Anomalous waiting-time distributions in postselection-free quantum many-body dynamics under continuous monitoring

Abstract

We investigate waiting-time distributions (WTDs) of quantum jumps in continuously monitored quantum many-body systems, whose unconditional dynamics lead to the trivial infinite-temperature state. We demonstrate that the WTD of a half-chain subsystem exhibits an anomalous tail, markedly deviating from the Poissonian distribution in stark contrast to that of the whole system. By analyzing the spectral properties of the superoperator $\mathscr L_0$, which is defined by removing the jump terms associated with the half-chain subsystem from the full Liouvillian, we find that the long-time behavior with the anomalous tail of the half-chain WTD is governed by the eigenvalue $λ_0\:(<0)$ with the largest real part. We further reveal a qualitative change in the system-size dependence of $λ_0$ as a function of the measurement strength: for sufficiently weak measurement, $λ_0$ decreases proportionally to the system size, while for strong measurement, $λ_0$ scales independently of the system size, signaling the persistence of the anomalous half-chain WTD in the thermodynamic limit. The WTD is extracted solely from the spacetime record of quantum jumps $\{t_i,x_i\}$ and can be experimentally accessed without postselection. Our work establishes a spectral framework for understanding nontrivial WTDs in subsystems of monitored quantum dynamics and provides a novel diagnostics to assess many-body effects on WTDs.

Figures (4)

• Figure 1: Schematic figure of our setup. We focus on the first and the second jumps (red crosses) in a half chain after the system reaches the steady state (see text) and calculate the probability distribution of the waiting time $\tau\equiv t_2-t_1$ along trajectory realizations.
• Figure 2: Eigenspectrum of the superoperator $\mathscr L_0$ for the Heisenberg model under continuous monitoring for (a) $L=6$, $\gamma=0.05$ and (b) $L=6$, $\gamma=1$. The eigenvalue with the largest real part, $\lambda_0$, is negative in contrast to the case of the Liouvillian. (c) System-size dependence of $\lambda_0$. For weak measurement strength, $\lambda_0$ decreases proportional to the system size, but for strong measurement, $\lambda_0$ is independent of the system size. The inset shows the colors for the measurement strengths $\gamma=0.05, 0.1, ..., 1$.
• Figure 3: Numerical results of the WTD for the Heisenberg model under continuous monitoring with $L=8$ for $\gamma=0.05$ [(a)], $0.5$ [(b)], and $1$ [(c)], demonstrating the emergence of the anomalous tail in the half-chain subsystem characterized by $\lambda_0$. Left panel shows the WTD of the half-chain subsystem (blue histogram). The red solid line denotes the Poissonian distribution, while the green solid line corresponds to an exponential distribution characterized by the eigenvalue $\lambda_0$ of the superoperator $\mathscr L_0$ (see text). The inset shows that the WTD of the whole system (blue histogram) is well described by the Poissonian distribution (red line). Right panels present enlarged views of the short- and long-time regimes of the half-chain WTD shown in the left panel. The parameter is set to $t_{\mathrm{ss}}=200$, and the number of trajectories is chosen to be $10^8$.
• Figure 4: Numerical results of the WTD for the Heisenberg model under continuous monitoring with $L=14$ for $\gamma=0.05$ [(a)], $0.5$ [(b)], and $1$ [(c)], demonstrating the emergence of the anomalous tail in the half-chain subsystem characterized by $\lambda_0$. Left panel shows the WTD of the half-chain subsystem (blue histogram). The red solid line denotes the Poissonian distribution, while the green solid line corresponds to an exponential distribution characterized by the eigenvalue $\lambda_0$ of the superoperator $\mathscr L_0$ (see text). The inset shows that the WTD of the whole system (blue histogram) is well described by the Poissonian distribution (red line). Right panels present enlarged views of the short- and long-time regimes of the half-chain WTD shown in the left panel. The parameter is set to $t_{\mathrm{ss}}=200$, and the number of trajectories is chosen to be $10^7$.