Real-time broadening of bath-induced density profiles from closed-system correlation functions

The Lindblad master equation is one of the main approaches to open quantum systems. While it has been widely applied in the context of condensed matter systems to study properties of steady states in the limit of long times, the actual route to such steady states has attracted less attention yet. Here, we investigate the nonequilibrium dynamics of spin chains with a local coupling to a single Lindblad bath and analyze the transport properties of the induced magnetization. Combining typicality and equilibration arguments with stochastic unraveling, we unveil for the case of weak driving that the dynamics in the open system can be constructed on the basis of correlation functions in the closed system, which establishes a connection between the Lindblad approach and linear response theory at finite times. In this way, we provide a particular example where closed and open approaches to quantum transport agree strictly. We demonstrate this fact numerically for the spin-1/2 $XXZ$ chain at the isotropic point and in the easy-axis regime, where superdiffusive and diffusive scaling is observed, respectively.

Figure 1

(a) Sketch of our setup. (b) Magnetization dynamics $\langle S_r^z\left(t\right)\rangle$ in the spin-1/2 $XXZ$ chain coupled to a single Lindblad bath, obtained from the full stochastic unraveling for anisotropy $\mathrm{\Delta }=1.5$, small coupling $\gamma /J=0.1$, weak driving $\mu =0.1$, and $N=20$ sites. (c) Corresponding spatial variance ${\mathrm{\Sigma }}^2\left(t\right)$ for $\mathrm{\Delta }=1.0$ and $\mathrm{\Delta }=1.5$. Additionally, a curve for large $\gamma /J=1.0$ is depicted for $\mathrm{\Delta }=1.5$. The dashed (dotted) fits indicate superdiffusive (diffusive) scaling. The saturation of ${\mathrm{\Sigma }}^2\left(t\right)$ at long times is due to finite $N$.

Figure 2

Test setting with artificial jump times ${\tau }_k=0,10,\cdots$. (a) Magnetization dynamics $d_{r_0}\left(t\right)$ at $r_0=N/2$ for a single trajectory with Haar-random initial state $\left|\psi \right(0)\rangle$ and weak driving $\mu \ll 1$. We here consider only the single Lindblad operator $L_1$. (b) Average over all possible trajectories with jump operators $L_1$ and $L_2$, weighted with the respective probabilities for $\mu =0.1$. In each case, numerical data (circles) are found to agree convincingly with the prediction in Eqs. (11) (curves) and (12) (crosses). Other parameters: $\mathrm{\Delta }=1.5$ and $N=20$. The dashed line indicates the long-time equilibration value of the correlation function, i.e., $0.5/N$

Figure 3

Analogous setup as in Fig. 2, but now for the full site dependence ${\overline{d}}_r\left(t\right)$ at various fixed times (a)–(d), which all lie in the middle of two jumps. Numerical data (circles) are in convincing agreement with the prediction in Eq. (11) (crosses). A Gaussian is also indicated in (a) for comparison. The striped area indicates the equilibrium background of the already induced magnetization.

Figure 4

Magnetization dynamics $\langle S_r^z\left(t\right)\rangle$ at different sites $r=r_0+l$ (curves), as generated by the full stochastic unraveling procedure (averaged over ${10}^5$ or more trajectories) for $\mathrm{\Delta }=1.5$ and $N=20$. This procedure is performed for the full $H_{\text{eff}}$ without any approximation. (a) Small $\gamma /J=0.1$ and (b) strong $\gamma /J=1.0$, both for weak $\mu =0.1$. (c) Strong $\mu =1.0$ and small $\gamma /J=0.1$. In all cases, we compare to the prediction (11) for $N=20$ and $N=36$ (circles).

Figure 5

Time evolution of the magnetization $\langle S_{r_0}^z\left(t\right)\rangle$ at the position $r_0=N/2$ of the local Lindblad operators, as given for weak driving $\mu =0.1$ by Eq. (11) with amplitudes according to Eq. (12). Curves for various values of the bath coupling $\gamma$ are obtained from the average over 10 000 different trajectories. The other model parameters are the same as in Figs. 2 and 3. A bath coupling $\gamma /J=0.1$ is comparable to the jump times in Fig. 2. (a) and (b) correspond to an initial state with and without local magnetization, respectively. In each case, data for $N=36$ sites are also depicted. (c) Full site dependence for $\gamma /J=0.1$ in (b).

Figure 6

Dynamics of the magnetization $\langle S_r^z\left(t\right)\rangle$ at various sites $r=r_0+l$ for anisotropies (a) $\mathrm{\Delta }=1.0$ and (b) $\mathrm{\Delta }=8.0$, as generated by the full stochastic unraveling procedure and as predicted by Eq. (11). Remaining model parameters: Small coupling $\gamma /J=0.1$, weak driving $\mu =0.1$, and system size $N=20$.

Figure 7

Time-dependent spatial variance ${\mathrm{\Sigma }}^2\left(t\right)$, as predicted by Eq. (11) for $D_{\text{closed}}/J=0.6$.

Figure 8

Analogous comparison as the one in Fig. 4, but now the initial pure state $\left|\psi \right(0)\rangle$ is not drawn at random.