2019/11/10 by Goldfriend, Tomer, Kurchan, Jorge
#Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1911.04011
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.