A Finite Temperature Treatment of Ultracold Atoms in a 1-D Optical   Lattice
  We consider the effects of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one dimensional optical lattice. We use a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism, considering both a translationally invariant lattice and one with additional harmonic confinement. In both cases we observe an upward shift in the critical temperature for Bose condensation. For the case with additional harmonic confinement, this is in contrast with results for the uniform gas. 
B. G. Wild, P. B. Blakie and D. A. W. Hutchinson