Quasiprobability distributions in dispersive optical bistability

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.