Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation

Weizhu Bao*, Dieter Jaksch, Peter Markowich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

364 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)318-342
Number of pages25
JournalJournal of Computational Physics
Volume187
Issue number1
DOIs
StatePublished - May 1 2003

Keywords

  • Approximate ground state solution
  • Bose-Einstein condensation (BEC)
  • Defocusing/focusing nonlinearity
  • Gross-Pitaevskii equation
  • Time-splitting spectral method

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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