Density and current statistics in boundary-driven monitored fermionic chains

We consider a one-dimensional system of noninteracting fermions featuring both boundary driving and continuous monitoring of the bulk particle density. Due to the measurements, the expectation values of the local density and current operators are random variables whose average behavior is described by a well-studied Lindblad master equation. By means of exact numerical computations, we go beyond the averaged dynamics and study their full probability distribution functions, focusing on the late-time stationary regime. We find that contrary to the averaged values, the spatial profiles of the median density and current are nontrivial, exhibiting qualitative differences as a function of the monitoring strength. At weak monitoring, the medians are close to the means, displaying diffusive spatial profiles. At strong monitoring, we find that the median density and current develop a domain-wall and single-peak profile, respectively, which are suggestive of a Zeno-like localization in typical quantum trajectories. While we are not able to identify a sharp phase transition as a function of the monitoring rate, our work highlights the usefulness of characterizing typical behavior beyond the averaged values in the context of monitored many-body quantum dynamics.

Figure 1

Cartoon of the setup. (a) A fermionic chain is subject to the Hamiltonian $H$, left/right boundary driving and continuous bulk monitoring, with strength $\mathrm{\Gamma }$ and $\gamma$, respectively. (b) The average state $\overline{\rho }$ is obtained by neglecting the measurement register and is described by the so-called dephasing model [69]. We compare the average density matrix with the conditional one ${\rho }_{\xi }$, depending on the history of measurement outcomes $\left\{{\xi }_{m,t}\right\}$. (c) At strong monitoring, the mean density displays a linear profile, whereas the locally typical density (associated with the median) is characterized by a domain-wall profile. (d) The average current is constant, while the locally typical current is inhomogeneous, displaying a single-peak profile.

Figure 2

Comparison between the average values of $n_m$, $j_m$ (orange solid lines) and those in a randomly generated, individual quantum trajectory (gray markers) for various measurement rates and a sufficiently late time, $t\gg {(L/\gamma )}^2$. Left panels: At low $\gamma$, the values in individual trajectories are typically close to the average ones. Right panels: For large $\gamma$, the mean values are strongly influenced by atypical configurations and fluctuations are more pronounced. Here we chose $\mathrm{\Gamma }=1$, while the system size is $L=160$.

Figure 3

Histograms corresponding to the probability distribution functions of the particle density $P\left(n_m\right)$ (top) and current $P\left(j_m\right)$ (bottom). The plots are obtained by sampling the dynamics at sufficiently large times. We consider a chain of size $L=160$ with $\mathrm{\Gamma }=1$ and vary the monitored strength $\gamma =0.2,0.4,0.8,1.2$. We compare the statistics at the boundaries $m=1,L$ and at different positions in the bulk, $m=L/8,L/4,L/2$. We have tested that the average density and current computed out of the data shown here match the analytic result for the Lindbladian steady values, up to the statistical accuracy.

Figure 4

(a) Typical density for various $\gamma$ (for $L=128$). We extract the finite-size fit at each $L$ via Eq. (27). (b) Scaling of $T_L$ vs the measurement rate $\gamma$. Our data show a crossover between a localized behavior with $\gamma \gtrsim 0.7$ and one with $\gamma \lesssim 0.7$ where the system is weakly delocalized.

Figure 5

(a) Typical current for various measurement rates for $L=96$. (b),(c) Log-log plots for $\langle \langle j_m\rangle \rangle$ for various $L$ and $\gamma$ at (b) $m=1$ and (c) $m=L/2$. We extract the estimate localization length via the fit in Eq. (28), neglecting $L\le L_{\min }$ [where the values of $L_{\min }$ are reported in (d)]. (d) Scaling of the exponents with $\gamma$ at various $L_{\min }$ for $m=1$ (blue lines) and $m=L/2$ (orange lines).

Figure 6

(a) Mode of $n_m$ for $L=128$ and various values of $\gamma$. The mode highlights a domain-wall shape, consistent with the qualitative analysis discussed in the main text. (b) Scaling of $T_L$ extracted from the mode vs the measurement rate $\gamma$. As for the median, our data show a crossover between a localized behavior and a delocalized one around $\gamma \sim 0.7$.