Activation rates in dispersive optical bistability with amplitude and phase fluctuations: A case without detailed balance

For the two-dimensional model of dispersive optical bistability put forward by Graham and Schenzle [Phys. Rev. A 23, 1302 (1981)] the activation rates of the metastable states at low noise are evaluated explicitly. The rates are calculated in terms of the mean first-passage time of a two-variable Fokker-Planck equation which does not obey detailed balance and which has a drift field not expressable as the gradient of the corresponding nonequilibrium potential. The forward rate, describing the transition from the state with low transmission to the state with high transmission, is exponentially decreased with increasing detuning ${\delta }^2$, whereas the backward rate is exponentially enhanced with increasing ${\delta }^2$. The prefactor of the rate exhibits a complicated dependence on the detuning parameter which for the absorptive case with zero detuning, $\delta =0\mathrm{}$, reduces to the familiar Kramers result of a Fokker-Planck system with a drift field derivable from a potential.