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In my talk I would like to present the details of complex semiclassical description of chaotic tunneling in a quasi-stationary regime realized after a sufficiently long time evolution and to discuss why the complex classical dynamical theory is inevitable to understand the tunneling process in the mixed phase space. Introducing basic theoretical tools for the complex semiclassical analysis and using a simple quantum map modelling a typical situation of tunnelling transition into an open space, I discuss characteristic features of the set of tunnelling trajectories giving major contributions to the semiclassical propagator. To extract physically significant subset of trajectories from the whole tunneling set a basic hypothesis, which is called the prunning tree hypothesis, is proposed. Based upon the hypothesis, the tunnelling wavefunction is constructed systematically, and it is compared with the fully quantum result in the quasi-stationary regime. The effect of Maslov index on the complicated interference patterns inherent in chaotic tunneling tail, which exhibit a sensitive dependence on the Planck constant, is demonstrated. Finally we interpret the contributing tunnelling trajectories in terms of the notions developed in complex dynamical theory. Relation between the tunnalling set and Julia set which is the closure of complexified stable-unstable manifolds of periodic orbits is emphasized. In particular, heteroclinic entanglement leading to a chaotic itenerancy of the tunelling trajectory over the imaginary domain of phase space, which is closely linked to a hyerarchical structure of the tunneling set, controls the far-field tunneling amplitude. An efficient semiclassical approximation which well reproduces the far-field amplitude is also proposed. |
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