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Journal of Mathematical Physics

Structure and representation of correlation functions and the density matrix for a statistical wave field in optics

E. C. G. Sudarshan, Narasimhaiengar Mukunda

Published Aug 1, 1979• 4 citations
Publisher

Abstract

A systematic structure analysis of the correlation functions of statistical quantum optics is carried out. From a suitably defined auxiliary two-point function we are able to identify the excited modes in the wave field. The relative simplicity of the higher order correlation functions emerge as a byproduct and the conditions under which these are made pure are derived. These results depend in a crucial manner on the notion of coherence indices and of unimodular coherence indices. A new class of approximate expressions for the density operator of a statistical wave field is worked out based on discrete characteristic sets. These are even more economical than the diagonal coherent state representations. An appreciation of the subtleties of quantum theory obtains. Certain implications for the physics of light beams are cited.

DOI
10.1063/1.524282

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A systematic structure analysis of the correlation functions of statistical quantum optics is carried out. From a suitably defined auxiliary two-point function we are able to identify the excited modes in the wave field. The relative simplicity of the higher order correlation functions emerge as a byproduct and the conditions under which these are made pure are derived. These results depend in a crucial manner on the notion of coherence indices and of unimodular coherence indices. A new class of approximate expressions for the density operator of a statistical wave field is worked out based on discrete characteristic sets. These are even more economical than the diagonal coherent state representations. An appreciation of the subtleties of quantum theory obtains. Certain implications for the physics of light beams are cited.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
In this paper we define the normalized coherence function of arbitrary order (m, n), in a manner which seems to be a natural generalization of that defined for the second-order coherence function. Both classical and quantized optical fields are considered and the results are compared. It is shown that for classical fields and also for quantized optical fields having nonnegative definite diagonal coherent state representations of the density operator, the modulus of these normalized coherence functions is bounded by the values 0 and 1. This definition differs from the one recently given by Glauber for quantized optical fields, where the normalized coherence functions may take arbitrarily large values even for fields having nonnegative definite diagonal representations of the density operator. Conditions for ``complete coherence,'' i.e., those under which the modulus of the normalized coherence function attains the limiting value 1, are discussed. Some consequences of stationarity and quasi-monochromaticity are also discussed.
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 comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive PPP representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without PPP representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.
The theory of quantum statistical mechanics is reviewed in the light of the quantum probability theory developed in recent years by mathematicians. We show that—which is our main motivation here—the former is a special model in the framework of the latter, by extending the characteristic function in the classical probability theory to the quantum probability theory based on the requirements of quantum statistical mechanics. We find that due to the fact that fundamental dynamical variables of a quantum system must respect certain commutation relations, one has to reform the classical characteristic function defined on a Euclidean space to a normalized, nonnegative definite function defined on a group which is generated by a Lie algebra formed from these commutation relations. Our key finding is that the group-theoretical characteristic function could be adopted to replace the density matrix commonly used in quantum statistical mechanics for representing a state of the quantum ensemble, and more than that, the new representation has some significant advantages in applications. To show its advantages compared with the conventional density matrix scenario, we illustrate its applications in: solving the master equation that describes the irreversible evolution of a quantum open system; determining the pointer states of a quantum open system; and expanding the density matrix with the pointer states. In particular, thanks to the quantum characteristic function, we are able to write down the solutions of the quantum-optical master equation explicitly, obtain its steady solutions and verify that the pointer states are exactly coherent states. Besides, the Caldeira-Leggett master equation for the quantum Brownian motion and its extension to the Lindblad type are also studied.