Complex trajectories in chaotic dynamical tunneling

We develop the semiclassical method of complex trajectories in application tochaotic dynamical tunneling. First, we suggest a systematic numerical techniquefor obtaining complex tunneling trajectories by the gradual deformation of theclassical ones. This provides a natural classification of the tunnelingsolutions. Second, we present a heuristic procedure for sorting out the leastsuppressed trajectory. As an illustration, we apply our technique to theprocess of chaotic tunneling in a quantum mechanical model with two degrees offreedom. Our analysis reveals rich dynamics of the system. At the classicallevel, there exists an infinite set of unstable solutions forming a fractalstructure. This structure is inherited by the complex tunneling paths and playsthe central role in the semiclassical study. The process we consider exhibitsthe phenomenon of optimal tunneling: the suppression exponent of the tunnelingprobability has a local minimum at a certain energy which is thus (locally) theoptimal energy for tunneling. We test the proposed method by comparison of thesemiclassical results with the results of the exact quantum computations andfind a good agreement.Comment: 21 pages, RevTeX style, 15 figures. Journal version; abstract, introduction and discussion modified, references adde