On dynamical tunneling and classical resonances

TL;DR

This paper links classical nonlinear resonances to dynamical tunneling, showing that phase space structures fully characterize tunneling phenomena and discussing their relation to quantum superexchange in various dynamical regimes.

Contribution

It establishes a direct relationship between classical phase space resonances and dynamical tunneling, providing explicit forms of dynamical barriers and clarifying their quantum counterparts.

Findings

Classical resonance islands determine dynamical tunneling behavior.

Resonance-assisted tunneling can be approximated in near-integrable systems.

Avoided crossings are not necessarily related to nonlinear resonances.

Abstract

This work establishes a firm relationship between classical nonlinear resonances and the phenomenon of dynamical tunneling. It is shown that the classical phase space with its hierarchy of resonance islands completely characterizes dynamical tunneling and explicit forms of the dynamical barriers can be obtained only by identifying the key resonances. Relationship between the phase space viewpoint and the quantum mechanical superexchange approach is discussed in near-integrable and mixed regular-chaotic situations. For near-integrable systems with sufficient anharmonicity the effect of multiple resonances {\it i.e.,} resonance-assisted tunneling can be incorporated approximately. It is also argued that the, presumed, relation of avoided crossings to nonlinear resonances does not have to be invoked in order to understand dynamical tunneling. For molecules with low density of states the resonance-assisted mechanism is expected to be dominant.