Out-of-time-order Correlators and Chaos in Quantum Billiards

TL;DR

This paper investigates quantum chaos in billiard systems by comparing classical and quantum Lyapunov exponents derived from out-of-time-order correlators, revealing geometry-dependent behaviors and temperature effects.

Contribution

It demonstrates the agreement between classical and quantum Lyapunov exponents in billiards using OTOCs and uncovers unique low-temperature phenomena linked to billiard geometry.

Findings

Classical and quantum Lyapunov exponents agree in billiard systems.

Sinai billiard exhibits sharp OTOC growth at low temperatures.

Ground state geometry influences quantum chaos indicators.

Abstract

We examine three billiard systems -- the cardioid, diamond (Superman), and Sinai billiards -- all of which are known to be classically chaotic. We compute their classical Lyapunov exponents, and using out-of-time-order correlators (OTOCs) in the semi-classical regime, we also derive their quantum Lyapunov exponents. We observe that the classical and quantum Lyapunov exponents are in agreement, strengthening the role of OTOCs as a diagnostic for quantum chaos in billiard systems. At very low temperatures, the OTOC of the Sinai billiard shows sharp growth, a phenomenon absent in the other two billiards. We identify the source of this anomalous behaviour in the geometry of the ground state wave function of the Sinai billiard, which is more sensitive to the curvature of the billiard compared to the other billiards. We also remark on the late-time behaviour of the OTOCs and how the scrambling/Ehrenfest time is related to the temperature of quantum billiards.