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arXiv:0811.1021 · Physical Review A

Bose-Einstein condensation of interacting gases in traps with and without optical lattice

Swarbhanu Chatterjee, A. E. Meyerovich

Published Mar 30, 2009• 1 citation
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Abstract

We compare effects of particle interaction on Bose-Einstein condensation in inhomogeneous traps with and without optical lattice inside. Interaction pushes normal particles away from the condensate droplet, which is located in the center of the trap, toward the periphery of the trap where the trapping potential is large. In the end, the remaining normal particles are squeezed to a quasi-two-dimensional (2D) shell around the condensate droplet, thus changing the effective dimensionality of the system. In the absence of the optical lattice, the index in the temperature dependence of the condensate density at the later stages of the process is close to 2 with a weak dependence on the number of trapped particles. In the presence of the lattice inside the trap, this simple picture breaks down and the index acquires a strong dependence on the number of particles inside the trap, gradually falling from a three-dimensional to a 2D value with an increase in the number of particles. This change in index is explained by the lattice-driven spread of the condensate droplet and the localization of the narrow-band particles by the trap potential.

DOI
10.1103/physreva.79.033628

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We compare effects of particle interaction on Bose-Einstein condensation in inhomogeneous traps with and without optical lattice inside. Interaction pushes normal particles away from the condensate droplet, which is located in the center of the trap, toward the periphery of the trap where the trapping potential is large. In the end, the remaining normal particles are squeezed to a quasi-two-dimensional (2D) shell around the condensate droplet, thus changing the effective dimensionality of the system. In the absence of the optical lattice, the index in the temperature dependence of the condensate density at the later stages of the process is close to 2 with a weak dependence on the number of trapped particles. In the presence of the lattice inside the trap, this simple picture breaks down and the index acquires a strong dependence on the number of particles inside the trap, gradually falling from a three-dimensional to a 2D value with an increase in the number of particles. This change in index is explained by the lattice-driven spread of the condensate droplet and the localization of the narrow-band particles by the trap potential.
Bose–Einstein condensation in a finite one-dimensional atomic Bose gas trapped in an optical lattice is studied within Bogoliubov’s approximation and then beyond this approximation, within the static fluctuation approximation. A Bose–Hubbard model is used to construct the Hamiltonian of the system. The effect of the potential strength on the condensate fraction is explored at different temperatures; so is the effect of temperature on this fraction at different potential strengths. The role of the number of lattice points (the size effect) at constant number density (the filling factor) is examined; so is the effect of the number density on the condensate fraction. The results obtained are compared to other published results wherever possible.
We study a strongly interacting array of Bose-Einstein condensates trapped in a one-dimensional (1D) optical lattice. The system is described by a nonstandard 1D Bose-Hubbard model in which both the tunneling matrix element and the on-site atomic interaction depend on the lattice site due to the interaction broadening of the local wave function and the system inhomogeneity. We quantitatively compare theoretical analyses based on the Gutzwiller approximation with experimental observations obtained using ytterbium atoms. We show that atomic states are highly number squeezed owing to strong interatomic interactions as the lattice potential becomes deeper. Furthermore, the calculated inhomogeneous collisional broadening of spectroscopic line shapes agrees well with high-resolution spectra measured by using the ultranarrow magnetic quadrupole ${}^{1}{S}_{0}{\ensuremath{-}}^{3}{P}_{2}$ transition.
Experiments on atomic Bose-Einstein condensates inside quasi-one-dimensional optical lattices and related developments in the realization of atom lasers are currently at the frontiers in atomic physics. We give a short review of theoretical progress in evaluating the coherent transport of matter in such configurations, which has been based on an adaptation of the Wannier function representation of quasi-particle states in periodic potentials and on the use of advanced numerical methods for the solution of the time-dependent Gross-Pitaevskii equation. We present various methods by which the band structure of the elementary excitations of a periodic condensate may be probed and describe in terms of Bloch oscillations the coherent emission of matter pulses from a condensate inside an optical lattice under the force of gravity. A harmonic force can be applied to a condensate inside a magnetic trap by a rapid displacement of the center of the trap and a transition of the condensate from superfluid to dissipative behavior is driven by superposing an optical lattice. Finally, in the superfluid regime the combination of a constant force and a harmonic force in the presence of an optical lattice drives Josephson-type oscillations of the condensate, leading to observable resonances and multimode behavior.
Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.
We investigate the combined effects of weak disorder and a two-dimensional (2D) optical lattice on the collective excitations of a harmonically trapped Bose-Einstein condensate (BEC) at zero temperature. Accordingly, we generalize the hydrodynamic equations of superfluid for a weakly interacting Bose gas in a 2D optical lattice to include the effects of weak disorder. Our analytical results for the collective frequencies beyond the mean-field approximation reveal the peculiar role of disorder, interplaying with the 2D optical lattice and interatomic interaction, on elementary excitations along the 3D to 1D crossover. In particular, consequences of disorder on the phonon propagation and surface modes are analyzed in detail. The experimental scenario is also proposed.