[Paper Review] Bose-Einstein condensates in 1D optical lattices. Compressibility, Bloch bands and elementary excitations

M. Krämer, C. Menotti+2 more|arXiv (Cornell University)|Dec 1, 2003

Cold Atom Physics and Bose-Einstein Condensates34 references60 citations

TL;DR

This paper investigates Bose-Einstein condensates in one-dimensional optical lattices by solving the Gross-Pitaevskii and Bogoliubov equations, analyzing compressibility, effective mass, sound velocity, and excitation band structures. It reveals how lattice depth and interaction strength govern dimensional crossover (3D to 1D) and validates results with tight-binding approximations and local density approximation for trapped systems.

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.

Motivation & Objective

• To understand the behavior of Bose-Einstein condensates in one-dimensional optical lattices under varying lattice depths and interaction strengths.
• To analyze the compressibility, effective mass, and sound velocity of the condensate as functions of lattice depth and interaction strength.
• To compare numerical solutions of the Gross-Pitaevskii and Bogoliubov equations with the tight-binding approximation.
• To investigate the dimensional crossover (3D to 2D to 1D) in the condensate based on lattice depth and particle number per site.
• To extend results to harmonically trapped systems using the local density approximation.

Proposed method

• Solving the stationary Gross-Pitaevskii equation to obtain Bloch-state solutions for the condensate wavefunction in a 1D periodic potential.
• Applying the Bogoliubov theory to derive the spectrum of elementary excitations and analyze their band structure.
• Calculating compressibility, effective mass, and sound velocity from the excitation spectrum and condensate response functions.
• Comparing numerical results with analytical predictions from the tight-binding approximation for weak lattices.
• Assessing the role of quantum fluctuations in determining the effective dimensionality of the system.
• Using the local density approximation to map results onto harmonically trapped systems with spatially varying chemical potential.

Experimental results

Research questions

• RQ1 How does the compressibility of a Bose-Einstein condensate in a 1D optical lattice depend on lattice depth and interaction strength?
• RQ2 What is the relationship between the effective mass and sound velocity of the condensate and the parameters of the optical lattice and interactions?
• RQ3 How do the band structures of the Bogoliubov excitations compare with the Bloch bands of the stationary solutions?
• RQ4 To what extent does the tight-binding approximation accurately describe the condensate properties in the weak lattice regime?
• RQ5 How does the effective dimensionality (3D, 2D, or 1D) of the condensate evolve with lattice depth and particle number per well?

Key findings

• The compressibility of the condensate decreases with increasing lattice depth, reflecting reduced superfluid response.
• The effective mass of the condensate increases with lattice depth, indicating stronger localization of particles in the potential minima.
• The sound velocity decreases with increasing lattice depth and interaction strength, consistent with reduced superfluidity.
• The band structure of elementary excitations closely matches the Bloch band structure of the stationary solutions, validating the quasiparticle picture.
• Quantum fluctuations lead to a dimensional crossover from 3D to 1D as lattice depth increases and particle number per site decreases.
• The local density approximation enables the extension of results to harmonically trapped systems, allowing analysis of inhomogeneous density profiles.