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

TL;DR

This paper explores how the growth of phase-space harmonics and out-of-time-ordered correlators reveals the transition from integrable to chaotic behavior in quantum systems, linking quantum signatures to classical chaos indicators.

Contribution

It establishes a quantitative connection between phase-space harmonic growth, out-of-time-ordered correlators, and classical Lyapunov exponents, providing a new method to detect chaos in quantum systems.

Findings

Harmonics and correlators grow exponentially in chaotic systems.

Growth rates are determined by classical Lyapunov exponents.

The approach can identify integrability to chaos crossover in many-body systems.

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.