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Physical Review A

Dynamical ordering: Applications to Fokker-Planck equations

Arkadiusz Orłowski, Krzysztof Wódkiewicz

Published Oct 1, 1994• 2 citations
Publisher

Abstract

A transformation of the linear second-order Fokker-Planck equations occurring in quantum optics is presented which overcomes difficulties connected with negative definiteness of the diffusion matrices. This transformation is based on a dynamical ordering of quantum quasiprobability distribution functions. The ordering depends on the parameters of the dynamical problem under consideration. An algorithm leading to exact solutions of the dynamically ordered Fokker-Planck equations with arbitrary initial conditions is given, and simple criteria, how to recognize squeezed light in arbitrary ordering, are presented. The problem of Fokker-Planck equations with time-dependent diffusion is also investigated.

DOI
10.1103/physreva.50.3401

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A transformation of the linear second-order Fokker-Planck equations occurring in quantum optics is presented which overcomes difficulties connected with negative definiteness of the diffusion matrices. This transformation is based on a dynamical ordering of quantum quasiprobability distribution functions. The ordering depends on the parameters of the dynamical problem under consideration. An algorithm leading to exact solutions of the dynamically ordered Fokker-Planck equations with arbitrary initial conditions is given, and simple criteria, how to recognize squeezed light in arbitrary ordering, are presented. The problem of Fokker-Planck equations with time-dependent diffusion is also investigated.
Th author introduces an exact and simple technique to convert the atomic part of the master equation for the density operator of a quantum optical system into a partial differential equation for a quasiprobability distribution in an arbitrary representation. He focusses on the special cases of two-level atoms and normal and symmetrical orderings, corresponding respectively to the P-function and Wigner representations. As an example of his results he derives the equation for the Wigner function of single-mode optical bistability with two-level atoms, and shows how he recovers the well known Fokker-Planck approximation.
A review of quasiprobability methods for transforming chemical and quantum-optical master equations into Fokker-Planck equations is presented. For cases where conventional representations lead to Fokker-Planck equations with non-positive-definite diffusion coefficients; e.g., sub-Poissonian statistics, a generalization of the representation involving an extension to the complex plane enables analytic results to be obtained for certain nonlinear chemical and optical processes. Alternatively, a different integration measure may be chosen which ensures a positive distribution and Fokker-Planck equation with positive-semidefinite diffusion coefficients. This enables stochastic differential equations to be defined. These methods are applied to two-photon absorption and dispersive bistability in quantum optics where nonclassical photon statistics arise and to two models of nonlinear chemical reactions where sub-Poissonian statistics occur.
In the general framework of the system size expansion of Van Kampen and Kubo, we consider the Fokker-Planck equation for a model of absorptive bistability in the bad-cavity limit. The physical system is described by the reduced atomic density operators after adiabatic elimination of the cavity field variables. Mapping of the master equation into ccc-number form according to the normal-ordering mapping scheme yields known results for the atomic fluctuations and correlation functions; however, it also leads to a Fokker-Planck equation with a non-positive-definite diffusion matrix. The symmetrical-order-mapping scheme eliminates this difficulty. The leading contribution to the system size expansion yields a Fokker-Planck equation for the symmetrical-ordered density function having a positive-definite diffusion matrix. The atomic expectation values and fluctuations previously derived from the quantum Langevin equations emerge naturally from this Fokker-Planck equation.
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.
The Lindblad master equation for the damped quantum harmonic oscillator is transformed into Fokker–Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal-ordering Q representation and the Wigner W representation. It will be proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker–Planck equations in the steady state, it will be shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients.