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Physical Review

Coherent-State Representations for the Photon Density Operator

Kevin E. Cahill

Published Jun 21, 1965• 86 citations
Publisher

Abstract

The "diagonal" $P$ representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the $P$ representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the $P$ representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the $P$ representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.

DOI
10.1103/physrev.138.b1566

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Density Operators and Quasiprobability Distributions

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The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the $P$ representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the $P$ representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter $s$ assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for $\mathrm{Res}<0$, has bounded eigenvalues for $\mathrm{Res}=0$, and has infinite eigenvalues for $s=1$. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter $s$ is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to $s=+1$, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the $P$ representation is appropriate.

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Regularization of the P Representation

Kevin E. Cahill

A representation is introduced for the density operator of the electromagnetic field that is suitable for all density operators and that reduces to the coherent-state $P$ representation when the latter exists. It expresses the density operator $\ensuremath{\rho}$ as the sum of four terms, each of which is a two-dimensional weighted integral over outer products of coherent states. The first integral has the form of the $P$ representation, i.e., the outer products are projection operators. The absence of singularities in this term is achieved by the presence of the three supplementary integrals which vanish when the density operator possesses the $P$ representation. In general, for stationary density operators, only the first two terms of the regularized $P$ representation are necessary. A simple prescription is given for obtaining the four weight functions of this representation from the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state and $\ensuremath{\rho}$ is the density operator. According to this prescription, the $P$ representation does not exist and one or more of the supplementary, regularizing terms is necessary when the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ contains a term that decreases more rapidly than $\mathrm{exp}(\ensuremath{-}{|\ensuremath{\alpha}|}^{2})$ as $|\ensuremath{\alpha}|\ensuremath{\rightarrow}\ensuremath{\infty}$. The regularized $P$ representation affords nonsingular integral expressions for all density operators and for most expectation values, including, when they are finite, those of the normally ordered products of the creation and annihilation operators, ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ and $a$. The construction and use of this representation is illustrated with the aid of simple examples in which the density operator does not possess the $P$ representation.

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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
The "diagonal" PPP representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) required to represent all density operators by means of the PPP representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the PPP representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the PPP representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) under which the expectation values of the normal ordered operators (a\ifmmode†\else\textdagger\fi)nam{({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}(a\ifmmode†\else\textdagger\fi)nam assume a particularly simple integral form are presented. It is shown that if P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.
The coherent-state PPP representation for the density operator of the electromagnetic field is studied for the case in which the density operator represents a pure state, $\ensuremath{\rho}=|\ensuremath{\psi}〉〈\ensuremath{\psi}|$. An exact and complete characterization is given of the states for which the PPP representation exists with a weight function $P(\ensuremath{\alpha})$ that is a tempered distribution. These states $|\ensuremath{\phi}〉$ form an exceedingly narrow class: each may be generated from a particular coherent state $|\ensuremath{\alpha}〉$ by the application of a finite number of creation operators, i.e., $|\ensuremath{\phi}〉=[{c}_{0}+{c}_{1}a\ifmmode\dagger\else\textdagger\fi{}+\ensuremath{\cdots}+{c}_{n}{(a\ifmmode\dagger\else\textdagger\fi{})}^{n}]|\ensuremath{\alpha}〉$, where $\ensuremath{\alpha}$ and the cn{c}_{n}cn​ are arbitrary complex numbers. For them the weight function $P(\ensuremath{\alpha})$ is a linear combination of the two-dimensional delta function and a finite number of its derivatives. For other pure states, the function $P(\ensuremath{\alpha})$ has singularities that are not compatible with the form of the PPP representation.
The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the PPP representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the PPP representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the PPP representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the PPP representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter sss assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for Res<0\mathrm{Res}<0Res<0, has bounded eigenvalues for Res=0\mathrm{Res}=0Res=0, and has infinite eigenvalues for s=1s=1s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter sss is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to s=+1s=+1s=+1, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the PPP representation is appropriate.
A representation is introduced for the density operator of the electromagnetic field that is suitable for all density operators and that reduces to the coherent-state PPP representation when the latter exists. It expresses the density operator $\ensuremath{\rho}$ as the sum of four terms, each of which is a two-dimensional weighted integral over outer products of coherent states. The first integral has the form of the PPP representation, i.e., the outer products are projection operators. The absence of singularities in this term is achieved by the presence of the three supplementary integrals which vanish when the density operator possesses the PPP representation. In general, for stationary density operators, only the first two terms of the regularized PPP representation are necessary. A simple prescription is given for obtaining the four weight functions of this representation from the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state and $\ensuremath{\rho}$ is the density operator. According to this prescription, the PPP representation does not exist and one or more of the supplementary, regularizing terms is necessary when the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ contains a term that decreases more rapidly than $\mathrm{exp}(\ensuremath{-}{|\ensuremath{\alpha}|}^{2})$ as $|\ensuremath{\alpha}|\ensuremath{\rightarrow}\ensuremath{\infty}$. The regularized PPP representation affords nonsingular integral expressions for all density operators and for most expectation values, including, when they are finite, those of the normally ordered products of the creation and annihilation operators, ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ and aaa. The construction and use of this representation is illustrated with the aid of simple examples in which the density operator does not possess the PPP representation.
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.