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Journal of Physics A Mathematical and General

The positive P representation and the laser equations

Sarben Sarkar, J.S. Satchell, H. J. Carmichael

Published Oct 1, 1986• 8 citations
Publisher

Abstract

The drift in the Fokker-Planck equation in the positive P representation describing a gas laser is shown to have a continuum of stable steady states. Only a small subset is physically acceptable. It is demonstrated that the diffusion does manage to average over the unphysical steady states to give correct correlation properties in the lasing region. However, in the chaotic region, the positive P representation equations have additional instabilities. As a consequence the positive P representation in the chaotic region leads to behaviour either different from known solutions or breaks down.

DOI
10.1088/0305-4470/19/14/015

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The drift in the Fokker-Planck equation in the positive P representation describing a gas laser is shown to have a continuum of stable steady states. Only a small subset is physically acceptable. It is demonstrated that the diffusion does manage to average over the unphysical steady states to give correct correlation properties in the lasing region. However, in the chaotic region, the positive P representation equations have additional instabilities. As a consequence the positive P representation in the chaotic region leads to behaviour either different from known solutions or breaks down.
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The problem of following the dynamical behavior of a quantum-mechanical system in the diagonal coherent-state representation is examined for those systems whose time evolution is specified by equations of motion for the coherent-state weight functional which resemble Fokker-Planck equations but have non-positive-definite diffusion matrices. A particular equation of this type describing a linear parametric process is considered in detail and several proposed generalizations of the diagonal representation, which include dynamical effects of the atomic system coupled to the electromagnetic field in simple models of a unimodal laser, are also briefly discussed.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
In this paper we discuss the influence of laser fluctuations on the hysteresis cycle in Optical Bistability. For the simplest case of an Absorptive Optical Bistability in a good cavity driven by an external laser light with non-white Gaussian amplitude fluctuations we derive a Fokker-Planck equation in the limit of large laser linewidth and good amplitude stabilization. This Fokker-Planck equation has a nonconstant diffusion term which shifts the most probable values of the steady state probability distribution from its deterministic positions. From the form of the steady state solution of the Fokker-Planck equation with non-white amplitude fluctuations we discuss the relative stability of the bistable branches. As in the case of quantum fluctuations the random amplitude of the laser field leads to a small range of values of the injected laser intensity in which the two bistable peaks have a comparable area. The transmitted field averaged with respect to the stationary distribution function coincides with one of the two deterministic branches except on the narrow transition region where we have large fluctuations.