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.

[1]  Luigi A. Lugiato,et al.  Time Evolution of an Optically Bistable System- Local Relaxation and Tunneling , 1980 .

[2]  L. Lugiato,et al.  Quantum-statistical treatment of the transient in absorptive optical bistability—local relaxation , 1980 .

[3]  D. Walls,et al.  Quantum theory of optical bistability. I. Nonlinear polarisability model , 1980 .

[4]  R. Gilmore,et al.  Catastrophe time scales and conventions , 1979 .

[5]  Dynamic behaviour of optical bistability , 1979 .

[6]  H. D. Vollmer,et al.  One-dimensional diffusion in soluble model potentials , 1979 .

[7]  W. Schieve,et al.  Stochastic-master-equation description of an optical bistability: A comparison with the microscopic theory , 1979 .

[8]  L. Lugiato.  Spectrum of transmitted and fluorescent light in absorptive optical bistability , 1979 .

[9]  H. Brand,et al.  Multiplicative stochastic processes in statistical physics , 1979 .

[10]  E. Garmire,et al.  Bistable Optical Devices: An Overview , 1979 .

[11]  H. Brand,et al.  Fluctuations in optical bistable systems , 1978 .

[12]  Photon statistics of a bistable absorber , 1978 .

[13]  H. Brand,et al.  A soluble stochastic model for first-order type nonequilibrium phase transitions , 1978 .

[14]  Derivation of a master equation for optical bistability , 1978 .

[15]  L. Lugiato,et al.  Photon Statistics and Spectrum of Transmitted Light in Optical Bistability , 1978 .

[16]  L. Lugiato,et al.  Instabilities for a coherently driven absorber in a ring cavity , 1978 .

[17]  L. Lugiato,et al.  Bistable absorption in a ring cavity , 1978 .

[18]  R. Gilmore,et al.  Dynamical approach to steady state and fluctuations in optically bistable systems , 1978 .

[19]  C. R. Willis.  Cooperative effects and fluctuations in optical bistability , 1977 .

[20]  N. Kampen,et al.  A soluble model for diffusion in a bistable potential , 1977 .

[21]  H. M. Gibbs,et al.  Differential Gain and Bistability Using a Sodium-Filled Fabry-Perot Interferometer , 1976 .

[22]  Irwin Oppenheim,et al.  Stochastic theory of multistate relaxation processes , 1967 .

[23]  R. Landauer.  Fluctuations in Bistable Tunnel Diode Circuits , 1962 .

[24]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .