Approximate mean-field equations of motion for quasi-2D Bose-Einstein condensate systems

TL;DR

This paper introduces an approximate method using a hybrid variational approach to simplify the 3D Gross-Pitaevskii equation for quasi-2D Bose-Einstein condensates, enabling efficient analysis of their dynamics.

Contribution

It develops a hybrid Lagrangian variational technique that reduces the 3D problem to an effective 2D equation coupled with a width evolution, improving computational efficiency.

Findings

The method accurately approximates 3D GPE solutions for quasi-2D condensates.

Application to ring-shaped traps demonstrates the method's effectiveness.

Comparison shows good agreement with full 3D numerical solutions.

Abstract

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, 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.