Dissipative Quantum Systems with Potential Barrier. General Theory and Parabolic Barrier

J. Ankerhold, H. Grabert, and G.-L. Ingold

Phys. Rev. E 51, 4267–4281 (1995) DOI: 10.1103/PhysRevE.51.4267

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 calculated explicitly near the barrier top in the temperature region where quantum effects become important. It is shown that a quasi-stationary state exists which describes a constant flux across the potential barrier and generalizes the classical Kramers steady state solution. This quantum flux state depends only on the parabolic approximation of the anharmonic barrier potential near the top. The parameter range where the solution is valid is investigated in detail. In particular, assuming that the flux state matches onto the equilibrium state on one side of the barrier within the harmonic range of the potential we gain a condition for the damping strength which reduces for very high temperatures to the result well-known from classical rate theory. Within the specified parameter range the decay rate out of a metastable state is calculated using the flux solution. It is equivalent with results of purely thermodynamic methods for primarily thermally activated barrier crossing and moderate to strong damping. The presented real time approach can be extended to lower temperatures and smaller damping.