Julia sets and chaotic tunneling: II

Chaotic tunneling is studied based on the complex semiclassical theory as a continuation of the previous paper (Shudo et al 2009 J. Phys. A: Math. Theor. 42 265101). In this paper, the nature of complex classical trajectories controlling chaotic tunneling is investigated. Combining the results of numerical investigations and rigorous mathematical considerations based on the theory of complex dynamical systems leads us to a fundamental mathematical theorem which relates the set of complex classical trajectories contributing to the tunneling probability, called the Laputa chains, and the chaotic component in the complexified phase space called the Julia set. In particular, we demonstrate that the mechanism for tunneling in non-integrable systems is controlled by a dense set of trajectories. This mechanism is radically different from the integrable system where a sparse set of instantons on invariant tori controls tunneling. The physical significance of claims in the fundamental mathematical theorem is numerically examined in detail. On the basis of the numerical studies and the ergodic nature of the Julia set, we finally propose a hypothesis which guarantees the existence of complexified trajectories contributing to the tunneling process in non-integrable systems. The hypothesis supports the following picture of chaotic tunneling: tunneling trajectories pass the dynamical barriers in the real space with the guidance of the stable and unstable sets of the Julia set in the complex space.