Tailoring Quantum Chaos With Continuous Quantum Measurements

TL;DR

This work investigates how continuous quantum measurements shape signatures of Hamiltonian quantum chaos as diagnosed by the spectral form factor (SFF). By formulating and solving a stochastic master equation for energy-monitoring, the authors show that typical quantum trajectories can enhance chaotic features relative to unitary evolution and dephasing, with the extent of enhancement tunable by the measurement strength $\gamma$ and efficiency $\eta$. The analysis in the Sachdev-Ye-Kitaev (SYK) model reveals a nonmonotonic dependence on $\gamma$ and a deeper, earlier ramp in the SFF with higher $\eta$, while the annealed approximation breaks down due to measurement backaction. Collectively, the results demonstrate a realistic, observer-controlled mechanism to tailor quantum chaos in open systems, with potential implications for quantum thermodynamics, scrambling, and quantum simulation.

Abstract

We investigate the role of quantum monitoring in the dynamical manifestations of Hamiltonian quantum chaos. Specifically, we analyze the generalized spectral form factor, defined as the survival probability of a coherent Gibbs state under continuous energy measurements. We show that quantum monitoring can tailor the signatures of quantum chaos in the dynamics, such as the extension of the ramp in the spectral form factor, by varying the measurement strength and detection efficiency. In particular, a typical quantum trajectory obtained by monitoring with unit efficiency exhibits enhanced quantum chaos relative to the average dynamics and to unitary evolution without measurements.

Figures (7)

• Figure 1: Enhancement of quantum chaos under continuous monitoring in an energy-dephasing process. The time evolution of the spectral form factor (SFF) is shown for different measurement strengths $\gamma$ in the SYK model with $N=26$ and $\beta=0$, averaged over 250 Hamiltonian ensembles and noise realizations. The light colored lines in the background depict various individual stochastic trajectories. The SFF for unitary evolution in a closed system ($\gamma = 0$) is shown by the gray curve.
• Figure 2: (a) Tailoring quantum chaos by tuning the measurement strength. The ramp duration in the SFF is characterized by the ratio $t_\mathrm{d}/t_\mathrm{p}$ of the plateau and dip times in the monitored SYK model, averaged over 250 stochastic trajectories. The ramp duration exhibits a non-monotonic behavior as a function of the measurement strength $\gamma$, reaching a maximum for $\gamma_{\rm opt}\sim\mathcal{O}(1)$, when $t_\mathrm{d}/t_\mathrm{p}$ is minimized. The dependence for small $\gamma$ on the inverse temperature is suppressed for values $\gamma>\gamma_{\rm opt}$. The limit of no jumps associated with null-measurement conditioning is shown in blue. (b) Ratio $t_\mathrm{d}/t_\mathrm{p}$ as a function of the measurement strength $\gamma$ for different values of the measurement efficiency $\eta$. In the absence of measurement backaction ($\eta = 0$), the dip time remains nearly constant in the weak measurement regime. By contrast, for finite efficiency ($\eta >0$), $t_\mathrm{d}$ exhibits a pronounced nonmonotonic dependence on $\gamma$.
• Figure S1: Evolution of SFF for different values of $\gamma$ in the SYK Hamiltonian with $N=26$. The figure illustrates the close correspondence between numerical simulations and analytical results.
• Figure S2: Time evolution of energy variance $\langle\Delta H^2\rangle_t$ under continuous monitoring for different measurement strengths $\gamma$. The light-colored curve represents individual stochastic trajectories, while the dark black line on it indicates the trajectory-averaged behavior, which is the average of 250 trajectory realizations.
• Figure S3: Evolution of SFF for different values of the measurement strength over a single SYK Hamiltonian ($N=26$) and a single noise trajectory. The case $\eta=0$ is associated with the mixed-state evolution resulting from pure dephasing in the absence of the innovation term. Fluctuations in time in the SFF are then suppressed. By contrast, at the level of single trajectories under continuous quantum measurements, the amplitude of the fluctuations increases with the value of $\eta$. The oscillatory behavior is maximal for $\eta=1$ when the conditioned quantum state is pure.