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Physical Review A

Transient behavior in absorptive optical bistability by the Hamilton-Jacobi method

Sarben Sarkar, J.S. Satchell

Published Apr 1, 1986• 3 citations
Publisher

Abstract

One-, two-, and five-dimensional Fokker-Planck equations for absorptive bistability are solved with use of small-noise asymptotic expansions, which are different from Gaussian linearized analysis. The cases studied are the bifurcation point for the start of hysteresis, where there is critical slowing down and the fluctuations are large, and the evolution of a steady-state distribution when the input field has a step change. The time evolution of the probability distribution is calculated.

DOI
10.1103/physreva.33.2517

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One-, two-, and five-dimensional Fokker-Planck equations for absorptive bistability are solved with use of small-noise asymptotic expansions, which are different from Gaussian linearized analysis. The cases studied are the bifurcation point for the start of hysteresis, where there is critical slowing down and the fluctuations are large, and the evolution of a steady-state distribution when the input field has a step change. The time evolution of the probability distribution is calculated.
We solve numerically the simplest Fokker-Planck equation which describes the effects of amplitude noise in absorbitive optical bistability. When the semiclassical solution exhibits critical slowing down, the probability distribution becomes two-peaked in a sizable time interval during the approach to the one-peaked steady-state distribution. The switching time undergoes considerable fluctuations, and on the average it is shorter than predicted by semiclassical theory. These noise effects seem accessible to experimental observation.
In this paper we discuss the influence of laser fluctuations on the hysteresis cycle in Optical Bistability. For the simplest case of an Absorptive Optical Bistability in a good cavity driven by an external laser light with non-white Gaussian amplitude fluctuations we derive a Fokker-Planck equation in the limit of large laser linewidth and good amplitude stabilization. This Fokker-Planck equation has a nonconstant diffusion term which shifts the most probable values of the steady state probability distribution from its deterministic positions. From the form of the steady state solution of the Fokker-Planck equation with non-white amplitude fluctuations we discuss the relative stability of the bistable branches. As in the case of quantum fluctuations the random amplitude of the laser field leads to a small range of values of the injected laser intensity in which the two bistable peaks have a comparable area. The transmitted field averaged with respect to the stationary distribution function coincides with one of the two deterministic branches except on the narrow transition region where we have large fluctuations.
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.
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