A semiclassical approach to study multiplicative noise induced rate processes from a metastable state

The theory of stochastic dynamics of an inhomogeneous quantum system is being explored in this work by calculating the decay rate from a metastable state of a Brownian particle under the influence of quantum state-dependent friction and diffusion. We begin with the appropriate c-number description of the concerned quantum Langevin equation with a multiplicative noise. We put forth certain important consequences that are ubiquitous for the situation of dissipative decay out of a metastable state. We illustrate the situation vis-á-vis the classical result that the non-linear nature of system-bath coupling results in a Langevin equation with state dependent dissipation and multiplicative noise. In this article, we establish that this non-linear interaction leaves its signature on the pre-exponential factor of the rate constant by invoking a semiclassical treatment of the barrier-crossing dynamics with a multiplicative noise and state-dependent dissipation. We also highlight the fact that an additional exponential factor, other than the Arrhenius one, incorporates the temperature dependence, which is deemed to be an effect that purely stems out of quantization. This quantum effect, interestingly, is seen to be entangled with dissipation.