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arXiv:2303.08047 · Physical review. E

Classical approach to equilibrium of out-of-time ordered correlators in mixed systems

Tomás Notenson, Ignacio García-Mata, Augusto J. Roncaglia, Diego A. Wisniacki

Published Jun 8, 2023• 10 citations
arXivHTMLPDF

Abstract

The out-of-time ordered correlator (OTOC) is a measure of scrambling of quantum information. Scrambling is intuitively considered to be a significant feature of chaotic systems, and thus, the OTOC is widely used as a measure of chaos. For short times exponential growth is related to the classical Lyapunov exponent, sometimes known as the butterfly effect. At long times the OTOC attains an average equilibrium value with possible oscillations. For fully chaotic systems the approach to the asymptotic regime is exponential, with a rate given by the classical Ruelle-Pollicott resonances. In this work, we extend this notion to the more generic case of systems with mixed dynamics, in particular using the standard map, and we are able to show that the relaxation to equilibrium of the OTOC is governed by generalized classical resonances.

DOI
10.1103/physreve.107.064207

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The out-of-time ordered correlator (OTOC) is a measure of scrambling of quantum information. Scrambling is intuitively considered to be a significant feature of chaotic systems, and thus, the OTOC is widely used as a measure of chaos. For short times exponential growth is related to the classical Lyapunov exponent, sometimes known as the butterfly effect. At long times the OTOC attains an average equilibrium value with possible oscillations. For fully chaotic systems the approach to the asymptotic regime is exponential, with a rate given by the classical Ruelle-Pollicott resonances. In this work, we extend this notion to the more generic case of systems with mixed dynamics, in particular using the standard map, and we are able to show that the relaxation to equilibrium of the OTOC is governed by generalized classical resonances.
Out-of-time-ordered correlators (OTOCs) measure the scrambling of information and are a powerful diagnostic of chaos. In chaotic quantum systems with a classical analog, their relaxation towards equilibrium is dictated by the classical Ruelle--Pollicott resonances. We investigate higher-order OTOCs in quantum maps, prototypical one-body systems with a classical limit showing all the essential features of chaotic dynamics. We demonstrate numerically that in the intermediate-time regime they decay exponentially at a rate that depends only on the parity of the order: odd-order OTOCs decay at the same rate as classical correlations, given by the leading Ruelle--Pollicott resonance, and even-order ones at twice that rate. Both are far slower than the naive expectation, under which the rate would grow in proportion to the order. We then mimic the effect of an environment by defining a coarse-grained superoperator whose effect is to introduce decoherence into the system, and find that the expected proportionality between rate and order is thereby restored. Finally, we show that at long times each higher-order OTOC becomes dominated by a single free cumulant, whose saturation value falls off inversely with the dimension of the Hilbert space, signalling the emergence of asymptotic freeness and highlighting the relevance of free probability theory as a framework for the study of quantum chaos.
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 is revealed 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 ℏ_{eff} size to the classical system we recover the OTOC decay rate, which is a consequence of the correspondence principle put forward in Phys. Rev. Lett. 108, 210605 (2012)10.1103/PhysRevLett.108.210605 and Phys. Rev. E 99, 042214 (2019)10.1103/PhysRevE.99.042214.
Two properties are needed for a classical system to be chaotic: exponential stretching and mixing. Recently, out-of-time order correlators were proposed as a measure of chaos in a wide range of physical systems. While most of the attention has previously been devoted to the short time stretching aspect of chaos, characterized by the Lyapunov exponent, we show for quantum maps that the out-of-time correlator approaches its stationary value exponentially with a rate determined by the Ruelle-Pollicot resonances. This property constitutes clear evidence of the dual role of the underlying classical chaos dictating the behavior of the correlator at different timescales.
We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify "information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into "regular" and "chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.
In classical dynamical systems, chaotic behavior is often associated with exponential sensitivity to initial conditions together with global phase-space structure. Translating this geometric concept to the strictly linear framework of quantum mechanics presents a conceptual puzzle. The out-of-time-ordered correlator (OTOC) is often motivated as the quantum analogue of the classical butterfly effect, but this slogan can hide important mathematical distinctions. This tutorial bridges the gap between applied mathematics and quantum information by detailing the mathematical machinery of the OTOC. We explore how classical sensitivity translates to operator non-commutativity, why standard two-point correlation functions fail to cleanly detect this sensitivity, and how the delocalization of quantum observables relates to classical notions of mixing. Crucially, we outline what the OTOC can and cannot diagnose, distinguishing between local instability and global chaos. Ultimately, we provide a precise and usable conceptual map, exploring how the Koopman-von Neumann formalism offers a framework to view classical and quantum dynamics through a shared linear perspective.
In classical dynamical systems, chaotic behavior is often associated with exponential sensitivity to initial conditions together with global phase-space structure. Translating this geometric concept to the strictly linear framework of quantum mechanics presents a conceptual puzzle. The Out-of-Time-Ordered Correlator (OTOC) is often motivated as the quantum analog of the classical butterfly effect, but this slogan can hide important mathematical distinctions. This tutorial bridges the gap between applied mathematics and quantum information by detailing the mathematical machinery of the OTOC. We explore how classical sensitivity translates to operator noncommutativity, why standard two-point correlation functions fail to cleanly detect this sensitivity, and how the delocalization of quantum observables relates to the classical notion of mixing. Crucially, we outline what the OTOC can and cannot diagnose, distinguishing between local instability and global chaos. Ultimately, we provide a precise and usable conceptual map, exploring how the Koopman–von Neumann formalism offers a framework to view classical and quantum dynamics through a shared linear perspective.