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

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.