Probing Slow Relaxation and Many-Body Localization in Two-Dimensional Quasiperiodic Systems

In a many-body localized (MBL) quantum system, the ergodic hypothesis breaks down, giving rise to a fundamentally new many-body phase. Whether and under which conditions MBL can occur in higher dimensions remains an outstanding challenge both for experiments and theory. Here, we experimentally explore the relaxation dynamics of an interacting gas of fermionic potassium atoms loaded in a two-dimensional optical lattice with different quasiperiodic potentials along the two directions. We observe a dramatic slowing down of the relaxation for intermediate disorder strengths. Furthermore, beyond a critical disorder strength, we see negligible relaxation on experimentally accessible time scales, indicating a possible transition into a two-dimensional MBL phase. Our experiments reveal a distinct interplay of interactions, disorder, and dimensionality and provide insights into regimes where controlled theoretical approaches are scarce.

Popular Summary

Most physical systems rapidly reach thermal equilibrium. Certain quantum-mechanical systems in the presence of strong applied disorder, however, can get stuck in nonequilibrium states forever. Such systems are said to be in the “many-body localized” phase. While this phase is well understood in one dimension, its nature and stability in higher dimensions is largely unexplored. How systems transition between the many-body localized phase and the more familiar “thermal” phase is also generally unclear. We experimentally explore these questions using ultracold potassium atoms loaded in quasiperiodic optical lattices.

Our setup mimics disorder but has some distinctive features, such as the lack of rare fluctuations present in disordered systems. We start our system far from equilibrium (with atoms loaded into alternate columns) and observe if the system preserves the memory of this pattern. Beyond a critical disorder strength, the relaxation of local observables remains incomplete up to extremely long time scales, which hints at the existence of a many-body localized phase in two dimensions. Moreover, we find that between a regime of fast relaxation at weak disorder and the apparent absence of relaxation at strong disorder, there exists a regime of extremely slow dynamics, in which local observables relax to their thermal equilibrium but do so on time scales that are much larger than the intrinsic time scales of our system.

Our work demonstrates absence of local thermalization in higher dimensions and paves the way for stabilizing exotic quantum phenomena, such as coherent quantum memories and topological phases, at temperatures at which they would be destroyed without disorder.

Figure 1

Schematic of the experiment. (a) The system is initialized in a striped density-wave pattern of fermionic ${}^{40}\mathrm{K}$ atoms in a random mixture of two spin states (red and blue) in a square lattice with tunneling matrix element $J$, quasiperiodic potential of strength $\mathrm{\Delta }$, and tunable on-site interactions $U$ between the different spins. The largest realized 2D system is composed of approximately $200\times 100$ sites with several thousand atoms. (b) For weak disorder strength, the system thermalizes quickly (green area), whereas at strong disorder it is likely to exhibit a many-body localized regime (blue). Close to the transition (red dot, ${\mathrm{\Delta }}_c$), a regime of slow relaxation is observed (red area), potentially caused by locally insulating regions.

Figure 2

Time evolution of an imprinted density-wave pattern in the interacting, two-dimensional Aubry-André model. We measure the time evolution of the imbalance between atom numbers on even and odd stripes for intermediate interactions $U=5J$ and varying disorder strength $\mathrm{\Delta }$. (a) At weak disorder ($\mathrm{\Delta }=1J$), the imbalance vanishes quickly within a couple of tunneling times, signaling ergodic dynamics. For intermediate disorder ($\mathrm{\Delta }=4J$), we observe a markedly slow relaxation. At even stronger disorders ($\mathrm{\Delta }=10J$), relaxation is absent up to a weak, previously measured [40] coupling to the environment. (b) The same time evolution is shown on a double logarithmic plot for additional values of the disorder strength. The solid lines denote fits to the model described in the main text [Eq. (2)]. In both plots, error bars denote the error of the mean from six individual experimental realizations. All times are in units of the tunneling time, $\tau =h/(2\pi J)$.

Figure 3

Closed-system imbalance and relaxation exponents near the critical regime. (a) Closed-system imbalance $\tilde{\mathcal{I}}(t)$, at short times ($10\tau$, green points) and at long times ($100\tau$, blue points) for fixed interaction strength of $U=5J$. Three dynamical regimes can be identified: (1) a rapidly thermalizing regime at low disorder, (2) a regime of slow relaxation at intermediate disorder (visualized by the gray area), and (3) a regime at strong disorder in which relaxation is absent on experimentally accessible time scales, consistent with a many-body localized phase. (b) The relaxation exponent $\zeta$ obtained from Eq. (2) decreases continuously with increasing disorder until a threshold value ${\mathrm{\Delta }}_c$, at which it vanishes (green points). A piecewise fit $\zeta \propto \{|\mathrm{\Delta }-{\mathrm{\Delta }}_c{|}^{\nu },0\}$ for $\zeta$ in the regime $\mathrm{\Delta }\in \{5J,12J\}$ yields ${\mathrm{\Delta }}_c=9J\pm 0.5J$ as the critical disorder strength of the possible MBL transition. The yellow line is an approximate upper bound for the relaxation exponent. The inset shows the same analysis for $\mathcal{I}(t)$, i.e., neglecting the weak coupling to the environment. Apart from a small offset (dashed line) consistent with the known background coupling [40], we still find a sharp change of the extracted exponents at ${\mathrm{\Delta }}_c$. Error bars denote the fit error and are typically smaller than the symbol size.

Figure 4

Interaction and energy density dependence. (a) Time evolution of the imbalance for fixed disorder strength $\mathrm{\Delta }=5J$ and vanishing $U\approx 0$ and finite interactions $U=5J$. While there is negligible relaxation in the noninteracting case, the interacting time trace shows slow relaxation. This indicates that the relaxation is inherently due to interactions. We also show the normalized total atom number $\mathcal{N}$ to show that atom loss is minimal on the time scales of the experiment [40]. (b) We measure the imbalance for fixed disorder strength $\mathrm{\Delta }=6J$ but starting from initial states with two different doublon fractions $D\approx 5\%$, 30%. In the large interaction limit, we observe a significantly higher imbalance for the larger doublon fraction. This can be qualitatively understood by the reduced mobility of a bound doublon. Each point is averaged over six individual experimental realizations, and error bars denote the error of the mean. Solid lines are guides to the eye.