Complexity of quantum motion and quantum-classical correspondence: A phase-space approach

Abstract

We discuss the connection between the out-of-time-ordered correlator and the number of harmonics of the phase-space Wigner distribution function. In particular, we show that both quantities grow exponentially for chaotic dynamics, with a rate determined by the largest Lyapunov exponent of the underlying classical dynamics, and algebraically—linearly or quadratically—for integrable dynamics. It is then possible to use such quantities to detect in the time domain the integrability-to-chaos crossover in many-body quantum systems.