Measurement-Induced Crossover of Quantum Jump Statistics in Postselection-Free Many-Body Dynamics

Abstract

We reveal a nontrivial crossover of subsystem fluctuations of quantum jumps in continuously monitored many-body systems, which have a trivial maximally mixed state as a steady-state density matrix. While the fluctuations exhibit the standard volume law $\propto L$ following Poissonian statistics for sufficiently weak measurement strength, anomalous yet universal scaling law $\propto L^α\:(α\sim 2.7$ up to $L=20)$ indicating super-Poissonian statistics appears for strong measurement strength. This drastically affects the precision of estimating the rate of quantum jumps: for strong (weak) measurement, the estimation uncertainty is enhanced (suppressed) as the system size increases. We demonstrate that the anomalous scaling of the subsystem fluctuation originates from an integrated many-body autocorrelation function and that the transient dynamics contributes to the scaling law rather than the Liouvillian gap. The measurement-induced crossover is accessed only from the postselection-free information obtained from the time and the position of quantum jumps and can be tested in ultracold atom experiments.

Figures (14)

• Figure 1: (a) Schematic figure of our setup. We count the number of jumps (red cross marks) in a half chain $N_\mathrm{jump}^\mathrm{half}$ and take its variance along trajectory realizations. (b) Table of the main results for the variance $V$ and estimation uncertainty $T_\mathrm{th}$ concerning the number of jumps. Subsystem fluctuation of quantum jumps (SFQJ) for strong measurement exhibits anomalous yet universal super-Poissonian statistics, and the estimation uncertainty is enhanced as the system size is increased, in stark contrast to the case for the weak measurement or the total fluctuation of quantum jumps.
• Figure 2: Numerical results for the variance of quantum jumps for the Heisenberg model ($J_z=1$) for $832$ trajectories and $T=990$. (a) Variance of net quantum jumps in the whole system [Inset: $V[N_\mathrm{jump}^\mathrm{total}(T)]/(\gamma L)$], (b) SFQJ, (c) $V[N_\mathrm{jump}^\mathrm{half}(T)]/L^2$ [Inset: $V[N_\mathrm{jump}^\mathrm{half}(T)]/(\gamma L^2)$], and (d) $V[N_\mathrm{jump}^\mathrm{half}(T)]/L$. The SFQJ exhibits a measurement-induced crossover of the system-size scaling. Data are plotted against $\gamma$ for system sizes $L=8, 10, \cdots, 18$ in (a)-(c) and against $L$ for measurement strengths $\gamma=0.05, 0.1, \cdots, 1$ from bottom to top in (d). We take the average over 20 time intervals for $t\in [200, 1190], [1190, 2180], \cdots, [19010, 2\times10^4]$steadystate.
• Figure 3: Numerical results for $V_\mathrm{anom}$ and $V_\mathrm{act}$ based on the quantum trajectory method for the Heisenberg model. (a) The leading contribution of $V_\mathrm{anom}$ and $V_\mathrm{act}$ changes at a critical measurement strength. (b) System-size scaling of $V_\mathrm{anom}/L$. Data are shifted by $2000$ as $L$ is increased as $L=8, 10, \cdots, 18$ in (a) to improve visualization, and measurement strengths are $\gamma=0.05, 0.1, \cdots, 1$ from bottom to top in (b). The parameters and methods are the same as in Fig. \ref{['fig_variance']}.
• Figure 4: Exponents $a$, $b$, and $c$ of the Liouvillian gap $\Delta\propto 1/L^a$ (blue), $\mathcal{C}_\mathrm{auto}^\mathrm{single}\propto L^b$ (green), and $\mathcal{C}_\mathrm{auto}^\mathrm{half}\propto L^c$ (red) obtained from the finite-size scaling analysis with $L=8, 10, 12, 14$ for the Heisenberg model. The anomalous scaling of SFQJ cannot be captured by the Liouvillian gap.
• Figure 5: (a) System-size scaling of $V_\mathrm{anom}$ and (b) the exponent $\alpha$, up to $L=20$ for $\gamma=1$ at half filling. We find the universal scaling $\propto L^\alpha$ ($\alpha\sim2.7$) irrespective of the values of $J_z$.