Finite-temperature treatment of ultracold atoms in a one-dimensional 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.

Figure 1

Translationally invariant lattice with ten atoms and ten sites. Overall condensate (a) and superfluid (b) fractions as a function of temperature for effective interaction strengths of 0.001 (dashed line), 1 (dot-dashed line), and 20 (solid line). The corresponding phase diagram is shown in panel (c). “S” represents the superfluid phase, and “N” the normal phase.

Figure 2

Translationally invariant lattice with ten atoms per site. Condensate fraction (a), and excitation spectrum as a function of temperature for effective interaction strengths of 0.0001 (dashed line), 0.1 (dot-dashed line), and 50 (solid line), (b) with the corresponding phase diagram (c). “S” represents the superfluid phase, and “N” the normal phase.

Figure 3

Inhomogeneous lattice with 41 lattice sites and ten atoms in total for $V_{\mathrm{eff}}=1$, at (a) $T=0\mathrm{nK}$, (b) $T=1\mathrm{nK}$, (c) $T=1.6\mathrm{nK}$. Circles represent the number of condensate atoms, squares represent the number of excited atoms and crosses the total number of atoms.

Figure 4

Overall condensate and superfluid fractions as a function of temperature, and the corresponding phase diagram for an optical lattice in a harmonic potential with 40 lattice sites (even case) and ten atoms. (a) Corresponds to the condensate fraction, (b) to the superfluid fraction for effective interaction strengths of 0.001 (dashed line), 0.5 (dot-dashed line), and 2 (solid line), and (c) to the phase diagram. “S” represents the superfluid phase, and “N” the normal phase.