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Journal of Physics B Atomic Molecular and Optical Physics

Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation

Sadhan K. Adhikari

Published Jan 13, 2003• 11 citations
Publisher

Abstract

We would like to correct two errors in the above paper. Please see the PDF file in the article options below for full details.
DOI
10.1088/0953-4075/36/2/501

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This project aims to study the behavior of a cloud of Rubidium-85 atoms under an attractive interaction potential when used to create a Bose-Einstein condensate. Efforts to describe the dynamics of an attractive condensate using only the Gross-Pitaevskii equation have been unable to describe all of the features of such a system, and so the mathematical approach used is a perturbative method of corrections to the Gross-Pitaevskii equation. A system of analytical equations to describe a condensate is derived herein, which is then arranged into a form suitable for numerical solution. The actual calculation is not performed, but details of the computational procedure are provided, as well as an analysis of the difficulties inherent in the calculation.
We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.
In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.