# Quantum Tunneling and Complex Dynamics in the Suris’s Integrable Map

## TL;DR

The paper explores quantum tunneling in an integrable map and finds subtle differences in tunneling behavior due to complex dynamics.

## Contribution

The study reveals that complex classical dynamics influence quantum tunneling effects not captured in real-plane analysis.

## Key findings

- Tunneling splitting in the integrable map and Hamiltonian system shows similar behavior with slight magnitude differences.
- Wave function tunneling tails differ significantly due to complex classical dynamics and branch points in the potential.
- Branch points in the potential function contribute to non-trivial tunneling tail behavior.

## Abstract

Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant differences. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role of yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.