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 most clear 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 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 i) in the semi-classical 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 $\sim \epsilon^{-1}$). ii) in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and iii) 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 $kT/\hbar$ may stay finite at a low temperature $T$.