Semiclassical description of resonance-assisted tunneling in one-dimensional integrable models

TL;DR

This paper develops a semiclassical framework using Hamiltonian normal forms and complex trajectories to accurately describe resonance-assisted tunneling in one-dimensional integrable models, matching numerical results.

Contribution

It introduces a systematic method to construct integrable models with resonance island chains and derives a semiclassical formula for tunneling splittings.

Findings

Analytical expression matches numerical splittings.

Method accurately controls size and position of resonance islands.

Provides a new semiclassical approach for resonance-assisted tunneling.

Abstract

Resonance-assisted tunneling is investigated within the framework of one-dimensional integrable systems. We present a systematic recipe, based on Hamiltonian normal forms, to construct one-dimensional integrable models that exhibit resonance island chain structures with accurately controlled sizes and positions of the islands. Using complex classical trajectories that evolve along suitably defined paths in the complex time domain, we construct a semiclassical theory of the resonance-assisted tunneling process. This semiclassical approach yields a compact analytical expression for tunneling-induced level splittings which is found to be in very good agreement with the exact splittings obtained through numerical diagonalisation.