Compressibility of a Fermionic Mott Insulator of Ultracold Atoms

We characterize the Mott insulating regime of a repulsively interacting Fermi gas of ultracold atoms in a three-dimensional optical lattice. We use in situ imaging to extract the central density of the gas and to determine its local compressibility. For intermediate to strong interactions, we observe the emergence of a plateau in the density as a function of atom number, and a reduction of the compressibility at a density of one atom per site, indicating the formation of a Mott insulator. Comparisons to state-of-the-art numerical simulations of the Hubbard model over a wide range of interactions reveal that the temperature of the gas is of the order of, or below, the tunneling energy scale. Our results hold great promise for the exploration of many-body phenomena with ultracold atoms, where the local compressibility can be a useful tool to detect signatures of different phases or phase boundaries at specific values of the filling.

Figure 1

Normalized compressibility versus density for the homogeneous 3D Hubbard model, shown for various interaction strenghts (a–d) and temperatures. The different curves were obtained using DQMC (closed symbols) and NLCE (open symbols). At half-filling, $\tilde{n}=1$, the compressibility vanishes for strong interactions and low temperatures as the system enters the Mott insulating regime.

Figure 2

(a) Azimuthally averaged column density (including both spin states) vs distance from the imaging axis $\rho$, for different values of $U_0/t_0$. Data points represent the average of eight individual realizations, with error bars corresponding to the standard deviation. The lines in (a) are obtained by integrating the density (calculated for $N=2\times {10}^5$ atoms at $T/t_0=0.6$) along the imaging axis. (b) Data points correspond to density profiles extracted from the column densities using the inverse Abel transform, where $r$ is the distance from the center of the trap. The lines in (b) show the density calculated for our trap along a body diagonal of the lattice.

Figure 3

Central density, ${\tilde{n}}_0$ vs atom number for various interaction strengths. The symbols show the average for a set of 5 to 10 independent realizations, with error bars indicating the standard deviation. The shaded regions are the results of numerical calculations for our trap at $T/t_0=0.6$ (solid, green) and 2.4 (crosshatched, gray), with the width of each region corresponding to a $\pm 14\%$ systematic uncertainty in the value of $U_0/t_0$, arising from the $\pm 5\%$ uncertainty in $V_0$. The red line is calculated at $T/t_0=1.0$, without considering the trap systematics. The calculated density becomes relatively insensitive to uncertainties in $U_0/t_0$ for the two larger values of $U_0/t_0$, which are deep in the Mott regime. For $T/t_0=0.6$ the total entropy per particle, $S/(N{k}_{\mathrm{B}})$, is between 0.5 and 1.0 for the ranges of $N$ and $U_0/t_0$ shown in the figure. A temperature of $T/t_0=2.4$ is chosen for comparison, as in this case $S/(N{k}_{\mathrm{B}})$ is between 1.5 and 2.4, which is similar to the range between 1.6 and 2.2 reported from the analysis of a previous experiment [37].

Figure 4

Normalized local compressibility, $\tilde{\kappa }$ versus density for different values of $U_0/t_0$. Closed symbols show the average of eight individual realizations with error bars indicating the standard deviation. The shaded regions are numerical calculations at $T/t_0=0.6$ (solid, green) and $T/t_0=2.4$ (crosshatched, gray) for $N=2\times {10}^5$, where the width of the region reflects a $\pm 14\%$ systematic uncertainty in $U_0/t_0$. The red line is calculated at $T/t_0=1.0$, without considering the trap systematics. With $N=2\times {10}^5$, the total entropy per particle at $T/t_0=0.6$ is approximately $S/(N{k}_{\mathrm{B}})=0.58$, 0.76, and 0.82 for $U_0/t_0=3.1$, 11.1, and 14.5, respectively; at $T/t_0=2.4$ it is approximately $S/(N{k}_{\mathrm{B}})=1.59$, 1.70, and 1.66, respectively.