Bose-Einstein condensates in 1D optical lattices: compressibility, Bloch bands and elementary excitations

TL;DR

This paper analyzes the properties of Bose-Einstein condensates in 1D optical lattices, focusing on their excitation spectrum, compressibility, and effective mass, with comparisons between numerical, analytical, and approximate models.

Contribution

It provides a comprehensive analysis of elementary excitations, band structure, and quantum fluctuation effects in BECs within 1D optical lattices, including the impact of lattice depth and interactions.

Findings

Compressibility and sound velocity depend on lattice depth and interaction strength.

The band structure of excitations aligns with tight-binding predictions.

Quantum fluctuations induce dimensional crossover effects.

Abstract

We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (``Bloch bands''). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.