A semiclassical correlation function approach to barrier tunneling

Kin’ya Takahashi, Kensuke S. Ikeda

D. Sokolovski, J. N. L. Connor

Joachim Ankerhold, Hermann Grabert

The time dependent density matrix of a system with potential barrier is studied using path integrals. The characterization of the initial state, which is assumed to be restricted to one side of the barrier, and the time evolution of the density matrix lead to a threefold path integral which is evaluated in the semiclassical limit. The semiclassical trajectories are found to move in the complex coordinate plane and barrier penetration only arises due to fluctuations. Both the form of the semiclassical paths and the relevant fluctuations change significantly as a function of temperature. The semiclassical analysis leads to a detailed picture of barrier penetration in the real time domain and the changeover from thermal activation to quantum tunneling. Deep tunneling is associated with quasizero modes in the fluctuation spectrum about the semiclassical orbits in the long time limit. The connection between this real time description of tunneling and the standard imaginary time instanton approach is established. Specific results are given for a double well potential and an Eckart barrier.

Niall D. Whelan

Kenneth G. Kay

We develop a treatment that divides phase space into regions bounded by surfaces and forms the energy-dependent semiclassical Green's function from short classical trajectory segments confined to the individual regions. We use this to generalize Bogomolny's quantization method to systems with barrier tunneling. Combined with an appropriate treatment of tunneling amplitudes, this yields a unified semiclassical framework that uses ordinary, real trajectories in classically allowed regions and imaginary-time trajectories in forbidden regions to obtain energies, decay rates, and wave functions for arbitrary states of multidimensional systems undergoing tunneling. Because the trajectory segments in the allowed region are short, components in the Green's function expression can be numerically evaluated using semiclassical initial value methods, yielding a promising approach for practical calculations.

Neepa T. Maitra, Eric J. Heller