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Physical Review

Density Operators and Quasiprobability Distributions

Kevin E. Cahill, Roy J. Glauber

Published Jan 25, 1969• 1,256 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.177.1882

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The problem of expanding a density operator \ensuremathρ\ensuremath{\rho}\ensuremathρ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) of the PPP representation, the Wigner distribution W(\ensuremathα)W(\ensuremath{\alpha})W(\ensuremathα), and the function 〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉, where ∣\ensuremathα〉|\ensuremath{\alpha}〉∣\ensuremathα〉 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(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) 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(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) 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†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\else\textdagger\fi. The Wigner distribution W(\ensuremathα)W(\ensuremath{\alpha})W(\ensuremathα) 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 〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉, 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(\ensuremathα,s)W(\ensuremath{\alpha},s)W(\ensuremathα,s) may be identified with the weight function P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα) of the PPP representation, with the Wigner distribution W(\ensuremathα)W(\ensuremath{\alpha})W(\ensuremathα), and with the function 〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉 when the order parameter sss assumes the values s=+1,0,\ensuremath−1s=+1, 0, \ensuremath{-}1s=+1,0,\ensuremath−1, respectively. The function W(\ensuremathα,s)W(\ensuremath{\alpha},s)W(\ensuremathα,s) is shown to be the expectation value of the ordered operator analog of the \ensuremathδ\ensuremath{\delta}\ensuremathδ 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(\ensuremathα,s)W(\ensuremath{\alpha},s)W(\ensuremathα,s) are exhibited as the order parameter sss is varied continuously from s=\ensuremath−1s=\ensuremath{-}1s=\ensuremath−1, corresponding to the function 〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉〈\ensuremathα∣\ensuremathρ∣\ensuremathα〉, to s=+1s=+1s=+1, corresponding to the function P(\ensuremathα)P(\ensuremath{\alpha})P(\ensuremathα). 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.
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 $P(\ensuremath{\alpha})$ 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 $P(\ensuremath{\alpha})$ 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.
The expansion of operators as ordered power series in the annihilation and creation operators aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required ccc-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter sss. In terms of this convention it is shown that for bounded operators the coefficients are finite when s>0s>0s>0, and the series are convergent when s>12s>\frac{1}{2}s>21​. For each value of the order parameter sss, a correspondence between operators and ccc-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator FFF is the one which results when the operators aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for FFF are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the PPP representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter sss.
Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the ``diagonal Hartree-Fock state projection'' representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the ``nondiagonal Hartree-Fock state projection'' representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.