Numerical method for evolving the dipolar projected Gross-Pitaevskii equation

TL;DR

This paper presents a spectral method for accurately evolving the dipolar projected Gross-Pitaevskii equation in Bose gases, efficiently handling mode restrictions and long-range interactions.

Contribution

It introduces a Hermite-polynomial spectral scheme with auxiliary states for efficient dipolar PGPE simulation, improving mode restriction and interaction evaluation.

Findings

Accurate implementation of mode restriction in dipolar PGPE

Efficient evaluation of dipolar interactions in reciprocal space

Derivation of Ehrenfest equations for angular momentum evolution

Abstract

We describe a method for evolving the projected Gross-Pitaevskii equation (PGPE) for an interacting Bose gas in a harmonic oscillator potential, with the inclusion of a long-range dipolar interaction. The central difficulty in solving this equation is the requirement that the field is restricted to a small set of prescribed modes that constitute the low energy c-field region of the system. We present a scheme, using a Hermite-polynomial based spectral representation, that precisely implements this mode restriction and allows an efficient and accurate solution of the dipolar PGPE. We introduce a set of auxiliary oscillator states to perform a Fourier transform necessary to evaluate the dipolar interaction in reciprocal space. We extensively characterize the accuracy of our approach, and derive Ehrenfest equations for the evolution of the angular momentum.