The stochastic Gross-Pitaevskii equation II

TL;DR

This paper derives a more precise stochastic Gross-Pitaevskii equation for Bose gases using a Wigner function approach, improving accuracy and numerical feasibility over previous models by explicitly handling thermalized modes and their interactions.

Contribution

It introduces a derivation of the stochastic Gross-Pitaevskii equation that avoids local conservation assumptions and explicitly models thermalized modes, enhancing accuracy and computational practicality.

Findings

Provides a derivation based on a quasi-classical Wigner function

Includes detailed treatment of thermalized modes above cutoff

Facilitates numerical implementation of the stochastic equation

Abstract

We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii equation, as introduced by Gardiner et al. (J. Phys. B 35,1555,(2002). The derivation does not rely on the concept of local energy and momentum conservation, and is based on a quasi-classical Wigner function representation of a "high temperature" master equation for a Bose gas, which includes only modes below an energy cutoff E_R that are sufficiently highly occupied (the condensate band). The modes above this cutoff (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provide noise and damping terms in the equation of motion for the condensate band, which we call the stochastic Gross-Pitaevskii equation. This approach is distinguished by the control of the approximations made in its derivation, and by the feasibility of its numerical implementation.