Quantification of the memory effect of steady-state currents from interaction-induced transport in quantum systems

Dynamics of a system in general depends on its initial state and how the system is driven, but in many-body systems the memory is usually averaged out during evolution. Here, interacting quantum systems without external relaxations are shown to retain long-time memory effects in steady states. To identify memory effects, we first show quasi-steady-state currents form in finite, isolated Bose- and Fermi-Hubbard models driven by interaction imbalance and they become steady-state currents in the thermodynamic limit. By comparing the steady-state currents from different initial states or ramping rates of the imbalance, long-time memory effects can be quantified. While the memory effects of initial states are more ubiquitous, the memory effects of switching protocols are mostly visible in interaction-induced transport in lattices. Our simulations suggest that the systems enter a regime governed by a generalized Fick's law and memory effects lead to initial-state-dependent diffusion coefficients. We also identify conditions for enhancing memory effects and discuss possible experimental implications.

Figure 1

Interaction-induced transport for probing (a) memory effects of initial states and (b) memory effects of switching protocols. (a) Two systems initially with different uniform interactions $U^{\text{small}}$ (dashed green) and $U^{\text{large}}$ (dashed orange) are quenched to the same interaction profile (solid blue). (b) Systems with the same initial interaction (dashed green) experience the same interaction imbalance (solid blue) switched on at different time scales. Inset: Linear switching protocols with different ramping times ($t_r=t_0, 7t_0$). (c, d) Interaction-induced transport in the BHM. (c) Particle density contour plot with $L=60$ sites, ${\overline{n}}_B=1, U=J$, and $\mathrm{\Delta }U=J$ on the left half of the lattice. The dashed line in (c) indicates the propagation of the low-density region from the left edge and the red star marks the time at which the density distortion affects the QSSC; as one can see the QSSC decreases at the gray star in (d). The red (light-blue) regime on the right (left) half of the system indicates the formation of density plateaus. (d) Current flowing through the interface of the interaction imbalance with different fillings and $U=\mathrm{\Delta }U=J$. The stars in (c) and (d) indicate the same time ($t=13t_0$).

Figure 2

QSSCs in the BHM with ${\overline{n}}_B=1/2$ and FHM with ${\overline{n}}_F=1/6$. Insets: The evolution of mass current, where the plateaus indicate the QSSC, and dashed lines indicate the time window in which the averages are taken. (a) BHM and (c) FHM with the same uniform initial interaction $U=J$ and different interaction imbalances. (b) BHM and (d) FHM with different uniform initial interactions and the same interaction imbalance, $\mathrm{\Delta }U=J$. A stronger initial interaction suppresses the QSSC value. Error bars are the standard deviation from the time average and they are within the symbol size if not shown.

Figure 3

Interaction-induced transport by a sudden interaction imbalance of (a, b) the BHM with ${\overline{n}}_B=1$ and (c, d) the FHM with ${\overline{n}}_F=1/2$. The system size is $L=60$ sites. Under a constant initial interaction $U=J$, the QSSCs as a function of the interaction imbalance are shown in (a) for the BHM and (c) for the FHM. By quenching the systems with the same $\mathrm{\Delta }U=J$ in the left half, the QSSCs decrease with the initial interaction as shown in (b) for the BHM and (d) for the FHM. The dashed line in (b) indicates the critical interaction of the 1D superfluid–Mott insulator transition. Insets: Current versus time for selected values labeled on the graph. Dashed lines indicate the time window in which the average is taken.

Figure 4

No QSSC can be observed if part of a bosonic system is noninteracting. (a) Currents through the middle of a lattice with $L=60$ sites and ${\overline{n}}_B=1/2$ versus time for two cases: initially noninteracting bosons experiencing an interaction imbalance (solid blue line) and initially interacting bosons with part of the system quenched to noninteracting bosons (dashed green line). (b) Particle density profiles at selected times for a system with an initial interaction $U=3J$ and an imbalance $\mathrm{\Delta }U=-3J$ applied to the right half. There are no steady structures in the central (shaded) region compared to Fig. 9. (c) Evolution of the particle density for $U=0$ and $\mathrm{\Delta }U=6J$ in the left half.

Figure 5

Memory effects of initial states for the BHM with (a) ${\overline{n}}_B=1/2$ and (c) ${\overline{n}}_B=1$ and for the FHM with (b) ${\overline{n}}_F=1/6$ and (d) ${\overline{n}}_F=1/2$. Squares correspond to the difference in the steady-state currents from two initial interactions $U^{\text{large}}=6J$ and $U^{\text{small}}$, which can be inferred from the $(U^{\text{large}}-U^{\text{small}})/J$ values, quenched to the same final interaction profile. Circles show the fidelity $\mathcal{F}$ between the two initial ground states.

Figure 6

Memory effects of switching protocols for the BHM with (a) ${\overline{n}}_B=1/2$ and (c) ${\overline{n}}_B=1$ and for the FHM with (b) ${\overline{n}}_F=1/6$ and (d) ${\overline{n}}_F=1/2$. Memory effects of switching protocols are probed with different finite ramping times $t_r=t_0$ and $7t_0$ for varying interaction imbalances but a fixed initial interaction, $U=J$ (red squares), and for different initial interactions but the same interaction imbalance, $\mathrm{\Delta }U=3J$ (green circles). A larger difference in the averaged steady-state currents indicates stronger memory effects. Starting with $U=J$, memory effects become stronger as the interaction imbalance increases. Memory effects are suppressed when the initial interaction is strong. For $t_r=t_0$ ($7t_0$), the averages are taken between $5t_0$ and $10t_0$ ($10t_0$ and $15t_0$).

Figure 7

Dynamics of the GPE for $N_b=10$ bosons under (a) a constant initial interaction $\tilde{U}=E_R$ and (b) a constant interaction imbalance $\mathrm{\Delta }\tilde{U}=E_R$. The results show decreasing QSSCs with increasing initial interaction or decreasing interaction imbalance. Statistical errors are within the symbol size and dashed lines indicates the regime where the average is taken. Inset in (a): Averaged currents for different interaction imbalances. Inset in (b): Amplitudes of the condensate wave function near the center of the system (shaded region) at time $t=80{\tilde{t}}_0$ for two initial interactions, $\tilde{U}=E_R$ (dashed red line) and $\tilde{U}=4E_R$ (solid blue line).

Figure 8

Memory effects of weakly interacting Bose gases described by the GPE. (a) Two examples under different initial conditions. Both cases reach the same final interaction configuration with ${\tilde{U}}^{\text{large}}$ in the left half and ${\tilde{U}}^{\text{small}}$ in the right half. In case (1), ${\tilde{U}}^{\text{large}}=3E_R$ (dashed-dotted purple line) and ${\tilde{U}}^{\text{small}}=E_R$ (thick solid blue line) are compared. In case (2), ${\tilde{U}}^{\text{large}}=1.5E_R$ (dashed green line) and ${\tilde{U}}^{\text{small}}=0.5E_R$ (thin solid red line) are compared. The two examples show that the QSSCs depend on the initial conditions. Inset: Initial profiles of the condensate wave functions for $\tilde{U}=0.5E_R$ (solid red line) and $\tilde{U}=1.5E_R$ (dashed green line), respectively. (b) Ramping of an interaction imbalance with finite time scales $t_r={\tilde{t}}_0/2, 10{\tilde{t}}_0$ for systems with $N_b=10$ bosons for the two cases $\tilde{U}=\mathrm{\Delta }\tilde{U}/2=E_R$ (top two lines) and $\tilde{U}=\mathrm{\Delta }\tilde{U}=2E_R$ (bottom two lines). QSSC values from different ramping times cannot be distinguished in either case.

Figure 9

Evolution of density profiles and diffusion coefficients of (a, c) the BHM with filling ${\overline{n}}_B=1/2$ and (b, d) the FHM with ${\overline{n}}_F=1/6$ on a 60-site lattice. A constant density gradient appears near the interface of the interaction imbalance (shaded region) when the steady-state current lasts. (a) and (b) start with $U=J$ and undergo an interaction imbalance quench $\mathrm{\Delta }U=J$ on the right half. (c) and (d) fix $\mathrm{\Delta }U=J$ (or $U=J$) and vary $U$ (or $\mathrm{\Delta }U$). Here $a_0$ is the lattice constant.

Figure 10

QSSC in potential-induced transport in the BHM with ${\overline{n}}_B=1/2$. The on-site potential energy in the left half of the system experiences a sudden change of $\mathrm{\Delta }V=J$. (a) Current versus time for two systems with different initial interactions $U=3J$ (dashed line) and $U=5J$ (solid line). (b) Density profiles at different times and (c) density contour for a system with initial interaction $U=5J$.

Figure 11

Currents in potential-induced transport of the BHM with $L=60$ sites and ${\overline{n}}_B=1/2$. (a) Currents at the interface of the potential imbalance versus time for different uniform $b$ interactions. Here $\mathrm{\Delta }V=J$. (b) QSSCs from systems with different uniform interactions, showing a clear dependence on the initial states. Insets: Results for the FHM with ${\overline{n}}_F=1/6$ and similar parameters. (c, d) Averaged steady-state values from different ramping times when the same potential imbalance is switched on linearly in time. $\mathrm{\Delta }V=2J$ in (c) and $\mathrm{\Delta }V=J$ in (d), and all symbols overlap within the error bars for each $U$. Memory effects of switching protocols are not resolvable here.