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Optical Engineering

Time Evolution of an Optically Bistable System- Local Relaxation and Tunneling

James D. Farina, Lorenzo M. Narducci, Jun‐Hui Yuan, L. A. Lugiato

Published Aug 1, 1980• 13 citations
Publisher

Abstract

After a brief review of some known steady state properties of an absorptive bistable system, we discuss in some details new aspects of the transient response that follows a sudden perturbation from steady state. The evolution of the perturbed bistable system towards its new steady state consists of two generally distinct phases: one is the local relaxation within each well of the bimodal potential, the second is the tunneling through the local maximum that separates the metastable and stable steady states. We give an analytic description of both phases of the evolution, and give explicit formulas for the local and global relaxation rates.

DOI
10.1117/12.7972545

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After a brief review of some known steady state properties of an absorptive bistable system, we discuss in some details new aspects of the transient response that follows a sudden perturbation from steady state. The evolution of the perturbed bistable system towards its new steady state consists of two generally distinct phases: one is the local relaxation within each well of the bimodal potential, the second is the tunneling through the local maximum that separates the metastable and stable steady states. We give an analytic description of both phases of the evolution, and give explicit formulas for the local and global relaxation rates.
A bistable absorber in the good-cavity limit and the Fokker-Planck equation describing the time evolution of the quasi-probability distribution for the transmitted field amplitude are considered. It is shown that the system evolves in two stages. The first consists of a local relaxation around the minimum (or minima) of a suitably defined potential. The second stage, which occurs over a much longer time scale, results from tunneling from one minimum to the other across the local maximum of the same potential. In this paper, only the first stage is analyzed. The time evolution of the expectation value of the transmitted field is described analytically. The accuracy of this approach is tested by numerical solutions of the Fokker-Planck equation.
We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter CCC. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.
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.
From a statistical viewpoint, the signature of optical bistability is the two-peaked character of its stationary probability distribution. However, the exceedingly long lifetime of the two metastable states makes the experimental observation very difficult. We recently showed (1) that the critical slowing down can produce the emergence of a double-peaked distribution during the transient, with a scenario that is accessible to experimental observation, a goal that has been brilliantly achieved by the Hannover group (2) using optically pumped sodium vapour. This phenomenon, called transient bimodality, is universal in the sense that it arises in all systems that, at a deterministic level, exhibit a long induction stage followed by a fast switching to the steady state (3). And, in fact, the Hannover team observed this phenomenon also in an electronic system (4).
This Letter comprises an experimental and theoretical investigation of the time evolution of a Fermi gas following fast and slow quenches of a one-dimensional optical double-well superlattice potential. We investigate both the local tunneling in the connected double wells and the global dynamics towards a steady state, i.e., a time-independent state. The local observables in the steady state resemble those of a thermal equilibrium state, whereas the global properties indicate a strong nonequilibrium situation.
The influence of noise on an optical bistable system in the vicinity of one of the bifurcation points is investigated by studying the transient response to a step input of light. The experiment is performed under absorptive conditions in a sodium-filled nonlinear Fabry-Perot in the bad cavity limit. Alternatively, a bistable electronic circuit (Schmitt trigger) is shown to exhibit the same type of behavior while admitting more detailed studies. On the monostable side of the bifurcation point, we observe a transient bimodality as well as a shift of the average switching delay to shorter times with increasing noise. Both results are in accord with a recent prediction by Broggi and Lugiato. We also present indications that the average switching time is smoothly connected with the lifetime of the metastable state on the bistable side of the bifurcation point.