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Physical Review A

Quasiprobability distributions in dispersive optical bistability

Karl Vogel, H. Risken

Published May 1, 1989• 70 citations
Publisher

Abstract

For the model of Drummond and Walls [J. Phys. A 13, 725 (1980)] describing dispersive optical bistability, the s-parametrized quasidistributions W(\ensuremath{\alpha},s) introduced by Cahill and Glauber [Phys. Rev. A 117, 1882 (1969)] are investigated. The equation of motion for W(\ensuremath{\alpha},s) generally has derivative terms up to third order. For the Q function (s=-1) and the P function (s=1) the third-order terms vanish. The equation of motion is then a pseudo-Fokker-Planck equation, i.e., a Fokker-Planck equation where the diffusion matrix is not positive definite or semidefinite. It is shown that the equation of motion with the inclusion of third-order-derivative terms can be solved by the matrix continued-fraction method. In particular, results for the Wigner function (s=0) are presented and compared with the results of the Q function (s=-1). Furthermore, the results are compared also with those obtained from the equation of motion for the Wigner function where the third-order-derivative terms have been neglected.

DOI
10.1103/physreva.39.4675

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For the model of Drummond and Walls [J. Phys. A 13, 725 (1980)] describing dispersive optical bistability, the s-parametrized quasidistributions W(\ensuremath{\alpha},s) introduced by Cahill and Glauber [Phys. Rev. A 117, 1882 (1969)] are investigated. The equation of motion for W(\ensuremath{\alpha},s) generally has derivative terms up to third order. For the Q function (s=-1) and the P function (s=1) the third-order terms vanish. The equation of motion is then a pseudo-Fokker-Planck equation, i.e., a Fokker-Planck equation where the diffusion matrix is not positive definite or semidefinite. It is shown that the equation of motion with the inclusion of third-order-derivative terms can be solved by the matrix continued-fraction method. In particular, results for the Wigner function (s=0) are presented and compared with the results of the Q function (s=-1). Furthermore, the results are compared also with those obtained from the equation of motion for the Wigner function where the third-order-derivative terms have been neglected.
For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.
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.
On the basis of the absorptive bistability model, including the fluctuations, we rederive in the good-cavity limit the Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted field amplitude. It is shown that on a semiclassical level different variants of the Fokker-Planck equation are possible. The equation of Bonifacio, Gronchi, and Lugiato is solved in terms of a path integral which involves a generalized Wiener measure. In order to calculate the averages of transmitted field moments as functions of time, a Monte Carlo integration scheme is used. This scheme consists in constructing random trajectories which result from the Langevin equation statistically equivalent to the derived Fokker-Planck equation. Numerical results in the bistable domain are presented for large values of the fluctuation parameter.
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.
Systems exhibiting dispersive optical bistability are considered as examples of nonequilibrium steady states lacking detailed balance. Methods for describing such states are developed. The Fokker-Planck equation for the steady-state distribution function of the transmitted electromagnetic field is derived. In the limit of small fluctuations the Fokker-Planck equation is reduced to the form of a Hamilton-Jacobi equation for a "nonequilibrium thermodynamic potential." This equation is solved in various approximations. The nonequilibrium thermodynamic potential acts like a free-energy for the first-order-type transition far from thermodynamic equilibrium lacking detailed balance. The potential entirely determines the steady-state distribution, the coexistence curve in the bistable domain, a Lyapunoff function of the deterministic equations of motion, and allows us to cast the deterministic equations of motion into the standard form of nonequilibrium thermodynamics.
In this paper the equations for quantum optical pulse propagation in nonlinear and dispersive single-mode fibers are presented in terms of two phase-space formulations based on the positive-P and the Wigner distributions. Included are the effects due to the coupling of the electromagnetic modes to the vibrational states of a vitreous silica fiber. By making use of the well-known equivalence of Fokker-Planck and Ito stochastic equations, we demonstrate two alternative methods for formulating the equations of motion as coupled stochastic c-number equations for the propagating field. The first method involves a representation of the density operator in terms of the positive P distribution function. This leads to exact stochastic equations of motion. The second method makes use of the Wigner distribution function. This method, which requires truncation of third-order derivative terms in the corresponding Fokker-Planck equation, is necessarily approximate. However, we discuss certain advantages to the Wigner approach that have made it the preferable method for exploratory work.