Dynamical tunneling-like effects in 1D classical systems

Dynamical tunneling occurs when a particle tunnels between two distinct classically trapped periodic regions of classical phase space that are not separated by a potential barrier. Although the dynamical tunneling has been observed in many multi-dimensional Hamiltonian systems, it has not been observed in 1D systems described by a single potential. In this paper, we show that classical trajectories of real potentials such as V1(x) = x4 exhibit dynamical tunneling-like behavior when energy or time is complex. It was found that the doubly periodic nature of the Jacobian elliptic functions is responsible for this dynamical tunneling-like behavior. The time spent in one region by the tunneling trajectory before crossing over to the other is found to be proportional to |(E03/4)/ΔE, where total energy E = E0 + iΔE with E0 < 0. Furthermore, we demonstrate that classical trajectories of the non-Hermitian system V2(x) = x4 + (1 + i)x show evidence of dynamical tunneling even for real energies. The role of complex time in dynamical tunneling is discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'. (paper)