Cooling Atomic Gases With Disorder

Cold atomic gases have proven capable of emulating a number of fundamental condensed matter phenomena including Bose-Einstein condensation, the Mott transition, Fulde-Ferrell-Larkin-Ovchinnikov pairing, and the quantum Hall effect. Cooling to a low enough temperature to explore magnetism and exotic superconductivity in lattices of fermionic atoms remains a challenge. We propose a method to produce a low temperature gas by preparing it in a disordered potential and following a constant entropy trajectory to deliver the gas into a nondisordered state which exhibits these incompletely understood phases. We show, using quantum Monte Carlo simulations, that we can approach the Néel temperature of the three-dimensional Hubbard model for experimentally achievable parameters. Recent experimental estimates suggest the randomness required lies in a regime where atom transport and equilibration are still robust.

Figure 1

Entropy per site $S/N$ as a function of $T/t_0$ for different site disorder strengths ${\mathrm{\Delta }}_{\mu }$ at $U/t_0=8$. $S$ is largely independent of disorder strength for ${\mathrm{\Delta }}_{\mu }/t_0=2$, $4\lesssim U/t_0=8$. For larger randomness, $S(T)$ decreases with ${\mathrm{\Delta }}_{\mu }$ so that if disorder is turned off adiabatically, the temperature $T$ decreases, as indicated by the horizontal arrow. The inset shows the double occupancy $D(T)$. Large disorder ${\mathrm{\Delta }}_{\mu }$ changes the sign of the slope $dD/dT$ from mostly positive, to mostly negative. Here, and in all subsequent figures, unless otherwise indicated, the lattice size is $6^3$, $n=1$, and the Trotter discretization is $\mathrm{\Delta }\tau =1/(20t_0)$. Up to 300 disorder realizations are used in the disorder averages.

Figure 2

Adiabats of the disordered 3D Hubbard model along a path which combines an increase of the interaction strength from $U/t_0=0$ to $U/t_0=8$ at fixed ${\mathrm{\Delta }}_{\mu }/t_0=16$ followed by a reduction of the site disorder. $T/t_0$ decreases along both trajectories, and, in particular, by about a factor of 3 at fixed $S/(N{k}_{\mathrm{B}})\equiv s/k_{\mathrm{B}}=\ln 2$ along the second path. For $s/k_{\mathrm{B}}=0.5$, the same reduction brings $T$ down to near $T_N$.

Figure 3

Entropy per site versus temperature for hopping disorder. Here, disorder cooling is strongest at lower entropies $s\sim 0.5$. The inset shows the entropy as a function of density of the clean system for ${\mathrm{\Delta }}_{\mu }/t_0=16$ and ${\mathrm{\Delta }}_t/t_0=4$ at fixed $U/t_0=8$ and $T/t_0=1$. Here, the entropy is obtained using $s(\mu ,T)={\int }_{-\infty }^{\mu }d\mu (\partial n/\partial T){|}_{\mu }$ [46], except for the three data points in black (darker shade) at $n=1$, which are obtained via integrating over $\beta$.

Figure 4

Antiferromagnetic structure factor $S_{\pi }$ as a function of $T$ as the site (top) and bond (bottom) disorder is varied. Site disorder drives $S_{\pi }$ to zero for ${\mathrm{\Delta }}_{\mu }\gtrsim U$ by destroying the magnetic moments $m^2=1-2D$, whereas singlet formation on bonds with large $J_{ij}\sim t_{ij}^2/U$ is induced by sufficiently large ${\mathrm{\Delta }}_t$ and also destroys the AF long-range order.

Figure 5

Final temperature $T_f/t_0$ resulting from adiabatically turning off disorder in a system with initial disorder strength ${\mathrm{\Delta }}_{\mu }/t_0$, shown on the horizontal axis, and two different initial temperatures $T_i/t_0=1$, 2. As ${\mathrm{\Delta }}_{\mu }$ increases beyond $U/t_0$, $T_f$ decreases. The Néel temperatures $T_N/t_0=0.19$, 0.35, 0.29 for $U/t_0=4$, 8, 12, respectively, are shown as dashed horizontal lines. The horizontal arrows are estimates for the onsets of MBL for (from left to right) $U/t_0=4.0$, 8.0, 12.0, obtained by linearly interpolating ${\mathrm{\Delta }}_c/12t_0$ vs $U/12t_0$ in Fig. 3 of Ref. [12].