Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime

Abstract

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given variant Planck's over 2pi regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.

Tractable Problems in Multidimensional Tunneling

“…This combination resulted, for the first time, in a semiclassical prediction of tunneling rates in generic mixed regular-chaotic systems that can be compared with the exact quantum rates on the level of individual peak structures [35]. This confirms the expectation that nonlinear resonances do indeed form the "backbone" behind non-monotonous substructures in tunneling rates.…”

“…And indeed very good agreement between this prediction and numerically computed tunneling rates was found for quantum maps that were designed such as to yield a "clean" mixed regular-chaotic phase space, containing a regular island and a chaotic region which both do not exhibit appreciable substructures [22], as well as for the mushroom billiard [23]. In more generic situations, where nonlinear resonances are manifested within the regular island, this approach yields reliable predictions for the direct tunneling of regular states at the regular-chaos border, and its combination with the resonance assisted mechanism described in the previous sections leads to good quantitative predictions for the tunneling rates for the states deep in the regular island [35]. We shall illustrate this on the example of the kicked rotor system in section 8.4.…”

“…A major advance in this context was achieved by improving the semiclassical evaluation of resonance-induced coupling processes in mixed systems, and by combining it with "direct" regular-to-chaotic tunneling [35]. This combination resulted, for the first time, in a semiclassical prediction of tunneling rates in generic mixed regular-chaotic systems that can be compared with the exact quantum rates on the level of individual peak structures [35].…”

“…Instead, we want to provide a simple, easy-to-implement, yet effective prescription how to compute the rates and time scales associated with tunneling processes solely on the basis of the classical dynamics of the system, without performing any diagonalization (not even any application) of the quantum Hamiltonian or of the time evolution operator. This prescription is based on chaos-and resonance-assisted tunneling in its improved form [35]. The main part of this contribution is therefore devoted to a detailed description of resonance-assisted tunneling and its combination with chaos-assisted tunneling in the sections 8.2 and 8.3, respectively.…”

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…Dynamical tunneling is a wave phenomenon which couples two distinct regions of raydynamical phase space (Davis and Heller, 1981); see also (Bäcker et al, 2008b;Löck et al, 2010). An example is the tunneling from a regular to the chaotic region in the phase space of the mushroom billiard, see Fig.…”