Bose-Einstein condensates in optical lattices: mathematical analysis and analytical approximate formulae

TL;DR

This paper analyzes the mathematical properties of Bose-Einstein condensates in optical lattices using the Gross-Pitaevskii equation, deriving explicit formulas and generalized variational solutions for ground states.

Contribution

It introduces a new formalism for explicit expressions of minimum energy and chemical potential, and generalizes the variational method for approximate solutions.

Findings

Ground states are orbitally stable for all interaction parameters.

Derived explicit formulas for energy and chemical potential.

Generalized variational method for higher-order approximations.

Abstract

We show that the Gross-Pitaevskii equation with cubic nonlinearity, as a model to describe the one dimensional Bose-Einstein condensates loaded into a harmonically confined optical lattice, presents a set of ground states which is orbitally stable for any value of the self-interaction (attractive and repulsive) parameter and laser intensity. We also derive a new formalism which gives explicit expressions for the minimum energy and the associated chemical potential. Based on these formulas, we generalize the variational method to obtain approximate solutions, at any order of approximation, for the energy, the chemical potential and the ground state.