Bose-Einstein condensation in an optical lattice

In this paper we develop an analytic expression for the critical temperature for a gas of ideal bosons in a combined harmonic lattice potential, relevant to current experiments using optical lattices. We give corrections to the critical temperature arising from effective mass modifications of the low energy spectrum, finite size effects, and excited band states. We compute the critical temperature using numerical methods and compare to our analytic result. We study condensation in an optical lattice over a wide parameter regime and demonstrate that the critical temperature can be increased or reduced relative to the purely harmonic case by adjusting the harmonic trap frequency. We show that a simple numerical procedure based on a piecewise analytic density of states provides an accurate prediction for the critical temperature.

Figure 1

Schematic diagram showing the combined harmonic lattice potential considered in this paper.

Figure 2

Schematic diagram of important energy scales. (a) Band structure of a translationally invariant 1D lattice of depth $4E_R$. Energy scales $e_0$ (black horizontal line), $w_0$ (gray dashed line), and $e_1$ (dash-dot line) identified (see text). (b) Correspondence of these energy scales to the combined potential. Small thick horizontal lines indicate the energies of ground band localized states.

Figure 3

Comparison of numerical smoothed density of states $\overline{g}\left(ϵ\right)$ (dots) to analytic density of states $\tilde{g}\left(ϵ\right)$ (black lines) for a 3D combined harmonic lattice potential. The lattice depth parameters are the same for each direction, i.e., $V_j=V_0$ for $j=1,2,3$. For reference, the characteristic energy scales $E_{\text{LE}}$, $e_{1j}$, $e_2$, and $E_{\text{bare}}$ are shown as gray curves. Isotropic harmonic trap taken with $\overline{\omega }=0.01{\omega }_R$. All energies are measured relative to ${ϵ}_g$.

Figure 4

Comparison of analytic and numerical critical temperatures. (a) Full numerical results for $T_c$ (squares), analytic results $T_{c0}$ (dotted), $T_{c1}$ (dashed), and simple numerical result using piecewise analytic density of states $T_{cN}$ (solid). (b) Energy scales compared to $T_{c0}$. (c) $\Delta N$ corrections. Calculation parameters: isotropic harmonic trap with $\overline{\omega }=0.025{\omega }_R$, and lattice depth parameters $V_j=8E_R$ for $j=1,2,3$.

Figure 5

Bose-Einstein condensation in a combined harmonic lattice potential. (a) Critical temperature as a function of lattice depth (all $V_j=V_0$) and total atom number for an isotropic harmonic trap with $\overline{\omega }=0.025{\omega }_R$. Isothermal levels spaced by $0.2E_R∕k$ shown as contour lines. (b) As for (a) but with $\overline{\omega }=0.05{\omega }_R$. (c) Condensate fraction vs temperature for the same parameters as (a) and $N=1\times {10}^5$ atoms.