Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations

H. Risken, Ch. Schmid, Wolfgang Weidlich

P. D. Drummond, Crispin W. Gardiner, Daniel F. Walls

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.

H. J. Carmichael

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

An alternative method was developed in the 1960s by Glauber, Haken and others based on the use of the solutions of Fokker-Planck equations (FPE) for quasi-probability distributions. The FPE works well for cases with stable fixed points, but difficulties with some of the quasi-probability distributions became apparent for the critical case. The master equation can be related to the FPE for a quasi-probability distribution by the use of a characteristic function. The characteristic function has the form of a generating function for the quasi-probability distribution. The evolution towards a steady state probability distribution for the Lorenz system provides a fascinating example of the ideas of algorithmic complexity in quantum chaos. Optical bistability has been extensively studied both experimentally and theoretically. Good agreement has been reached between theory and experiment for steady state quantities, but the quantum statistical predictions of the theory are only beginning to be explored in experiment.

H. J. Carmichael

Kurt Germey, F.‐J. Schütte, R. Tiebel +1 more

Abstract An externally driven light mode in a ring cavity may show multistable behaviour if linear losses and dispersive nonlinear coupling compete with each other. The paper considers the trilinear (sum frequency) system with two externally driven modes which shows bistability simultaneously in these two modes. It is described in a fully quantum manner. The master equation and the Fokker‐Planck equation for the Glanmber‐Sudarshan quasi‐probalbility are deduced with appropriate consideration of fluctuations. A closed steady solution of the Fokker‐Planck equation is given for one mode only under strong restrictive assumptions.