Influence of classical resonances on chaotic tunneling

Abstract

Dynamical tunnelling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunnelling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunnelling yields good overall agreement with the quantum-tunnelling rates, and indicates that partial barriers within the chaos might play a prominent role.

Tractable Problems in Multidimensional Tunneling

“…This description is partly based on previous publications of ours [27,28,31,32,33], but contains also some new aspects and significant improvements especially concerning the determination of the matrix elements associated with resonance-induced couplings. Moreover, partial barriers in the chaotic domain are now incorporated into the scheme of resonance-assisted tunneling, which generally gives rises to an effective enhancement of the size of the regular islands under consideration.…”

“…[31,32,33,34] resonance-assisted couplings were incorporated in an approximate manner into the framework of chaos-assisted tunneling in order to provide a quantitative theory for the regular-to-chaotic coupling rate. In this context, it is assumed that the dominant coupling between regular states within and chaotic states outside the island is provided by the presence of a nonlinear resonance within the island.…”

“…A straightforward implementation of this idea yields good agreement with the exact tunneling rates as far as their average decay with 1/ in the deep semiclassical limit is concerned. Moreover, individual plateaus and peak structures could be traced back to the influence of specific nonlinear resonances, not only for double-well-type tunneling in closed or periodic systems [31,32,33], but also for tunneling-induced decay in open systems [34]. However, the predictive power of this method was still rather limited, insofar as individual tunneling rates at given system parameters could be over-or underestimated by many orders of magnitude.…”

“…Up to now, and in our previous publications [27,28,31,32,33], we completely neglected the action dependence of the coupling coefficients V k (I). This approximation should be justified in the semiclassical limit of extremely small , where resonance-assisted tunneling generally involves multiple coupling processes [28] and transitions across individual resonance chains are therefore expected to take place in their immediate vicinity in action space.…”

“…8.9 and 8.10 involve a few differences as compared to some of our previous publications [31,32,33,34]. To start with, (I) the action dependence of the coupling coefficients associated with the resonances has been included [see Eq.…”