Solving for the Wigner Function

Hong-Yi Fan, Qin Guo

In quantum optics theory the Wigner function W(α) of a density operator p can be obtained by perfoming the integration over the p's P representation W(α) =? 2d 2 z πP(z)e -2(α*-z*)(α-z) . In this paper we derive the P representation from the Wigner function by solving this integration equation. A formula indicating how to obtain P(z) from W(α) is presented.

Jens Peder Dahl

Cheuk-Yin Wong

Previously, an explicit solution for the time evolution of the Wigner function was presented in terms of auxiliary phase space coordinates which obey simple equations that are analogous with, but not identical to, the classical equations of motion. They can be solved easily, and their solutions can be used to construct the time evolution of the Wigner function. In this paper, the usefulness of this explicit solution is demonstrated by solving a numerical example in which the Wigner function has strong spatial and temporal variations as well as regions with negative values. It is found that the explicit solution gives a correct description of the time evolution of the Wigner function. We examine next the pseudoparticle method which uses classical trajectories to evolve the Wigner function. We find that the lowest-order pseudoparticle approximation reproduces the general features of the time evolution, but there are deviations. We show how these deviations can be systematically reduced by including higher-order correction terms in powers of ℏ 2 .

Cheuk-Yin Wong

Abstract. Previously, an explicit solution for the time evolution of the Wigner function was presented in terms of auxiliary phase space coordinates which obey simple equations that are analogous with, but not identical to, the classical equations of motion. They can be solved easily and their solutions can be utilized to construct the time evolution of the Wigner function. The usefulness of this explicit solution is demonstrated by solving a numerical example in which the Wigner function has strong spatial and temporal variations as well as regions with negative values. It is found that the explicit solution gives a correct description of the time evolution of the Wigner function. We examine next the pseudoparticle approximation which uses classical trajectories to evolve the Wigner function. We find that the pseudoparticle approximation reproduces the general features of the time evolution, but there are deviations. We show how these deviations can be systematically reduced by including higher-order correction terms in powers of ¯h 2.

Maurice A. de Gosson