Long time evolution for a one-dimensional Fokker-Planck process: Application to absorptive optical bistability

Using the Bonifacio-Lugiato mean field model of an absorptive bistable device, we discuss in some detail the long time dynamics and the influence of quantum noise on the metastable states of a bi-stable system. We develop our analysis along two seemingly different lines: first, by means of a suitable generalization and reinterpretation of a method due to Kramers, we obtain an explicit analytic expression for the escape time out of the metastable state, which is valid not only near steady state, but through the long time evolution of the system as well; next, we calculate the same escape time by the so-called mean first passage time method. The two results are found to differ by a numerical factor of two. In our discussion we have made an effort to explore details of both the original Kramers' scheme and of the first passage time method that needed clarification. We have also related the global long time evolution to the escape processes out of the metastable states.