Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap

TL;DR

This paper introduces an efficient spectral Galerkin method using harmonic oscillator basis functions and Gauss-Hermite quadrature to numerically solve the time-dependent Gross-Pitaevskii equation for Bose-Einstein condensates in harmonic traps.

Contribution

The paper presents a novel spectral algorithm combining harmonic oscillator basis and Gauss-Hermite quadrature for improved simulation of Bose-Einstein condensate dynamics.

Findings

Accurate simulation of condensate breathing modes

Efficient computation for harmonic trap potentials

Validation against known analytical solutions

Abstract

We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a non-linear Schroedinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we present an efficient algorithm by making use of a spectral Galerkin method, using a basis set of harmonic oscillator functions, and the Gauss-Hermite quadrature. We apply this algorithm to the simulation of condensate breathing and scissors modes.