Numerical difficulties with the positive P representation: an Illustrative example

Abstract

Nonclassical states of the electromagnetic field are states for which the Glauber-Sudarshan P function does not exist as a positive-definite nonsingular probability density. A classical stochastic description for systems that produce nonclassical states can be recovered, however, using the positive P representation. But difficulties can arise in numerical simulations of nonlinear stochastic models derived using this representation.1 This paper addresses the unresolved question of whether the difficulties are purely numerical or fundamental to the representation. Recently we used the positive P representation to provide an appealing picture of the nonclassical states produced by a degenerate parametric oscillator.2 The resolution of difficulties in that problem by a reflecting boundary condition seemed rather model dependent. We analyze the quantized anharmonic oscillator in the positive P representation. Correct operator averages are derived from the stochastic model analytically. However, an explicit construction of the positive P function shows that there are inherent difficulties for numerical simulations. Unconstrained diffusion in phase space occurs, and for long times the important trajectories for recovering correct moments are those that diffuse far from the origin. Even if numerical instability can be avoided, an enormous number of trajectories would be required to cover phase space sufficiently to obtain accurate averages.