Tailoring Quantum Chaos With Continuous Quantum Measurements

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.

💡 Research Summary

In this work the authors investigate how continuous quantum measurements, specifically continuous monitoring of the system’s energy, affect the dynamical signatures of quantum chaos as diagnosed by the spectral form factor (SFF). The SFF, traditionally defined as the fidelity between a coherent Gibbs state and its time‑evolved counterpart, displays a characteristic dip‑ramp‑plateau structure in chaotic systems. By extending this definition to open dynamics, the authors introduce a “generalized SFF” that remains meaningful under stochastic evolution.

The dynamics under continuous measurement are described by a stochastic master equation (SME). The SME contains a deterministic Liouvillian part, which includes the Hamiltonian and an energy‑dephasing term proportional to the measurement strength γ, and a stochastic innovation term that accounts for the back‑action of the measurement outcomes. The measurement efficiency η controls how much of the measurement record is actually recorded: η = 1 corresponds to perfect detection (pure‑state quantum trajectories), while η = 0 reduces the evolution to a purely dephasing Lindblad master equation.

The authors derive an explicit expression for the generalized SFF in terms of a “dephased partition function” that depends on the stochastic Wiener process Wt. This formulation shows that each quantum trajectory remains pure, while ensemble averaging over many trajectories introduces decoherence.

To illustrate the theory, the Sachdev‑Ye‑Kitaev (SYK) model with N = 26 Majorana fermions is used as a maximally chaotic testbed. The Hamiltonian is random quartic, and the authors average over 250 disorder realizations and many stochastic trajectories. The results are presented in two main figures. Figure 1 compares three dynamics: (i) the non‑Hermitian “no‑jump” evolution (effective when jumps are rare), (ii) the dissipative Lindblad evolution (energy dephasing only), and (iii) the full stochastic evolution with jumps. For weak measurement strength (γ ≪ 1) the stochastic trajectories follow the no‑jump dynamics, showing a rapid decay into the dip and an extended ramp, i.e., an enhancement of chaotic signatures. As γ increases to order unity, the dip‑to‑plateau ratio t_d/t_p reaches a minimum, indicating maximal ramp extension. For γ ≫ 1 the ramp shortens again because frequent jumps wash out spectral correlations.

Figure 2 explores the dependence on measurement strength and efficiency. The ratio t_d/t_p exhibits a non‑monotonic behavior versus γ, with an optimal γ_opt ≈ O(1). The optimal point is largely independent of temperature for γ > γ_opt. Varying η shows that at η = 0 (no back‑action) the dip time is almost constant, while for η > 0 the dip time varies strongly with γ, producing the most pronounced enhancement at η = 1 where each trajectory remains pure.

The physical mechanism is identified as a Gaussian filter e^{‑2γtE_n^2} acting on the energy spectrum at the trajectory level. This filter suppresses high‑energy contributions, thereby sharpening the universal spectral correlations that give rise to the ramp. In the weak‑measurement regime the filter is mild, preserving chaos; at strong measurement it becomes too aggressive, erasing correlations.

A key conceptual contribution is the demonstration that realistic stochastic trajectories (including measurement records) can amplify quantum chaos without the exponential suppression associated with post‑selecting no‑jump trajectories. This makes experimental observation of long‑time SFF features feasible in platforms such as cold atoms, superconducting qubits, or NMR, where continuous weak measurement of energy (or related observables) is possible.

Finally, the authors discuss the role of the annealed approximation: averaging the ratio of stochastic quantities is not equivalent to the ratio of averages, and its breakdown is responsible for the observed enhancement beyond simple dephasing.

In summary, the paper establishes continuous energy measurement as a tunable knob for tailoring quantum chaotic dynamics. By adjusting measurement strength γ and detection efficiency η, one can either suppress or amplify the dip‑ramp‑plateau structure of the SFF, offering a practical route to control chaos in many‑body quantum systems and opening new avenues for exploring the interplay of measurement‑induced decoherence and intrinsic chaotic behavior.