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Physical Review A

Phase-space procedure for calculating multitime correlation functions of the electromagnetic field

Paul D. Lett, Rupamanjari Ghosh, Stephen R. Friberg, L. Mandeļ

Published Aug 1, 1984• 2 citations
Publisher

Abstract

It is pointed out that a method recently described by Hillery and Zubairy can be generalized to yield the multitime correlation function in any order for an interacting quantum field in the form of a phase-space average. The correlations are expressible as $c$-number integrals with the help of the coherent-state propagator, but the associated phase-space density is complex.

DOI
10.1103/physreva.30.890

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It is pointed out that a method recently described by Hillery and Zubairy can be generalized to yield the multitime correlation function in any order for an interacting quantum field in the form of a phase-space average. The correlations are expressible as ccc-number integrals with the help of the coherent-state propagator, but the associated phase-space density is complex.
Using the recently developed phase-space techniques for the treatment of quantum-mechanical problems, we set up a procedure for calculating multitime-correlation functions in terms of the joint distribution functions. The correlation functions are then expressed simply as integrals over the associated phase space. Explicit expressions are given for these joint distribution functions, in terms of Green's functions of the ccc-number equations of motion for the phase-space equivalent of the density operator. Using these joint distribution functions, an exact regression theorem is rederived, and the connection with the multitime correspondence between classical and quantum stochastic processes is discussed.
We consider the evaluation of multitime-correlation functions for quantum statistical systems characterized by dynamical variables of the angular-momentum type (such as, for example, paramagnetic or nuclear spins or collections of two-level atoms), and show how the atomic coherent-state representation of Arecchi et al. can be used to express the multitime averages in terms of phase-space integrals. Our calculation suggests a convenient classical-quantum correspondence rule for angular-momentum degrees of freedom.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.
In this paper we formulate a new stochastic description of quantum mechanics in phase space. The theory of phase-space representations of quantum mechanics, initiated by Wigner, Groenewold, and Moyal and systematized recently by Agarwal and Wolf is essentially a single-time theory, in that it deals only with the quantum-mechanical joint distribution functions for position and momentum at a single instant of time. We develop a natural multitime extension of such a single-time theory. We consider a class of multitime phase-space distribution functions such that an arbitrary quantum multitime correlation function can be expressed as a phase-space average of the form encountered in classical stochastic theories. We study the nonclassical features of these multitime distribution functions and show that they may be considered as characterizing a generalized stochastic process in phase space. We demonstrate that the multitime distribution functions that correspond to Hamiltonian evolution of isolated quantum systems satisfy a certain condition that may be regarded as characterizing a generalized Markov process. We also investigate certain special features of the generalized stochastic processes that characterize the evolution of open systems.
We describe a general approach for the analysis of "phase-space" or "ccc-number" descriptions of complex quantum systems. Our approach is similar to that recently developed by Agarwal and Wolf, but is based upon a formalism developed by Lax. We show that the ccc-number description of atomic operator equations employed by Gordon in his theory of the laser can be derived from an operator ordering formalism. A general method for deriving a Fokker-Planck equation in any ccc-number association scheme from a given Markovian operator equation is shown. The formalism is employed to find expressions for arbitrary multitime averages in terms of two-time conditional probabilities alone.
Derivation of two-time second-order correlation function by following approaches such as stochastic differential equation, coherent-state propagator, and quasi-statistical distribution function is presented. In the process, the time dependence of the operators is transferred to the density operator by making use of trace operation in which the coherent state propagator and QQQ-function that represent the quantum system under consideration are expressed in terms of different time parameters. Even though the number of resulting integrations are found to be large, the accompanying implementation turns out to be straightforward in view that the associated ccc-number functions are Gaussian by nature. In relation to the asserted possibility of rewriting the result of one of the approaches in terms of the other, the presented derivation is expected to lay a strong foundation for viable technique of calculating correlations of various moments at different times that can be deployed in revealing quantum correlations.