Coherent Tunneling and Instantons in Presence of Classical Chaos

The influence of classical chaos on dynamical tunneling1) in the semiclassical regime of quantum mechanics was observed in the pioneer work by Lin and Ballentine.2) Namely, it was detected in the numerical experiment that regions of chaotic motion in the classical phase space of the model system (periodically driven quartic double well) can enhance the rate of quantum tunneling between KAM-islands in several orders of magnitude compared to the ordinary tunneling in completely integrable case. The coherent nature of the tunneling was also emphasized (no leakage of the probability into the chaotic zone). The opposite impact of classical chaos on dynamical tunneling was found in the Ref.3) On the example of the model system with double-well potential affected by the periodic in time perturbation it was numerically demonstrated that for specific parameter values the tunneling between symmetry related KAM-islands can be coherently destructed i.e. the wave packet can be almost completely localized in one of the wells. For the theoretical explanation of the chaos assisted tunneling the multi-level model ∗) was proposed and tested for the two coupled quartic oscillators.4) The explanation of the tunneling rate enhancement observed in the numerical simulations is the replacement of usual two-level tunneling dynamics by more complicated three-level tunneling mechanism, appeared at the avoided level crossing of primarily regular (quasi-)energy doublet and chaotic eigenstate.5) Detailed description of numerical technique which can be used for simulation of chaos assisted tunneling can be found in the Ref.6) Possible applications of the chaos assisted tunneling are the active product selection in chemical reactions and the dynamical tunneling control for various physical systems.7) The Refs.2), 3), 4), 5), 6), 7) are clearly indicate the existing priority of numerical simulations for the investigation of the chaos assisted tunneling regime. A very few works devoted to the analytical investigation of this phenomenon do exist. Namely, the path integral formalism was primarily used for the analytical description of the