Dispersive optical bistability for large photon numbers and low cavity damping

For a quantum-mechanical model of dispersive optical bistability we derive in the classical limit, i.e., for large photon numbers, a two-variable Fokker-Planck equation. For low cavity damping this Fokker-Planck equation can be approximated by a one-variable Fokker-Planck equation which is solved numerically. If, in addition, the number of thermal photons is small compared with the number of photons inside the cavity, an analytical result for the transition rate is obtained.