Complex-classical mechanism of the tunnelling process in strongly coupled 1.5-dimensional barrier systems

The fringed tunnelling, which can be observed in strongly coupled 1.5-dimensional barrier systems as well as in autonomous two-dimensional barrier systems, is a manifestation of intrinsic multi-dimensional effects in the tunnelling process. In this paper, we investigate such an intrinsic multidimensional effect on the tunnelling by means of classical dynamical theory and semiclassical theory, which are extended to the complex domain. In particular, we clarify the underlying classical mechanism which enables multiple tunnelling trajectories to simultaneously contribute to the wavefunction, thereby resulting in the formation of the remarkable interference fringe on it. Theoretical analyses are carried out in the low-frequency regime based upon a complexified adiabatic tunnelling solution, together with the Melnikov method extended to the complex domain. These analyses reveal that the fringed tunnelling is a result of a heteroclinic-like entanglement between the complexified stable manifold of the barrier-top unstable periodic orbit and the incident beam set. Tunnelling particles reach the real phase plane very promptly, guided by the complexified stable manifold, which gives quite a different picture of the tunnelling from the ordinary instanton mechanism. More fundamentally, the entanglement is attributed to a divergent movement of movable singularities of the classical trajectory, namely, to a singular dependence of singularities on its initial condition.