Complexified stable and unstable manifolds and chaotic tunneling

Abstract

Quantum tunneling in nonintegrable systems is studied based on the complex semiclassical analysis in the time domain. The theory of complex dynamical systems predicts that complexified stable and unstable manifolds form the skeleton of the dynamics in phase space, and mixing and ergodicity hold even for the mixed system. After surveying some rigorous results derived for the Henon map, we numerically verify that some fundamental properties thus predicted can actually be observed, and then investigate natures of complexified stable and unstable manifolds since they guide the complex orbits controlling the tunneling transition in the mixed phase space. In particular, for a properly designed scattering map, we examine the imaginary action associated with the stable manifolds for the periodic orbits in the real plane and find that the orbits tending to the sticky zone of the KAM region provide the most dominant contribution in the semiclassical sum. It is worth emphasizing that the behavior of such dominant orbits is affected by the presence of chaos in the complex plane, which is an aspect far beyond the scope of the dynamics in the real plane.