The decay of a metastable state of a system coupled to a heat-bath environment is studied. A functional-integral method is presented allowing for the calculation of decay rates at finite temperatures and in the presence of dissipation. The theory is utilized to determine the rate for a wide range of parameters. The temperature extends from the region where the decay is thermally activated down to very low temperatures where the system decays by tunneling from its ground state in the metastable well. The range of damping parameters covers the region from weakly damped to heavily overdamped motions. It is found that the transition between thermally activated decay and tunneling occurs near a crossover temperature $T_0$ which decreases with increasing damping strength. Well above $T_0$ the rate follows the classical Arrhenius law where the preexponential factor is affected by the frequency-dependent damping. As $T_0$ is approached, quantum corrections to the classical rate formula become increasingly important. In the vicinity of $T_0$ the rate follows a scaling law describing the crossover between thermally activated and quantum-mechanical decay. In the region below $T_0$ the decay rate can be determined analytically only in limiting cases. For a system with Ohmic dissipation and a cubic potential, accurate numerical calculations are presented exhausting the range of parameters not covered by analytical results.

• Received 26 March 1987

DOI:https://doi.org/10.1103/PhysRevB.36.1931