Probing Transport and Slow Relaxation in the Mass-Imbalanced Fermi-Hubbard Model

Constraints in the dynamics of quantum many-body systems can dramatically alter transport properties and relaxation timescales even in the absence of static disorder. Here, we report on the observation of such constrained dynamics arising from the distinct mobility of two species in the one-dimensional mass-imbalanced Fermi-Hubbard model, realized with ultracold ytterbium atoms in a state-dependent optical lattice. By displacing the trap potential and monitoring the subsequent dynamical response of the system, we identify suppressed transport and slow relaxation with a strong dependence on the mass imbalance and interspecies interaction strength, consistent with eventual thermalization for long times. Our observations demonstrate the potential for quantum simulators to provide insights into unconventional relaxation dynamics arising from constraints.

Popular Summary

While a few isolated quantum systems are known to retain the memory of initial states for arbitrarily long times, most quantum systems with many particles exhibit quick relaxation to local equilibrium after an external perturbation. In contrast, interacting mixtures of heavy and light quantum particles are believed to exhibit finite but extremely slow relaxation. For these systems, the combination of the strong interactions and vastly different masses creates dynamical constraints, which we experimentally probe in the motion of ultracold atom ensembles. For strong interactions and large mass imbalance, we identify a particularly slow relaxation of the system.

In our experiments, ytterbium atoms in two distinct electronic states take on the role of heavy and light particles. We first perturb the system by displacing its position and then monitor the dynamics in the recorded atomic density. For finite mass imbalance and strong interactions between the particles, the light particles remain at the initial position of the system, signaling a suppression of transport. Eventually, they move toward the final position of the system, albeit extremely slowly. The associated timescale is much slower than naive expectations for individual particles or interacting ensembles would suggest, and our systematic study shows that this timescale changes unusually strongly when adjusting the interaction strength or mass imbalance.

Our work sheds light on how strong interactions can lead to extremely slow relaxation dynamics in isolated quantum many-body systems. Such an unusual behavior can not only be found in mixtures of heavy and light particles but also in other systems that exhibit strong dynamical constraints.

Figure 1

Transport in the mass-imbalanced Fermi-Hubbard model. (a) Illustration of the mass-imbalanced Fermi-Hubbard model with on-site interaction $U$ and hopping $t_L\gg t_H$ of the light (blue circles) and heavy (red circles) particles, respectively. We also show the atomic states relevant for the experimental implementation. (b) Schematic representation of the different steps in our transport measurement (top row) and integrated densities of the light atoms (bottom row) from experimental snapshots. (c) Scenario for the transient regime following a quench in the mass-imbalanced Fermi-Hubbard model for a variable inverse hold time (vertical axis). Here, the dashed line marks the crossover between unconventional relaxation dynamics at early times (question mark) and diffusion at late times. The orange arrows indicate a trajectory in the regime of our experiment.

Figure 2

Wannier-Stark localization in the absence of interactions. (a) Fraction of light atoms $N_r/N$ transported to the right half of the system (blue circles) in the absence of heavy atoms and for variable confinement $\kappa /t_L$ determined by the lattice depth $V_L$. Each point is the average of 2–5 measurements, and error bars indicate the uncertainty of $\kappa /t_L$ (partly smaller than the marker size). The solid line corresponds to our numerical calculation [42], and the insets show raw atomic column densities for $\kappa /t_L=1.7(1)\times {10}^{-2}$ and $9(1)\times {10}^{-2}$ with the dashed lines indicating the boundary that determines the atom count $N_r$ in the right half of the system. (b) Probability density ${|{\mathrm{\Psi }}_n(x)|}^2$ and eigenenergy $E_n/t_L$ of the lowest-lying single-particle eigenstates for $V_L=9E_{\mathrm{rec}}$ ($\kappa /t_L=4.9\times {10}^{-2}$). We add a small linear potential to lift the degeneracy of the localized states.

Figure 3

Constrained early-time dynamics in the interacting heavy-light mixture. Fraction of light atoms $N_r/N$ transported to the right half of the system (blue markers) for variable heavy-light interaction strength $U$, fixed hopping ratio $t_H/t_L=0.104(7)$, and confinement $\kappa /t_L=1.7(1)\times {10}^{-2}$. Each point is the average of 3–4 measurements, and error bars (partly smaller than the marker size) denote the standard error of the mean of $N_r/N$ and the uncertainty of $U/t_L$. The bottom panels show raw atomic column densities for $U/t_L=-4.8(4)$, 0.02(4), and 2.9(2) (from left to right, hexagonal markers), with the dashed lines indicating the boundary that determines the atom count $N_r$ in the right half of the system. The inset of the main panel shows the numerical matrix-product-state simulations of a single tube with the Hubbard parameters and confinement strength of the experiment, and ${\mathcal{N}}_{\mathrm{sim}}=5$ atoms of each species.

Figure 4

Late-time relaxation dynamics. Density deviation $\delta n(\tau )$ of the light atoms with respect to the initial atom distribution after translating the trap minimum by $\mathrm{\Delta }x$ [see Eq. (2)], for a variable hold time $\tau >0$, as illustrated in the bottom-right schematic of panel (b). The time traces correspond to three heavy-light interaction strengths (see legend) with (a) $t_H/t_L=0.102(6)$ and (b) 0.34(3). Each data point is calculated from the average of 3–5 atomic densities and an independent measurement of $n(x,0)$. Error bars (partly smaller than the marker size) denote the uncertainty of $\tau$, determined from the estimated uncertainty of the hopping amplitude $t_L$, and standard error of $\delta n(\tau )$, determined from jackknife resampling [42]. The colored lines show exponential fits, as discussed in the main text, and in panel (a), a three-point moving average for $U/t_L\simeq -10$ as a guide to the eye. We show our numerical calculation for $U=0$ as a gray band, with the height derived from the estimated systematic uncertainty of $\delta n(\tau )$ [42]. The two central panels show integrated atomic densities, which are used to calculate $\delta n(\tau =330\hslash /t_L)$ in (a).