Dissipative Quantum Systems with Potential Barrier . General Theory and Parabolic

We study the real time dynamics of a quantum system with potential barrier coupled to a heat-bath environment. Employing the path integral approach an evolution equation for the time dependent density matrix is derived. The time evolution is evaluated explicitly near the barrier top in the temperature region where quantum eeects become important. It is shown that there exists a quasi-stationary state with a constant ux across the potential barrier. This state generalizes the Kramers ux solution of the classical Fokker-Planck equation to the quantum regime. In the temperature range explored the quantum ux state depends only on the parabolic approximation of the anharmonic barrier potential near the top. The parameter range within which the solution is valid is investigated in detail. In particular, by matching the ux state onto the equilibrium state on one side of the barrier we gain a condition on the minimal damping strength. For very high temperatures this condition reduces to a known result from classical rate theory. Within the speciied parameter range the decay rate out of a metastable state is calculated from the ux solution. The rate is shown to coincide with the result of purely thermodynamic methods. The real time approach presented can be extended to lower temperatures and smaller damping.