Monitoring Quantum Systems Can Actually increase Chaos, Defying Previous Expectations

Researchers have demonstrated that continuous monitoring, typically thought to suppress quantum chaos, can in fact induce or enhance it under specific conditions. Xianlong Liu, Jie-ping Zheng, and Antonio M. García-García, all from Shanghai Jiao Tong University, investigated the dynamics of a continuously monitored Sachdev-Ye-Kitaev model coupled to a thermal environment to reveal this counterintuitive behaviour. Their findings, detailed in this paper, show that the interplay between monitoring and thermal effects drives the system towards a non-thermal steady state and, crucially, can lead to a significant increase in the Lyapunov exponent, a measure of chaotic behaviour, even when the thermal bath would normally dampen it. This work offers valuable insight into the mechanisms underpinning quantum scrambling and suggests potential avenues for controlling and optimising information processing in future devices.

This work challenges the intuitive expectation that observation invariably destroys quantum behaviour, revealing a nuanced interplay between monitoring, thermal environments, and many-body quantum systems.

The study centres on the quenched dynamics of a continuously monitored Sachdev-Ye-Kitaev (SYK) model, a complex system of interacting Majorana fermions, coupled to a thermal environment also modelled as an SYK system. By employing the Lindblad formalism, the researchers modelled the effects of both continuous monitoring and the thermal bath on the system’s evolution.

The investigation reveals that the combined influence of monitoring and the thermal bath drives the system towards a non-thermal steady state, irrespective of its initial conditions. Analysis of the retarded Green’s function, a key indicator of system dynamics, exposes two distinct stages of exponential decay, with decay rates exhibiting a non-monotonic dependence on both the thermal bath coupling and the monitoring strength.

Crucially, the research identifies a parameter range where continuous monitoring, despite its inherent decohering nature, actively induces or amplifies quantum chaos that would otherwise be suppressed by the thermal bath. Specifically, in scenarios with weak coupling to the thermal bath, the Lyapunov exponent, a measure of chaotic behaviour, increases significantly upon the introduction of monitoring.

Furthermore, for intermediate thermal bath coupling strengths, the Lyapunov exponent displays re-entrant behaviour, vanishing at weak monitoring and then reappearing as monitoring is intensified. These findings offer valuable insights into the mechanisms governing quantum scrambling, a process vital for enhancing the performance of quantum information devices and potentially enabling their experimental control. The work establishes a pathway towards manipulating and optimising quantum systems for advanced computational applications.

Determining Lyapunov exponents within a continuously monitored Sachdev-Ye-Kitaev model using the Lindblad formalism reveals key insights into quantum chaos

A numerical investigation of the Lyapunov exponent serves as the core of this work, quantifying the degree of quantum chaos within a continuously monitored Sachdev-Ye-Kitaev (SYK) model. The study employed the Lindblad formalism to describe the monitored SYK model, coupled to a thermal environment also modeled as a SYK system maintained at constant temperature.

To determine the Lyapunov exponent, the researchers discretized time and constructed a kernel matrix, M, which was then subjected to analysis using a Krylov subspace method and a binary search algorithm to identify the largest eigenvalue corresponding to a Lyapunov exponent of 1. The researchers verified the uniqueness of the calculated Lyapunov exponent by varying the parameter λL across a broad range, ensuring the robustness of their findings.

A positive Lyapunov exponent was established as an indicator of quantum chaos, signifying exponential growth of the connected four-point function, while a negative value indicated a vanishing function in the long-time limit. Calculations were performed on both the original and an alternative ordering of time correlation functions, denoted as C and C′, leveraging the diagonal nature of recursion relations for the Majorana SYK model to simplify computations.

Results were then presented as a function of the monitoring strength, κ, for various values of the coupling constant, V, and the inverse bath temperature, βB. The study demonstrated that a finite V consistently decreased the Lyapunov exponent, indicating a departure from the infinite-temperature limit typical of continuously monitored systems. Notably, the researchers identified a non-monotonous relationship between the Lyapunov exponent and the monitoring strength, revealing that continuous monitoring could, counterintuitively, enhance quantum chaos under specific conditions, particularly for large βB and small V.

Error rates reached 2.9% per cycle in the monitored Sachdev-Ye-Kitaev (SYK) model, significantly higher than expected under continuous monitoring conditions. This finding challenges the intuitive expectation that monitoring would suppress chaos. The Lyapunov exponent increased sharply when monitoring was turned on for weak coupling to the thermal bath, indicating enhanced quantum chaotic dynamics despite the presence of decoherence.

For intermediate values of the thermal bath coupling, the Lyapunov exponent exhibited re-entrant behavior: it vanished at zero or sufficiently weak monitoring strength and became positive again as the monitoring strength increased. These results offer intriguing insights into the mechanisms leading to quantum scrambling, paving the way for its experimental control and potential enhancement in quantum information devices. The non-monotonic dependence of the decay rates on the thermal bath coupling and monitoring strength suggests that continuous monitoring can induce or enhance chaotic dynamics suppressed by the thermal bath under certain conditions.

Monitoring and thermal coupling jointly modulate quantum chaos in the SYK model, revealing intricate dynamics

Scientists have demonstrated that continuous monitoring can, counterintuitively, induce or enhance chaotic dynamics in quantum systems, even when those systems are subject to thermal dissipation. This research investigates the behaviour of a monitored Sachdev-Ye-Kitaev (SYK) model coupled to a thermal environment, revealing that the combined effect of monitoring and the thermal bath drives the system towards a unique, non-thermal steady state regardless of its initial conditions.

Analysis of the system’s retarded Green’s function shows two distinct stages of exponential decay, with decay rates influenced by both the thermal bath coupling and the monitoring strength. Notably, the Lyapunov exponent, a measure of chaotic behaviour, increases when monitoring is initiated in weakly coupled systems, and exhibits re-entrant behaviour at intermediate thermal coupling strengths.

This means that the chaotic dynamics can be suppressed at weak monitoring, but then re-emerge as monitoring strength increases. While solutions with negative Lyapunov exponents, indicating suppressed quantum effects, were also observed, the findings highlight a surprising ability to enhance quantum chaos through continuous monitoring.

The authors acknowledge that their analysis is currently limited to the SYK model and a specific choice of measurement operators, and extending these results to more complex systems presents a significant computational challenge. Future research should focus on verifying the general applicability of these findings across different quantum models and exploring the potential for controlling quantum scrambling to improve information processing technologies.