Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

Abstract

For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonanceassisted enhancement of regular-to-chaotic tunneling in a non-perturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.

Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

“…Note, that opening the system in Λ leads to decay rates showing qualitatively the same results as the weights w 0 [48]. In experiments, the weight can be related to quality factors of optical modes in microcavities or to decay rates of resonance states.…”

“…However, due to tunneling, there is also non-vanishing probability in the other, classically forbidden, regions. We introduce the weight in these regions as a quantitative measure of resonance-assisted tunneling which shows similar characteristics as known from two dimensional quantum maps [48]. A perturbative description of resonance-assisted tunneling is obtained for both single coupled resonances and double resonances.…”

“…The effect of resonance-assisted tunneling can be measured in various ways. Following the studies of regularto-chaotic tunneling or partially open quantum systems we measure the weight of |ψ 0 on the opposite side of the resonance s = (s 1 , s 2 ), accounting for the geometry of phase space [48]. In order to determine this weight we formally follow Ref.…”

“…In order to determine this weight we formally follow Ref. [48] and introduce a region Λ in action space given by…”

“…As Λ is dynamically separated from the quantizing torus I 0 , the weight of the state |ψ 0 on Λ measures the strength of tunneling over the resonance channel. For the quantum system this translates into the projector [48]…”