A Hybrid Lagrangian Variation Method for Bose-Einstein Condensates in Optical Lattices

TL;DR

This paper introduces a hybrid variational method that simplifies solving the Gross--Pitaevskii equation for Bose-Einstein condensates in optical lattices, enabling faster and accurate analysis of systems with rapid potential variations.

Contribution

A novel hybrid Lagrangian variational technique that reduces the computational complexity of modeling BECs in optical lattices by coupling a quasi-1D GP equation with transverse Gaussian widths.

Findings

The method accurately models BEC dynamics in optical lattices.

It enables non-adiabatic manipulation of condensates.

The approach simplifies complex 3D problems into coupled equations.

Abstract

Solving the Gross--Pitaevskii (GP) equation describing a Bose--Einstein condensate (BEC) immersed in an optical lattice potential can be a numerically demanding task. We present a variational technique for providing fast, accurate solutions of the GP equation for systems where the external potential exhibits rapid varation along one spatial direction. Examples of such systems include a BEC subjected to a one--dimensional optical lattice or a Bragg pulse. This variational method is a hybrid form of the Lagrangian Variational Method for the GP equation in which a hybrid trial wavefunction assumes a gaussian form in two coordinates while being totally unspecified in the third coordinate. The resulting equations of motion consist of a quasi--one--dimensional GP equation coupled to ordinary differential equations for the widths of the transverse gaussians. We use this method to investigate how an optical lattice can be used to move a condensate non--adiabatically.