Quantum Lyapunov exponent in dissipative systems

TL;DR

This paper investigates the behavior of the out-of-time order correlator (OTOC) in dissipative quantum systems, revealing its relation to classical Lyapunov exponents and how noise influences its decay, bridging quantum chaos and classical dynamics.

Contribution

It introduces a novel analysis of OTOC decay in open quantum systems with dissipation, connecting quantum decay rates to classical Lyapunov exponents and noise effects.

Findings

OTOC decay rate relates to classical Lyapunov exponent

OTOC is more sensitive than other measures in distinguishing chaos

Adding Gaussian noise recovers classical decay behavior

Abstract

The out-of-time order correlator (OTOC) has been widely studied in closed quantum systems. However, there are very few studies for open systems and they are mainly focused on isolating the effects of scrambling from those of decoherence. Adopting a different point of view, we study the interplay between these two processes. This proves crucial in order to explain the OTOC behavior when a phase space contracting dissipation is present, ubiquitous not only in real life quantum devices but in the dynamical systems area. The OTOC decay rate is closely related to the classical Lyapunov exponent -- with some differences -- and more sensitive in order to distinguish the chaotic from the regular behavior than other measures. On the other hand, it reveals as a generally simple function of the longest lived eigenvalues of the quantum evolution operator. We find no simple connection with the Ruelle-Pollicott resonances, but by adding Gaussian noise of $\hbar_{\text{eff}}$ size to the classical system we recover the OTOC decay rate, being this a consequence of the correspondence principle put forward in [Physical Review Letters 108 210605 (2012) and Physical Review E 99 042214 (2019)]