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arXiv:cond-mat/0702315 · Physical Review A

Interacting bosons in an optical lattice: Bose-Einstein condensates and Mott insulator

Oleksandr Fialko, Ch. Moseley, K. Ziegler

Published May 31, 2007• 8 citations
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Abstract

A dense Bose gas with hard-core interaction is considered in an optical lattice. We study the phase diagram in terms of a special mean-field theory that describes a Bose-Einstein condensate and a Mott insulator with a single particle per lattice site for zero as well as for nonzero temperatures. We calculate the densities, the excitation spectrum, and the static structure factor for each of these phases.

DOI
10.1103/physreva.75.053616

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A dense Bose gas with hard-core interaction is considered in an optical lattice. We study the phase diagram in terms of a special mean-field theory that describes a Bose-Einstein condensate and a Mott insulator with a single particle per lattice site for zero as well as for nonzero temperatures. We calculate the densities, the excitation spectrum, and the static structure factor for each of these phases.
Several models of a strongly interacting Bose gas in an optical lattice are studied within the functionalintegral approach. The one-dimensional Bose gas is briefly discussed. Then the Bose-Einstein condensate and the Mott insulator of a three-dimensional Bose gas are described in mean-field approximation, and the corresponding phase diagrams are evaluated. Other characteristic quantities, like the spectrum of quasiparticle excitations and the static structure factor, are obtained from Gaussian fluctuations around the mean-field solutions. We discuss the role of quantum and thermal fluctuations, and determine the behavior of physical quantities in terms of density and temperature of the Bose gas. In particular, we study the dilute limit, where the mean-field equation becomes the Gross-Pitaevskii equation. This allows us to extend the Gross-
Abstract A strongly interacting Bose gas in an optical lattice is studied using a hard‐core interaction. Two different approaches are introduced, one is based on a spin‐1/2 Fermi gas with attractive interaction, the other one on a functional integral with an additional constraint (slave‐boson approach). The relation between fermions and hard‐core bosons is briefly discussed for the case of a one‐dimensional Bose gas. For a three‐dimensional gas we identify the order parameter of the Bose‐Einstein condensate through a Hubbard‐Stratonovich transformation and treat the corresponding theories within a mean‐field approximation and with Gaussian fluctuations. This allows us to evaluate the phase diagram, including the Bose‐Einstein condensate and the Mott insulator, the density‐density correlation function, the static structure factor, and the quasiparticle excitation spectrum. The role of quantum and thermal fluctuations are studied in detail for both approaches, where we find good agreement with the Gross‐Pitaevskii equation and with the Bogoliubov approach in the dilute regime. In the dense regime, which is characterized by the phase transition between the Bose‐Einstein condensate and the Mott insulator, we discuss a renormalized Gross‐Pitaevskii equation. This equation can describe the macroscopic wave function of the Bose‐Einstein condensate in the dilute regime as well as close to the transition to the Mott insulator. Finally, we compare the results of the attractive spin‐1/2 Fermi gas and those of the slave‐boson approach and find good agreement for all physical quantities.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice. We present a model of this phenomenon which we are able to analyze rigorously. The system is a hard core lattice gas at half of the maximum density and the optical lattice is modeled by a periodic potential of strength $\ensuremath{\lambda}$. For small $\ensuremath{\lambda}$ and temperature, BEC is proved to occur, while at large $\ensuremath{\lambda}$ or temperature there is no BEC. At large $\ensuremath{\lambda}$ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase. The interparticle interaction is essential for this transition, which occurs even in the ground state. Surprisingly, the condensation is always into the p=0\mathbf{p}=\mathbf{0}p=0 mode in this model, although the density itself has the periodicity of the imposed potential.
A grand-canonical system of interacting bosons is considered to study phase transitions of ultracold atoms in an optical lattice. The phase diagram is discussed in terms of a matrix-like order parameter, representing a symmetric phase (Mott insulator) and a symmetry-broken phase (Bose-Einstein condensate). An exact solution was found by introducing NNN colors and taking the limit N→∞N\to\inftyN→∞.
Bose-Einstein condensation (BEC) in cold gases can be turned on and off by an external potential, such as that presented by an optical lattice.We present a model of this phenomenon which we are able to analyze rigorously.The system is a hard core lattice gas at half-filling and the optical lattice is modeled by a periodic potential of strength λ.For small λ and temperature, BEC is proved to occur, while at large λ or temperature there is no BEC.At large λ the low-temperature states are in a Mott insulator phase with a characteristic gap that is absent in the BEC phase.The interparticle interaction is essential for this transition, which occurs even in the ground state.Surprisingly, the condensation is always into the p = 0 mode in this model, although the density itself has the periodicity of the imposed potential.
The Bose-Hubbard model of a twofold degenerate Bose gas is studied in an optical lattice with one particle per site and virtual tunneling to empty and doubly occupied sites. An effective Hamiltonian for this system is derived within a continued-fraction approach. The ground state of the effective model is studied in mean-field approximation for a modulated optical lattice. A dimerized mean-field state gives a Mott insulator whereas the lattice without modulations develops long-range correlated phase fluctuations due to a Goldstone mode. This result is discussed in comparison with the superfluid and the Mott-insulating state of a single-component hard-core Bose gas.