Quantum suppression of chaotic tunneling

The interplay between chaotic tunneling and dynamical localization in mixed phase space is investigated. Semiclassical analysis using complex classical orbits reveals that tunneling through torus regions and transport in chaotic regions are not independent processes, rather they are strongly correlated and described by complex orbits with both properties. This predicts a phenomenon analogous to the quantum suppression of classical diffusion: chaotic tunneling is suppressed as a result of dynamical localization in chaotic regions. This hypothesis is confirmed by numerical experiments where the effect of destructive interference is attenuated. Tunneling phenomena are purely quantum effects. Nevertheless its nature is strongly influenced by underlying classical dynamics. In particular nontrivial aspects appear in the dynamical tunneling in mixed-type phase space, in which quasi-periodic and chaotic trajectories coexist [1]. Tunneling transitions between quasi-doublet states are enhanced by chaotic states [2] and the existence of nonlinear resonances also leads to a qualitative change of tunneling processes [3, 4]. The problem of quantum tunneling in multidimensional systems or more specifically in nonintegrable systems first raised in [5] would be of fundamental importance in quantum mechanics, and an approach taken there has recently been extended in [6]. However, our understanding for multidimensional tunneling is still far from complete. A primary difficulty lies in the fact that dynamical tunneling in chaotic systems takes place in very complicated phase space; the structure of classical phase space itself is not an easily understandable object. Another reason would be that dynamical tunneling proceeds in complex environments in the sense of wave phenomena. Scarring [7] or dynamical localization [8, 9], or other types of invariant structure become sources of partial structures often observed in wave functions. It is not clear at all to what extent these various wave effects are independent of each other. These aspects make it difficult to evaluate the tunneling rate between torus to chaotic regions quantitatively. The trajectory description would be one of promising strategies to understand quantum phenomena of chaotic systems. In particular, the semiclassical analysis is now recognized as an efficient approach to this end, and there are indeed a bunch of numerical tests supporting its validity. Concerning dynamical tunneling, it was shown that the complex semiclassical theory works fairly well and explains the mechanism of tunneling penetration out of quasi-periodic regions to chaotic seas [10–12]. In such a treatment, Quantum suppression of chaotic tunneling 2 the classical dynamics is extended to complex phase space, and the trajectories on the Julia set is most responsible for reproducing tunneling wavefunctions [13]. The aim of the present Letter is, based on the arguments predicted by the complex trajectory description of chaotic tunneling, to show that dynamical tunneling in mixed phase space and dynamical localization are strongly correlated to each other, so that the destruction of coherence, or more precisely the destruction of destructive interference in chaotic regions, not only induces delocalization of wavefunction in chaotic regions, but also causes strong enhancement of tunneling transition. In other words, “genuine” chaotic tunneling is suppressed by dynamical localization in surrounding chaotic regions, which is entirely analogous mechanism as dynamical localization suppresses classical diffusion [8, 9]. The result must serve as further understandings of amphibious states [14, 15], recently discovered quantum states that show the failure of semiclassical wavefunction hypothesis [16]. We first provide evidence for why we can predict that chaotic tunneling is tightly correlated with the dynamical localization process. As found in [11, 13], an exponentially large number of complex orbits appear as the contributors of the timedomain semiclassical propagator, and they indeed have almost equal weights in the semiclassical sum, which means that chaotic tunneling occurs as a consequence of superposition of exponentially many component waves. To demonstrate it, we here employ an area-preserving map