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

By Weizhu Bao, Dieter Jaksch, Peter A. Markowich. 2003

552 citations recorded by the source corpus.

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 an 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.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/52c585407fe9c417f3bb593809f3ca0921a1c6a2

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap (2009, methodology): “Other methods for numerical solution of the time-dependent GP equation have also appeared in the literature [7,14,16,23,26,27,28,29,32,42,43,47,48].”
Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates (2005, background): “where β2≈β 2 =β √ γz/2π and V2(x,y)= 12 ( γ(2) xx 2 +γ(2) yy 2 ) [1, 7, 8].”
Spectral and High-Order Methods with Applications (2006, methodology): “Our numerical experiments also show that our scheme with M = 30 gives similar numerical results as the TSSP method [8] for this example, with 5132 grid points over the box [−8, 8]2 and time step k = 0.”
Mathematical Models and Numerical Methods for Bose-Einstein Condensation (2012, methodology): “For a cigar-shaped condensate [31,148,150] ωy ωx, ωz ωx, ⇐⇒ γy 1, γz 1, (1.”