Equilibration of quantum chaotic systems

TL;DR

This paper investigates the applicability of the quantum ergodic theorem to quantum chaotic systems, demonstrating through numerical examples that such systems relax to high-entropy equilibrium states with quantum-specific fluctuation characteristics.

Contribution

The paper provides the first numerical verification of the quantum ergodic theorem in quantum chaotic systems, highlighting differences in fluctuation behavior compared to classical systems.

Findings

Quantum chaotic systems relax to high-entropy equilibrium states.

Quantum fluctuations around equilibrium are exponential, unlike Gaussian classical fluctuations.

The equilibrium state resembles the classical micro-canonical ensemble.

Abstract

Quantum ergordic theorem for a large class of quantum systems was proved by von Neumann [Z. Phys. {\bf 57}, 30 (1929)] and again by Reimann [Phys. Rev. Lett. {\bf 101}, 190403 (2008)] in a more practical and well-defined form. However, it is not clear whether the theorem applies to quantum chaotic systems. With the rigorous proof still elusive, we illustrate and verify this theorem for quantum chaotic systems with examples. Our numerical results show that a quantum chaotic system with an initial low-entropy state will dynamically relax to a high-entropy state and reach equilibrium. The quantum equilibrium state reached after dynamical relaxation bears a remarkable resemblance to the classical micro-canonical ensemble. However, the fluctuations around equilibrium are distinct: the quantum fluctuations are exponential while the classical fluctuations are Gaussian.