Semiclassical time evolution of the density matrix and tunneling

The time dependent density matrix of a system with potential barrier is stu died using path integrals. The characterization of the initial state, which is assumed to be restricted to one side of the barrier, and the time evolu tion of the density matrix lead to a threefold path integral which is evalu ated in the semiclassical limit. The semiclassical trajectories are found t o move in the complex coordinate plane and barrier penetration only arises due to fluctuations. Both the form of the semiclassical paths and the relev ant fluctuations change significantly as a function of temperature. The sem iclassical analysis leads to a detailed picture of barrier penetration in t he real time domain and the changeover from thermal activation to quantum t unneling. Deep tunneling is associated with quasizero modes in the fluctuat ion spectrum about the semiclassical orbits in the long time limit. The con nection between this real time description of tunneling and the standard im aginary time instanton approach is established. Specific results are given for a double well potential and an Eckart barrier.