalism for coherence theory is developed in terms of the (quasi) probability functional of the radiation field. The formalism is closely connected to the P representation of the density operator. For the fields confined within a spatial volume of finite size, this connection is expressed as a multidimensional Fourier integral. A certain type of correspondence, illustrated by explicit examples, between the classical and the quantum mechanical description of the field statistics is established. The recent results of Titulaer and Glauber concerning the (1, 1)th-order coherent fields are analyzed from the point of view of the functional formulation.

Journal of Mathematical Physics•Jan 1, 1970•3 citations

Quantum distribution theory and partially coherent radiation

Marlan O. Scully, Muhammad Suhail Zubairy

As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.

Cambridge University Press eBooks•Sep 4, 1997•2 citations

The Electromagnetic Field

Kurt Gottfried

In this chapter, the authors prove that the free electromagnetic field is equivalent to an infinite set of uncoupled one-dimensional harmonic oscillators, one oscillator being associated with each value of the propagation vector k and polarization mode α. In order to quantize the electromagnetic field the authors shall have to express the field equations in Hamiltonian form and impose the canonical commutation rules on the canonically conjugate variables. Because the electromagnetic field possesses an infinite number of degrees of freedom it is also possible to formulate the theory in terms of a very different set of dynamical variables, the field operators themselves. To be more precise, the field operators only have off-diagonal matrix elements in the representation in which the photon occupation numbers are diagonal. In a state with a definite number of photons the expectation value of any field strength vanishes, and it is only meaningful to discuss fluctuations in the field strengths.

CRC Press eBooks•Mar 9, 2018•12 citations

Characteristic States of the Electromagnetic Radiation Field

Marvin M. Miller, Eli A. Mishkin

It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides $|n〉$, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators $b$ and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables $x$, $p$, and $H$, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states $|n〉$. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator $a$ is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.

Physical Review•Dec 23, 1966•24 citations
A preliminary account of the quantum theory of coherence for electromagnetic radiation is presented using the method of Schwinger's variational action principle. The generating functional for the quantum expectation values of arbitrary products of field operators for the free electromagnetic field subject to any homogeneous boundary conditions is explicitly given in terms of the dyadic Green's function for the corresponding classical problem. Although thermal expectation values are generated directly by the process of functional differentiation in this formulation, it is possible to generate expectation values for any free field state by employing projection operations. In particular Glauber's coherent state expectation values may be obtained. Results for the radiation field coupled only to c-number sources may trivially be extracted from the free field formulation. (Author)
Abstract : The report summarizes researches concerned with theoretical studies of coherence effects in electromagnetic fields, with special reference to the electromangetic range of the spectrum. Various properties of fluctuating fields were investigated and second and higher order coherence effects were analyzed. Relation between the classical and quantum theory of coherence was also considered. A degree of coherence of arbitrary order was introduced which has some advantages over that introduced earlier by other authors. Further, diagonal coherent state representation of quantum operators was studied. Some related investigations in statistical mechanics, and in diffraction theory of the electromagnetic field were also carried out. (Author)
Methods are developed for discussing the photon statistics of arbitrary fields in fully quantum-mechanical terms. In order to keep the classical limit of quantum electrodynamics plainly in view, extensive use is made of the coherent states of the field. These states, which reduce the field correlation functions to factorized forms, are shown to offer a convenient basis for the description of fields of all types. Although they are not orthogonal to one another, the coherent states form a complete set. It is shown that any quantum state of the field may be expanded in terms of them in a unique way. Expansions are also developed for arbitrary operators in terms of products of the coherent state vectors. These expansions are discussed as a general method of representing the density operator for the field. A particular form is exhibited for the density operator which makes it possible to carry out many quantum-mechanical calculations by methods resembling those of classical theory. This representation permits clear insights into the essential distinction between the quantum and classical descriptions of the field. It leads, in addition, to a simple formulation of a superposition law for photon fields. Detailed discussions are given of the incoherent fields which are generated by superposing the outputs of many stationary sources. These fields are all shown to have intimately related properties, some of which have been known for the particular case of blackbody radiation.
A functional formalism for coherence theory is developed in terms of the (quasi) probability functional of the radiation field. The formalism is closely connected to the P representation of the density operator. For the fields confined within a spatial volume of finite size, this connection is expressed as a multidimensional Fourier integral. A certain type of correspondence, illustrated by explicit examples, between the classical and the quantum mechanical description of the field statistics is established. The recent results of Titulaer and Glauber concerning the (1, 1)th-order coherent fields are analyzed from the point of view of the functional formulation.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
In this chapter, the authors prove that the free electromagnetic field is equivalent to an infinite set of uncoupled one-dimensional harmonic oscillators, one oscillator being associated with each value of the propagation vector k and polarization mode α. In order to quantize the electromagnetic field the authors shall have to express the field equations in Hamiltonian form and impose the canonical commutation rules on the canonically conjugate variables. Because the electromagnetic field possesses an infinite number of degrees of freedom it is also possible to formulate the theory in terms of a very different set of dynamical variables, the field operators themselves. To be more precise, the field operators only have off-diagonal matrix elements in the representation in which the photon occupation numbers are diagonal. In a state with a definite number of photons the expectation value of any field strength vanishes, and it is only meaningful to discuss fluctuations in the field strengths.
It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides ∣n〉|n〉∣n〉, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators bbb and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables xxx, ppp, and HHH, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states ∣n〉|n〉∣n〉. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator aaa is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.