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.
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.
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) quantifies information scrambling in quantum systems and serves as a key diagnostic of quantum chaos. In one-body systems with a classical counterpart, the relaxation of the OTOC is governed by Ruelle-Pollicott resonances. For many-body systems lacking a semiclassical limit, recent studies have identified an analogous role played by the Liouvillian spectrum of weakly open extensions of the dynamics, where the slowest decay rate-the Liouvillian gap-encodes relaxation. Here we study the kicked Ising spin chain and show that the long-time exponential decay of the OTOC in the isolated system occurs at a rate equal to twice this intrinsic gap. This correspondence is demonstrated across parameter regions exhibiting distinct level spacing statistics, indicating that the Liouvillian spectrum provides a robust framework for characterizing relaxation and irreversibility in closed many-body quantum systems.
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at t<t_{E} to its monotonic decrease with time at t>t_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator, C(t), for the classical and quantum kicked rotor -- a textbook driven chaotic system -- and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength, where no classical chaos exists globally. In this case, the Lyapunov exponent vanishes, while the OTOC's growth rate may remain finite corresponding to disconnected local chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time tE: transitioning from a time-independent value of t−1lnC(t) at ttE. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.
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.
Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time, but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent -- defined by the evolution of the 4-point out-of-time-order correlator (OTOC) -- of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that {\em i)} in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε1/3, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε−1). {\em ii)} in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and {\em iii)} for sufficiently small perturbations the ε1/3 dependence is also suppressed -- another purely quantum effect which we explain. Several numerical examples which demonstrate the theoretical predictions are given. The implication for the results to the behavior of real near-integrable systems, and for quantum limits on chaos are briefly discussed.