Degenerate Fermi gas in a combined harmonic-lattice potential

In this paper we derive an analytic approximation to the density of states for atoms in a combined optical lattice and harmonic trap potential as used in current experiments with quantum degenerate gases. We compare this analytic density of states to numerical solutions and demonstrate its validity regime. Our work explicitly considers the role of higher bands and when they are important in quantitative analysis of this system. Applying our density of states to a degenerate Fermi gas, we consider how adiabatic loading from a harmonic trap into the combined harmonic-lattice potential affects the degeneracy temperature. Our results suggest that occupation of excited bands during loading should lead to more favorable conditions for realizing degenerate Fermi gases in optical lattices.

Figure 1

Schematic diagram comparing the translationally invariant lattice to the combined harmonic lattice considered in this paper.

Figure 2

Energy level spacing for the 1D eigenspectrum relevant to the experimental setup in Ref. 16. (a) $V_0=5E_R$, $\omega =0.024{\omega }_R$. (b) $V_0=10E_R$, $\omega =0.033{\omega }_R$. Parameters derived for ${}^{40}\mathrm{K}$ with lattice made from counter propagating $\lambda =826\mathrm{nm}$ lasers with harmonic confinement arising from the focused lasers (beam waist taken to be $50\mu \mathrm{m}$). Labels $A–E$ explained in text.

Figure 3

The critical site number for localization, as defined in Eq. (6), as a function of lattice depth and harmonic trap frequency. Contour lines are shown for $n_{\mathrm{crit}}=1$, 3, 5, and 7.

Figure 4

Density of states for $\overline{\omega }=0.051{\omega }_R$. Numerical results for the density of states are shown using various markers (as labeled in the figure). The analytic density of states given in Eq. (10) is plotted for the cases with $V⩾5E_R$ (solid line), using the expression ${ϵ}_{\text{gap}}=2\sqrt{V_0{E}_R}-E_R$ for the band gap energy. The harmonic oscillator density of states is also shown (dashed line). The averaging interval used for the smoothed density of states is $\Delta ϵ=0.303E_R$.

Figure 5

The density of states for a combined harmonic trap and lattice in the loose trap limit. Isotropic harmonic trap with trap frequency $\overline{\omega }=0.01{\omega }_R$. The critical energy, given by ${ϵ}_{\mathrm{crit}}=m{\overline{\omega }}^2{a}^2{n}_{\mathrm{crit}}^2∕2$, is shown as a dashed vertical line and labeled by its corresponding lattice depth. The averaging interval used for the smoothed density of states is $\Delta ϵ=0.10E_R$.

Figure 6

Temperature of an ideal Fermi gas isentropically loaded into a combined harmonic-lattice potential. The ratio of the final degeneracy temperature (after loading) to the initial degeneracy temperature in the harmonic trap are shown for various initial temperatures and lattice depths for the case (a) $N=50\times {10}^3$ and (c) $N=250\times {10}^3$. In (e) the Fermi energy is shown as a function of lattice depth for $N=50\times {10}^3$ (circles), $N=250\times {10}^3$ (crosses). (b), (d), and (f) correspond to (a), (c), and (e), respectively, but are calculated using the analytic density of states (10). For all results we have used an isotropic harmonic trap of frequency of $\overline{\omega }=0.04{\omega }_R$.

Figure 7

$N_c$ for the case of ${}^{40}\mathrm{K}$ in a lattice formed by focused lasers (solid) $\lambda =826\mathrm{nm}$ with $70\mu \mathrm{m}$ waist, (dashed) $\lambda =1200\mathrm{nm}$ with $50\mu \mathrm{m}$ waist.

Figure 8

Degeneracy temperature for isentropic loading of ${}^{40}\mathrm{K}$ into an optical lattice. (a) $N=50000$ and (b) $N=750000$. Parameters correspond to solid line in 7. Additional harmonic confinement of frequency $\omega =0.005{\omega }_R$ is superimposed on the confinement due to the focused beam waist to make the spectrum well behaved as $V_0\to 0$.

Figure 9

Comparison between numerically determined energy gap for a 1D sinusoidal lattice (solid), with the upper and lower limits corresponding to the band gap at the center and the edge of the Brillouin Zone. The harmonic approximation, $\hslash {\omega }_{\mathrm{Latt}}=2\sqrt{V_0{E}_R}$ (circles) and the estimate treating the quartic expansion perturbatively, ${ϵ}_{\mathrm{gap}}=2\sqrt{V_0{E}_R}-E_R$ (crosses).