Properties of the stochastic Gross-Pitaevskii equation: Projected Ehrenfest relations and the optimal plane wave basis

TL;DR

This paper analyzes the stochastic Gross-Pitaevskii equation, deriving Ehrenfest relations with projector effects, and explores how basis choice impacts the description of finite-temperature Bose-Einstein condensates, identifying an optimal plane wave basis.

Contribution

It derives Ehrenfest relations for the SGPE including projector effects and compares basis choices, identifying an optimal plane wave basis for accurate finite-temperature BEC descriptions.

Findings

Ehrenfest relations for SGPE include growth noise and projector effects.

The validity condition for neglecting projector terms is more stringent than previously thought.

The optimal plane wave basis improves the accuracy of equilibrium property calculations.

Abstract

We investigate the properties of the stochastic Gross-Pitaevskii equation describing a condensate interacting with a stationary thermal cloud derived by Gardiner and coworkers. We find the appropriate Ehrenfest relations for the SGPE, including the effect of growth noise and projector terms arising from the energy cutoff. This is carried out in the high temperature regime appropriate for the SGPE, which simplifies the action of the projectors. The validity condition for neglecting the projector terms in the Ehrenfest relations is found to be more stringent than the usual condition of validity of the truncated Wigner method or classical field method -- which is that all modes are highly occupied. In addition it is required that the overlap of the nonlinear term with the lowest energy eigenstate of the non-condensate band is small. We show how to use the Ehrenfest relations along with the corrections generated by the projector to monitor dynamical artifacts arising from the cutoff. We also investigate the effect of using different bases to describe a harmonically trapped BEC at finite temperature by comparing the condensate fraction found using the plane wave and single particle bases. We show that the equilibrium properties are strongly dependent on the choice of basis. There is thus an optimal choice of plane wave basis for a given cut-off energy and we show that this basis gives the best reproduction of the single particle spectrum, the condensate fraction and the position and momentum densities.