Quasi-integrable systems are slow to thermalize but may be good scramblers

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time-the clearest example being the Solar System-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 four-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. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one: it scales as epsilon(1/3), with epsilon being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is similar to epsilon(-1)). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the epsilon(1/3) dependence is also suppressed-another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/h may stay finite at a low temperature T.