Chaos and Quantum Tunneling

TL;DR

This paper reviews dynamical tunneling in mixed Hamiltonian systems, exploring how chaos influences quantum tunneling and clarifying the conditions under which chaos enhances tunneling probabilities.

Contribution

It provides a comprehensive overview of phenomenological perspectives and classical mechanics approaches to dynamical tunneling, clarifying the chaos-tunneling relationship.

Findings

Chaos-assisted tunneling can enhance tunneling probabilities.

Resonance effects play a significant role in dynamical tunneling.

Classical mechanics extended into the complex domain offers new insights.

Abstract

In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.