Stochastic resonance: Nonperturbative calculation of power spectra and residence-time distributions

We examine the response of a finite-temperature two-state system to periodic driving using time-dependent transition rate theory. This system can exhibit the phenomenon of stochastic resonance, where raising the temperature increases the signal-to-noise ratio of the response. We obtain the power spectrum and the distribution of residence times nonperturbatively for any transition rates that are periodic in time. Given the drive period ${\mathit{T}}_{\mathit{s}}$, the power spectrum is the Fourier transform of the sum of ‘‘signal,’’ which is periodic in time with period ${\mathit{T}}_{\mathit{s}}$, and ‘‘noise,’’ which is the product of an exponential and a function periodic with period ${\mathit{T}}_{\mathit{s}}$. The residence-time distribution is the product of an exponential and a function that is periodic with period ${\mathit{T}}_{\mathit{s}}$. Both the power spectrum and the residence-time distribution can be calculated exactly given the dependence of the transition rates on the control parameter (e.g., asymmetry or temperature). We calculate the characteristics of stochastic resonance for a two-state system with activated transition rates and for a quantum-mechanical dissipative two-level system.