Classical mechanism of multidimensional barrier tunneling

Multidimensionality of systems significantly affects on tunneling phenomena observed. In particular, if a system under consideration is classically nonintegrable, very complicated tunneling phenomena, so called chaotic tunneling, are observed. The aim of this short review is to explain the underlying classical mechanism of multidimensional barrier tunneling by using the semiclassical method based on classical dynamics extended to the complex domain, i.e. complex semiclassical method. The tunneling probability of multidimensional barrier systems is still well reproduced by using the complex semiclassical method even in the chaotic tunneling regime, in which a characteristic tunneling phenomenon, i.e. the fringed tunneling, is observed. However the classical trajectories guided by complexified stable and unstable manifolds dominantly contribute to the tunneling probability, which gives quite a different picture of the tunneling from that given by the ordinary instanton mechanism