Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation
Abstract
We study the numerical solution of the time-dependent Gross-Pitaevskii equation (GPE) describing a Bose-Einstein condensate (BEC) at zero or very low temperature. In preparation for the numerics we scale the 3d Gross-Pitaevskii equation and obtain a four-parameter model. Identifying `extreme parameter regimes', the model is accessible to analytical perturbation theory, which justifies formal procedures well known in the physical literature: reduction to 2d and 1d GPEs, approximation of ground state solutions of the GPE and geometrical optics approximations. Then we use a time-splitting spectral method to discretize the time-dependent GPE. Again, perturbation theory is used to understand the discretization scheme and to choose the spatial/temporal grid in dependence of the perturbation parameter. Extensive numerical examples in 1d, 2d and 3d for weak/strong interactions, defocusing/focusing nonlinearity, and zero/nonzero initial phase data are presented to demonstrate the power of the numerical method and to discuss the physics of Bose-Einstein condensation.
Keywords
Cite
@article{arxiv.cond-mat/0303239,
title = {Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation},
author = {Weizhu Bao and Dieter Jaksch and Peter A. Markowich},
journal= {arXiv preprint arXiv:cond-mat/0303239},
year = {2009}
}
Comments
33 pages, 11 figures. Journal of Computational Physics, to appear