Nonequilibrium Mass Transport in the 1D Fermi-Hubbard Model

We experimentally and numerically investigate the sudden expansion of fermions in a homogeneous one-dimensional optical lattice. For initial states with an appreciable amount of doublons, we observe a dynamical phase separation between rapidly expanding singlons and slow doublons remaining in the trap center, realizing the key aspect of fermionic quantum distillation in the strongly interacting limit. For initial states without doublons, we find a reduced interaction dependence of the asymptotic expansion speed compared to bosons, which is explained by the interaction energy produced in the quench.

Figure 1

Schematics of the expansion experiment. Top: Initial state of the harmonically trapped two-component Fermi gas with (a) singlons (red) and doublons (blue) and (b) only singlons in an optical lattice. After quenching to lower lattice depths and removing the harmonic trap, fermions expand in a homogeneous 1D lattice; $J=h·0.54(3)\mathrm{kHz}$ denotes the tunnel coupling, $U$ refers to the on-site interaction strength and $d$ is the lattice constant. The expansion dynamics is dominated by first-order processes: (a) the resonant exchange of singlon and doublon positions leads to quantum distillation, (b) the dynamical formation of doublons results in reduced asymptotic expansion velocities. Bottom: Time-dependent density-matrix renormalization group (tDMRG) simulations of the atomic density $⟨{\widehat{n}}_i⟩$ for $U=20J$ as a function of time $t$ in units of the tunneling time $\tau =\hslash /J$.

Figure 2

Dynamical phase separation of singlons and doublons. Half-width-at-half-maximum (HWHM) size $R_{s,d}$ of singlon ($s$, red) and doublon ($d$, blue) clouds as a function of time for (a) $U=5J$ and (b) $U=20J$. The dashed lines illustrate the hypothetical expansion of a noninteracting doublon cloud with effective tunneling $J_{\mathrm{eff}}$ [53]. Insets: Number of atoms on singly and doubly occupied sites, $N_s$ and $N_d$, as a function of time. (c) Experimental snapshots of the integrated line densities for singlon and doublon clouds, ${\rho }_s(x)$ and ${\rho }_d(x)$, at $t=0$ (left) and $t=40\tau$ (right). (d) Ratio $N_d^c/N_s^c$ of atom numbers on doubly and singly occupied sites in the central region of the cloud (red rectangle in the inset) as a function of time for $U=20J$. Every data point is averaged over four measurements and error bars represent the standard error of the mean. Solid lines are guides to the eye.

Figure 3

tDMRG results for the relative doublon cloud size. Main panel: Relative doublon cloud size $\mathrm{\Delta }{R}_d(t)$ for three different initial uniform densities $n$ at $U/J=20$. The initial state consists of 12 singlons, 4 doublons, and $\{0,4,8\}$ holons, respectively [53]. The solid lines end at time $t_{\max }$, when the width of the singlon cloud increased to $\mathrm{\Delta }{R}_s=0.8$. This value corresponds to the experimental one at $t=40\tau$. Inset: Experimental data for $\mathrm{\Delta }{R}_d$ as a function of the interaction strength at $t=40\tau$, which was evaluated using linear fits to the time traces $R_d(t)$ as shown in Figs. 2, 2 [53].

Figure 4

Radial expansion velocities ${\mathit{v}}_{\mathit{r}}$. Experiment (circles) and tDMRG simulations for fermions (red dark-shaded diamonds) and bosons (green light-shaded diamonds, from Ref. [19]) as a function of $U/J$. Solid lines are guides to the eye. The grey dashed line indicates $v_r=\sqrt{2}d/\tau$ in the limiting cases $U/J=0$ and $U/J\to \infty$. All initial states in the numerical simulations have a uniform average density of $n=1$ in a box with initial size $L_{\mathrm{init}}=10$.