Instantons revisited: dynamical tunnelling and resonant tunnelling

TL;DR

This paper develops a semiclassical method using complex trajectories to accurately analyze various tunnelling phenomena, including dynamical and resonant tunnelling, beyond traditional approaches.

Contribution

It introduces an instanton-like approach with complex-time paths that extends tunnelling analysis to complex scenarios where standard Wick rotation fails.

Findings

Accurately reproduces tunnelling effects in double-well potentials.

Extends applicability to dynamical and resonant tunnelling situations.

Discusses potential for multidimensional and chaotic tunnelling analysis.

Abstract

Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.