High-temperature properties of fermionic alkaline-earth-metal atoms in optical lattices

We calculate experimentally relevant properties of trapped fermionic alkaline-earth-metal atoms in an optical lattice, modeled by the SU$\left(N\right)$ Hubbard model. We employ a high-temperature expansion that is accurate when the temperature is larger than the tunneling rate, similar to current regimes in ultracold atom experiments. In addition to exploring the Mott insulator-metal crossover, we calculate final temperatures achieved by the standard experimental protocol of adiabatically ramping from a noninteracting gas, as a function of initial gas temperature. Of particular experimental interest, we find that increasing $N$ for fixed particle numbers and initial temperatures gives substantially colder Mott insulators after the adiabatic ramping, up to more than a factor of 5 for relevant parameters. This cooling happens for all $N$, fixing the initial entropy, or for all $N\lesssim 20$ (the exact value depends on dimensionality), at fixed, experimentally relevant initial temperatures.

Figure 1

Observables as a function of distance to trap center $r$ for $N=4$ at $T/U=1/15$, $\mu /U=3.0$, and in three dimensions. Top: Density (solid: atomic limit; dashed: $t/U=0.04$). Although the value of the $t/T$ is not deep in the regime where the high-temperature expansion is valid, such a large value was chosen so the effects would be visible in the density profile. Insets: Density vs $\mu$ for $N=2$, 5, and 10, showing $N$ and $n$ dependence. Left inset: Zoom around the $n=1$ shell. Right inset: Zoom around $n=1$, 3, 5 for $N=2$, 5, 10 with density and $\mu$ shifted by $n_s=0$, 2, 4 and ${\mu }_s=0$, 2, 4 so that the Mott shells nearly overlap on the plot. Bottom: Entropy (solid: atomic limit; dashed: $t/U=0.016$; thin horizontal: $T\ll U$ deep Mott and deep metal limits discussed in text).

Figure 2

Adiabatic loading: Final temperature $T/U$ on a log scale as a function of initial temperature $T_i/\omega$ for $N=2$, 4, 6 (top to bottom at low temperatures) at lattice depth $V_0=18E_R$, in three dimensions for $t/U=0$ (solid) and $t/U=0.07$ (dashed) calculated with the second-order high-temperature series. Parameters are fixed to experimental values for ${}^{173}$Yb [21]: We find $U/\omega =50$, $a/\ell =0.4$, and $\mathcal{N}=5\times {10}^4$ particles using standard optical lattice results for $U$ [20]. The experimental initial gas temperature is $T_i/\omega =5.2$ ($T_i/\mu =0.14$ for $N=6$). Parameters are roughly the same for other alkaline-earth-metal atom experiments. Inset: Density profiles after adiabatic loading for $T_i/\omega =7.3$ ($T_i/\mu =0.14$, 0.17, and $0.20$ for $N=2$, 4, 6, respectively).