Quantum Lyapunov exponents beyond continuous measurements

Abstract

Quantum systems, when interacting with their environments, may exhibit non-equilibrium states that are tempting to be interpreted as quantum analogs of chaotic attractors. However, different from the Hamiltonian case, the toolbox for quantifying dissipative quantum chaos remains limited. In particular, quantum generalizations of Lyapunov exponents, the main quantifiers of classical chaos, are established only within the framework of continuous measurements. We propose an alternative generalization based on the unraveling of quantum master equation into an ensemble of 'quantum trajectories', by using the so-called Monte Carlo wave-function method. We illustrate the idea with a periodically modulated open quantum dimer and demonstrate that the transition to quantum chaos matches the period-doubling route to chaos in the corresponding mean-field system.It is one of the pillar concepts of Chaos theory that complex deterministic dynamics is rooted in the local instability which forces two initially close trajectories to diverge. This divergence is conventionally quantified with Lyapunov exponents (LEs), a powerful tool to quantify dynamical chaos. The history of attempts to generalize LEs to quantum dynamics is nearly as old as the history of Quantum Chaos. Most of this history is about the Hamiltonian limit, where the spectral theory of Quantum Chaos [1] was established first. The corresponding generalizations range from early ideas to use quasi-probability functions and define quantum LEs in terms of a "distance" between them [2-4] to very recent advances based on out-of-time correlation functions [5][6][7]. When a quantum system is open and its dynamics is modeled with a quantum master equation [8], the evolution of the system's density operator can be unraveled into an ensemble of evolving trajectories, each one described by a wave function [8]. Dynamics of these wave functions is essentially stochastic; therefore, LEs could be introduced in a more intuitive way than in the Hamiltonian limit. But will so-defined exponents make sense? Here we define a particular type of quantum LEs and give a positive answer to this question. Since quantum trajectories [9] are not just a formal trick but a part of reality, e.g., in optical [10] and microwave [11] cavity systems, we believe that our results will be of interest to the theoreticians (and, hopefully, to the experimentalists) dealing with these systems.

Chaotic spin-photonic quantum states in an open periodically modulated cavity

“…To calculate the largest Lyapunov exponent (LE), we use the recently developed method based on a parallel evolution of fiducial and auxiliary trajectories, ψ f (t) and ψ a (t), under Eq. ( 5) 12 , in the spirit of the classical LE ideology 26 . The distance between the trajectories is calculated as the absolute difference between the two corresponding observables θ .…”

“…In our recent work 17 we showed that a photonic mode of an open and periodically modulated Kerr-nonlinear cavity can exhibit transitions from regular dynamics to chaos. A degree of chaos is quantified with quantum Lyapunov exponent 12 . These transitions are also associated with modification of the probability distribution of photon emission waiting times, which changes its intermediate asymptotic from the exponential (regular dynamics) to a power-law (chaos) decay.…”

Quantum Lyapunov exponents and complex spacing ratios: two measures of Dissipative Quantum Chaos

“…( 4), is initiated again, until the next quantum jump occurs, etc. The largest quantum Lyapunov exponent 8 is calculated as the average rate of exponential growth of the distance between the base ψ b (t) and perturbed ψ v (t) quantum trajectories that evolve according to the Eq. ( 4), in full analogy with the classical definition 23 .…”

“…In our recent works [8][9][10] , we proposed an alternative way to quantify the degree of chaos in the evolution determined by Lindbladian L . Namely, we introduced a quantum version of Lyapunov exponents (LEs) which is based on the idea of unraveling the master equation, Eqs.…”

“…The concept of Lyapunov exponents which allows to quantify instability of the system dynamics by following the corresponding trajectories in the system space space, is a pillar of classical Dissipative Chaos 1 . In a series of recent works, we generalized this concept to the dynamics of open quantum systems [8][9][10] . The idea is to unravel the evolution determined by a Markovian generator into an ensemble of quantum trajectories and then to use (in)stability characteristics of these trajectories as the basis to decide whether the original open quantum dynamics is regular (or chaotic).…”