Bath-Induced Decay of Stark Many-Body Localization

We investigate the relaxation dynamics of an interacting Stark-localized system coupled to a dephasing bath, and compare its behavior to the conventional disorder-induced many body localized system. Specifically, we study the dynamics of population imbalance between even and odd sites, and the growth of the von Neumann entropy. For a large potential gradient, the imbalance is found to decay on a timescale $\tau$ that grows quadratically with the Wannier-Stark tilt. For the noninteracting system, it shows an exponential decay, which becomes a stretched exponential decay in the presence of finite interactions. This is different from a system with disorder-induced localization, where the imbalance exhibits a stretched exponential decay also for vanishing interactions. As another clear qualitative difference, we do not find a logarithmically slow growth of the von Neumann entropy as it is found for the disordered system. Our findings can immediately be tested experimentally with ultracold atoms in optical lattices.

Figure 1

Dynamics of the population imbalance between even and odd sites, $I$, for a noninteracting system from the initial charge-density-wave state. (a) $I$ as a function of $\gamma t$ for $M=8$. (b) $-\log (I)$ as a function of the scaled time $\tilde{t}=\gamma t(J/r{)}^2$ for $M=8$. The black dot-dashed line denotes the simple expression given in Eq. (8). The inset shows the stretching exponent $\beta$ of the fitting curve $I=I_0{e}^{-(t/t_0{)}^{\beta }}$ as a function of field gradient $r$. (c) $-\log (I)$ at $r=15J$ for different system sizes $M$. The black dot-dashed line is the fitting curve for large systems in the inset based on the two-parameter exponential decay function $I=I_0{e}^{-t/t_0}$. For all the plots, the solid lines denote the results obtained by numerical integration of the master equation (2) ($M=8$) or by time evolving a density matrix in a tensor-product form with Trotter Gates using the ITensor library [56] ($M\ge 20$). The dashed lines and dotted lines depict the approximated imbalance in real space $I$ [Eq. (3)] and with respect to the Wannier-Stark eigenstates $\tilde{I}$ [Eq. (7)], respectively, obtained by using the classical rate equations (6). They overlap with ($M=8$) or smoothly connect to ($M\ge 20$) the solid lines at late times. The dissipation rate is $\gamma =0.1J$.

Figure 2

Dynamics of the imbalance for interacting systems (orange solid line) from the initial charge-density-wave state. As a comparison, the result for the noninteracting case is shown as a blue line. The black dot-dashed line in (a) is the approximated result obtained from the simple equations in (9) by using $⟨n_i⟩={\sum }_k|{\psi }_{ik}{|}^2⟨{\tilde{n}}_k⟩$. The dashed lines in (b) are fitting curves based on a stretched exponential function $I_0{e}^{-(t/t_0{)}^{\beta }}$. The inset shows the stretching exponent $\beta$ as a function of interaction strength $V$. The parameters are $M=8$, $\gamma =0.1J$, $r=9J$, $\epsilon =2.2$.

Figure 3

Growth of the von Neumann entropy $S(t)$ in time for a chain at half filling from the initial charge-density-wave state. In (a), the dotted lines are the results for $V=0$, and the solid lines are the results for $V=J$. The inset figure shows the entropy as a function of the scaled time $\tilde{t}$. In (b), the entropies for the Stark-localized system and the disorder-localized system are compared. For the disorder model, the on-site potential is $W_i=w_i$, with $w_i$ being a random number uniformly distributed in the interval $[-W,W]$. The results are averaged over 100 disorder realizations. The inset shows the entropy as a function of the scaled time $\tilde{t}$. For the disorder system, the scaled time is defined as $\tilde{t}=\gamma t(J/W{)}^2$ [32]. The field strengths $W$ and $r$ are chosen so that the single-particle localization lengths for the two systems are comparable. The system size for (a) and (b) are $M=8$. (c) shows the normalized entropy for different system sizes at $r=6J$. The interaction strength for (b) and (c) is $V=J$. The dissipation rate is $\gamma =0.1J$.

Figure 4

Comparison of dynamics of the imbalance (a) and entropy (b) from the initial charge-density-wave state for different on-site potentials $W_i$ (inset). The blue solid line is the result for a linear potential with $W_i=-ri$. The orange dashed line depicts the result for a quadratic potential with $W_i=-ri+\alpha (i/M{)}^2$. The parameters are $M=8$, $\gamma =0.1J$, $V=J$.