Disorder-controlled relaxation in a three-dimensional Hubbard model quantum simulator

Understanding the collective behavior of strongly correlated electrons in materials remains a central problem in many-particle quantum physics. A minimal description of these systems is provided by the disordered Fermi-Hubbard model (DFHM), which incorporates the interplay of motion in a disordered lattice with local interparticle interactions. Despite its minimal elements, many dynamical properties of the DFHM are not well understood, owing to the complexity of systems combining out-of-equilibrium behavior, interactions, and disorder in higher spatial dimensions. Here, we study the relaxation dynamics of doubly occupied lattice sites in the three-dimensional DFHM using interaction-quench measurements on a quantum simulator composed of fermionic atoms confined in an optical lattice. In addition to observing the widely studied effect of disorder inhibiting relaxation, we find that the cooperation between strong interactions and disorder also leads to the emergence of a dynamical regime characterized by disorder-enhanced relaxation. To support these results, we develop an approximate numerical method and a phenomenological model that each capture the essential physics of the decay dynamics. Our results provide a theoretical framework for a previously inaccessible regime of the DFHM and demonstrate the ability of quantum simulators to enable understanding of complex many-body systems through minimal models.

Figure 1

Experimental setup. (a) A nonequilibrium doublon population for atoms in two spin states is prepared in a 3D cubic lattice using an interaction quench. (b) The atoms are described by the disordered Fermi-Hubbard model [Eq. (1)], which involves tunneling with amplitude $t$, doublon formation with interaction energy $U$, and a local disordered energy offset $\epsilon$ with an exponential distribution $\rho \left(\epsilon \right)$ characterized by disorder strength $\mathrm{\Delta }$. (c) While the full dynamics of the system involve the interplay of doublons and single atoms, the case of an isolated doublon is useful for intuition about the dynamical regimes. For weak disorder (compared to $U$), the decay of doublons into pairs of single atoms is suppressed by the gap between the doublon and single-particle energies. Moderate disorder can enhance the decay rate by creating resonant pairs of sites with an energy difference comparable to $U$. However, sufficiently strong disorder can hinder decay by suppressing diffusion to these resonant sites.

Figure 2

Examples of experimental and numerical doublon relaxation data at (a) $\mathrm{\Delta }/U=0.07$, (b) 1.1, (c) 2.7, and (d) 4.7. All data are taken at $U/12t=1.8$ (corresponding to a $12E_R$ lattice depth in the experiment). Sample traces of the doublon population vs time for different disorder strengths are shown for experimental measurements (points); the error bars give the statistical standard error of the mean for averaging over multiple measurements. The fits used to extract $\tau$ are shown using solid lines. The dashed lines denote the fit values for the equilibrium doublon population, which is typically 10% of the total atom number [25] and which decays over time due to overall number loss. The corresponding GDTWA simulations [25] are displayed as shaded regions that show the standard error of the mean from disorder and trajectory averaging. The horizontal axis for (d) (the strongest disorder case) is compressed to bring the slow decay into view. At the highest disorder the doublon lifetime $\tau$ becomes closest to the overall number loss ${\tau }_{\text{loss}}$, potentially affecting the fitted value for the equilibrium doublon fraction [25].

Figure 3

Dynamical regimes of doublon relaxation. The doublon lifetime $\tau$ shows a strong nonmonotonic dependence on disorder, first decreasing (green region, labeled 1) and then increasing (blue region, labeled 2) with larger disorder strength. The measurements (points) are compared to numerical GDTWA simulations (crosses). The error bars for the experimental data provide the standard error of the fit used to determine $\tau$, while the error bars for the GDTWA numerical data are smaller than the symbols. The solid line represents a fit of the measurements to the analytical reaction-diffusion model with a diffusion constant that decreases monotonically with disorder. The lifetime for overall number loss in the experiment ${\tau }_{\text{loss}}$ (dashed-dotted line) is independent of disorder and more than an order of magnitude slower than the doublon lifetime at all but the highest disorder value. (a)–(d) indicate the values of $\mathrm{\Delta }/U$ corresponding to Figs. 2–2.