Genuine Quantum Chaos and Physical Distance Between Quantum States

TL;DR

This paper introduces a new physical distance measure between quantum states that reveals genuine quantum chaos, allowing the identification of chaotic and regular regions in quantum systems.

Contribution

It proposes a novel physical distance between quantum states that uncovers quantum chaos despite the linearity of quantum mechanics.

Findings

Physical distance can be very small between orthogonal states

Quantum Lyapunov exponent and chaos measure are defined using this distance

Quantum Poincaré sections distinguish regular and chaotic dynamics

Abstract

We show that there is genuine quantum chaos despite that quantum dynamics is linear. This is revealed by introducing a physical distance between two quantum states. Qualitatively different from existing distances for quantum states, for example, the Fubini-Study distance, the physical distance between two mutually orthogonal quantum states can be very small. As a result, two quantum states, which are initially very close by physical distance, can diverge from each other during the ensuing quantum dynamical evolution. We are able to use physical distance to define quantum Lyaponov exponent and quantum chaos measure. The latter leads to quantum analogue of the classical Poincar\'e section, which maps out the regions where quantum dynamics is regular and the regions where quantum dynamics is chaotic. Three different systems, kicked rotor, three-site Bose-Hubbard model, and spin-1/2 XXZ model, are used to illustrate our results.