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Journal of Physics A Mathematical and General

Calculating generating functions from characteristic functions, with application to quantum optics

Edward B. Rockower, N. B. Abraham

Published Oct 1, 1978• 10 citations
Publisher

Abstract

A method is found for calculating the number generating function of a harmonic oscillator density matrix directly from the quantum characteristic function. This technique is used to interpret thermal noise in a model for an optical detector. The results clearly reveal an earlier misinterpretation and emphasise that the terminal noise is a statistically independent amplitude in the detector/harmonic oscillator which is superposed with an amplitude having the same statistical nature as the incident optical field.

DOI
10.1088/0305-4470/11/10/007

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A method is found for calculating the number generating function of a harmonic oscillator density matrix directly from the quantum characteristic function. This technique is used to interpret thermal noise in a model for an optical detector. The results clearly reveal an earlier misinterpretation and emphasise that the terminal noise is a statistically independent amplitude in the detector/harmonic oscillator which is superposed with an amplitude having the same statistical nature as the incident optical field.
A new operator derivation of the relation giving the photon-number generating function, G(\ensuremath{\gamma}), in terms of the quantum characteristic function, C(\ensuremath{\xi},${\ensuremath{\xi}}^{\mathrm{*}}$), is presented. The inverse problem is then solved, calculating C(\ensuremath{\xi},${\ensuremath{\xi}}^{\mathrm{*}}$) directly from G(\ensuremath{\gamma}). Because G(\ensuremath{\gamma}) contains less phase information than C(\ensuremath{\xi},${\ensuremath{\xi}}^{\mathrm{*}}$), we can either assume that the field has a completely random phase (e.g., a stationary field) or be content with calculating the phase average of C(\ensuremath{\xi},${\ensuremath{\xi}}^{\mathrm{*}}$). We then derive an expression giving G(\ensuremath{\gamma}) for the superposition of two arbitrary, independent fields in terms of the individual G(\ensuremath{\gamma})'s for each (stationary) field. A number of examples illustrate the methods, including a determination of the quantum characteristic function for a field with K-distributed amplitude fluctuations.
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.
In this paper, we consider the photoelectron counting distribution when the wave field incident on the detector is a mixture of harmonic signal and thermal noise. An expression for the factorial moment-generating function of the number of counts is obtained in terms of the eigenvalues of an associated integral equation. This expression is then used to evaluate the factorial cumulants of the number of counts. The results are valid for an arbitrary counting interval. The extension to photoelectron counting with several detectors is considered in paper II of the present investigation.
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The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.