Lyapunov Exponent and Out-of-Time-Ordered Correlator’s Growth Rate in a Chaotic System

Abstract

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, → 0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the fourpoint 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 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 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 tE: transitioning from a time-independent value of t −1 ln C(t) at t < tE to its monotonous decrease with time at t > tE. 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.Introduction. -One of the central goals in the study of quantum chaos is to establish a correspondence principle between classical and quantum dynamics of classically chaotic systems [1][2][3][4][5][6][7]. Several previous works [7][8][9][10][11] have attempted to recover fingerprints of classical chaos in quantum dynamics. In particular, Aleiner and Larkin [12] showed the existence of a semiclassical "quantum chaotic" regime attributed to the delay in the onset of quantum effects (due to weak localization) revealing the key measure of classical chaos -the Lyapunov exponent (LE). Recently, the subject of quantum chaos has been revived by the discovery of an unexpected conjecture that puts a bound on the growth rate of an outof-time-ordered four-point correlator (OTOC) [13,14]. OTOC was first introduced by Larkin and Ovchinnikov to quantify the regime of validity of quasi-classical methods in the theory of superconductivity [15]. The growth rate of OTOC appears to be closely related to LE. Recent works have proposed experimental protocols to probe OTOC in cold atom and cavity QED setups [16]. Several recent preprints have employed OTOC as a probe to characterize many-body-localized systems [17].

Dynamical many-body localization in an integrable model

“…For example, a recent work [32] considers interacting kicked Dirac particles with individual Hamiltonians, H 0 = 2πασ x p + Mσ z , and provides a simple argument that this nonintegrable system also exhibits MBL. First, this model also exhibits localization when α's are generic distinct irrationals.…”

Out-of-time-order correlators in quantum mechanics

“…In fact, such a small time scale does not show up in ref. [41], in which the system without the boundary was considered. It has been also known for quantum fidelity or Loschmidt echo (see [42] for a review) that generally the time region for reproducing the classical Lyapunov behavior is quite limited.…”

“…As an example of time-dependent Hamiltonian systems, an OTOC for a kicked rotor system has been studied in ref. [41]. 4 If one can estimate the OTOC by an analytic way although it is very unlikely for ordinary chaotic systems, its exponential growth might be observed in a short time scale.…”

Onset of many-body chaos in the O(N) model

“…Furthermore, in systems with a semiclassical limit, the growth of scrambling, as measured by these special correlation functions, can be heuristically related to the growth of chaos in the corresponding classical model as measured by classical Lyapunov exponents [22]. However, there are some important subtleties with this connection and it remains incompletely understood [23]. Because the growth of scrambling describes the effective loss of memory of the initial state and because of its relation to growth of classical chaos, the onset of scrambling can be regarded as a fully quantum avatar of the growth of chaos.…”