Finite-temperature theory of superfluid bosons in optical lattices

Abstract

A practical finite temperature theory is developed for the superfluid regime of a weakly interacting Bose gas in an optical lattice with additional harmonic confinement. We derive an extended Bose-Hubbard model that is valid for shallow lattices and when excited bands are occupied. Using the Hartree-Fock-Bogoliubov-Popov mean-field approach, and applying local density and coarse-grained envelope approximations, we arrive at a theory that can be numerically implemented accurately and efficiently. We present results for a threedimensional system, characterizing the importance of the features of the extended Bose-Hubbard model and compare against other theoretical results and show an improved agreement with experimental data. PACS numbers: 67.85.Hj, 03.75.Hh, 05.30.Jp III. MEAN-FIELD APPROXIMATION A. Mean-field approach: condensate and non-condensateWe assume that the local number of condensate atoms is either macroscopic or zero [57,58], so that the field operator,Ψ(r), can be separated into a c-number condensate component (the order parameter), Φ(r), and a non-condensate field operator,ψ(r), defined by the usual broken symmetry approach, Φ(r) ≡ Ψ (r) ,ψ(r) ≡Ψ(r) − Φ(r) so that ψ (r) = 0.The assumption that Φ(r) is a c-number is inaccurate near the edges of the condensate, where the local condensate density, |Φ(r)| 2 , is small and just below the critical temperature, since fluctuations are important in such regions.

Geometric scale invariance as a route to macroscopic degeneracy: Loading a toroidal trap with a Bose or Fermi gas

“…An important effect of interactions is the depletion of the condensate at zero temperature, known as the quantum depletion. While the quantum depletion is generally very small in harmonic traps (typically < 1% [30]), other potentials such as lattices can considerably enhance this [31,32]. By visual inspection of Fig.…”

“…Here we obtain a formal series expansion of the density of states in the toroidal potential, in the scale-invariant regime. We make use of a decomposition of the general density of states problem into a sequence of convolutions of densities of states corresponding to each additive term in the single particle energy [32]. We can write, for example…”

Adiabatic loading of rotating bosons into 1D optical lattice

“…The requirements of this type of loading can be understood from the entropy considerations [16,20]. For the rotating condensate in an optical lattice, the normalized entropy per particle is given by [29] S…”

“…A good semiclassical approximate [22,23,24,25,26] is provided. The advantage of the semiclassical approach lies in its simplicity, in comparison to the quantum-mechanical calculations (Bose Hubbard model) [27,28], and its generality allows the treatment of the finite temperature regime [29,30]. Our approach can be summarized as follow: a conventional method of quantum mechanics is used to calculate the localized spectrum of this system [31,32,33,34], in which the classical analogy approach first used by Fetter [35] is employed.…”

Honeycomb optical lattices with harmonic confinement

“…4 and shows a remarkable agreement with the numerically calculated DOS, apart from the edge states, which are not captured by the semiclassical estimate. The efficacy of the LDA was demonstrated for a three-dimensional (3D) cubic lattice with a harmonic confining potential in [34], and the method should be valid in any optical lattice potential as long as the condition a osc a 0 is satisfied. Based on the semiclassical estimate, we can explain the following features of the DOS:…”