Eur. Phys. J. D 27, 247-261 (2003)
https://doi.org/10.1140/epjd/e2003-00284-4

Bose-Einstein condensates in 1D optical lattices

Compressibility, Bloch bands and elementary excitations

¹ Dipartimento di Fisica, Università di Trento and BEC-INFM, 38050 Povo, Italy
² Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, 25121 Brescia, Italy
³ Kapitza Institute for Physical Problems, 117334 Moscow, Russia

Corresponding author: ^a Kraemer@science.unitn.it

Received: 14 May 2003
Revised: 15 July 2003
Published online: 15 December 2003

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.