Disorder-Induced Localization in a Strongly Correlated Atomic Hubbard Gas

We observe the emergence of a disorder-induced insulating state in a strongly interacting atomic Fermi gas trapped in an optical lattice. This closed quantum system, free of a thermal reservoir, realizes the disordered Fermi-Hubbard model, which is a minimal model for strongly correlated electronic solids. We observe disorder-induced localization of a metallic state through measurements of mass transport. By varying the lattice potential depth, we detect interaction-driven delocalization of the disordered insulating state. We also measure localization that persists as the temperature of the gas is raised. These behaviors are consistent with many-body localization, which is a novel paradigm for understanding localization in interacting quantum systems at nonzero temperature.

Figure 1

Schematic representation of experimental geometry and disordered lattice. The atoms are cooled in a magnetic trap (copper) and an optical dipole trap formed from 1064 nm laser beams (gray lines). Optical lattice laser beams (red) superimposed on the trap form a cubic lattice potential. A 532 nm optical speckle field (green) is focused onto the atoms using a 1.1 $f$-number lens (gray hemisphere). Atoms in two hyperfine states (red and blue spheres) are trapped in the disordered lattice potential (false color) formed at the intersection of all the laser beams. A two-dimensional representation of the lattice is shown for clarity. The imaging direction is along the [111] direction of the lattice and is indicated by a dashed line.

Figure 2

(a) The c.m. velocity of the atom gas measured after an applied impulse for $s=4$ (blue squares), 5 (red circles), 6 (green triangles), and 7 $E_R$ (orange diamonds). Sample images used to determine $v_{\mathrm{c}.\mathrm{m}.}$ are shown in false color for $s=4$ $E_R$ for $\mathrm{\Delta }=0$ $E_R$ (i) and $\mathrm{\Delta }=1.46$ $E_R$ (ii). The field of view for all absorption images used in this work is 0.54 mm. The projection of the Brillouin zone onto the imaging plane, which is a hexagon because of the imaging and lattice-beam geometry, is indicated using solid black lines. The blue dotted line is the exponential fit used to determine ${\mathrm{\Delta }}_c$ for $s=4$ $E_R$; the arrow indicates ${\mathrm{\Delta }}_c$ for $s=4$ $E_R$. The error bars are the standard error in the mean for the 7–9 experimental runs that are averaged for each data point. (b) Images taken at $s=4$ with $\mathrm{\Delta }=0$ $E_R$ (iii) and $\mathrm{\Delta }=1.46$ $E_R$ (iv) without an impulse. The quasimomentum $\tilde{q}$ projected along the vertical axis in the imaging plane is measured in units of the maximum allowed quasimomentum in the Brilloun zone $q_{\max }$. (c) Traces through images (iii) (solid blue line) and (iv) (blue shaded region) showing the measured optical depth (OD).

Figure 3

Interaction induced delocalization. The characteristic disorder strength ${\mathrm{\Delta }}_c/12t$ is shown for varying interaction strength $U/12t$, which is controlled by tuning the lattice potential depth $s$. The blue arrow indicates an interaction-driven insulator-metal transition. ${\mathrm{\Delta }}_c$ varies by less than 20% across this range, while $t$ changes by a factor of 2.2. The error bars show the uncertainty in the fit to the data in Fig. 2 used to determine ${\mathrm{\Delta }}_c$. The percolation threshold is shown as a dashed line, and a linear fit to the data as a red solid line.

Figure 4

Temperature dependence of localization explored through measurements of $v_{\mathrm{c}.\mathrm{m}.}$. Data are shown for the marginally localized case at the lowest temperature with $\mathrm{\Delta }$ fixed to ${\mathrm{\Delta }}_c=1$ $E_R$ at $s=4$ $E_R$ (closed circles) and for $\mathrm{\Delta }=0.4$ $E_R$ (open circles). Representative images are shown in false color for gases without an applied impulse for $T_{\text{lat}}\approx 30\mathrm{nK}$ (i) and $T_{\text{lat}}\approx 70\mathrm{nK}$ (ii) for $\mathrm{\Delta }=1$ $E_R$. Slices are shown through these images using the scheme from Fig. 2 for $\mathrm{\Delta }=1$ $E_R$ (black line) and $\mathrm{\Delta }=0.4$ $E_R$ (red line) in (iii). The gray band corresponds to our resolution limit $v_{\text{res}}$ for center-of-mass motion, and the red band is the 95% confidence interval for a linear fit (red line) to the $\mathrm{\Delta }=1$ $E_R$ data. The error bars represent the standard error for the 12–48 experimental runs that were averaged for each data point.