Complex Semiclassical Description of Scattering Problem in Systems with 1.5 Degrees of Freedom

The tunneling effect in multidimensional systems may be greatly influenced by the underlying chaotic dynamics which is generic in more than one-dimensional systems. Aiming at a classical dynamical description of chaotic tunneling, we have developed in the present paper a basic formulation for the complex-domain semiclassical wave-matrix of systems with 1.5 degrees of freedom, namely, periodically time-dependent 1D-scattering systems, which can be regarded as a minimal model of multidimensional systems. Using an autonomous 2D system equivalent to the periodically time-dependent system, the semiclassical expression of the wave-matrix described in the 1D time-dependent picture is derived. In the latter half of the paper, our semiclassical formulation is examined with two simple examples, i.e., an oscillating wall and an oscillating barrier. The semiclassical wave-matrix is constructed, paying particular attention to some problems which emerge by extending classical dynamics to complex domain, and the results are compared with the fully quantum counterparts. The semiclassical results agree quite well with the corresponding quantal results if complex classical trajectories are fully taken into account. The relationship between the semiclassical wave-matrix and Miller's classical S-matrix is also discussed.