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Physical Review A

Time-dependent fluctuations and phase hysteresis in dispersive bistability

Jacek Chrostowski, Andrzej Zardecki, Claude A. Delisle

Published Jul 1, 1981• 11 citations
Publisher

Abstract

Dispersive optical bistability with fluctuations is analyzed in the mean-field approximation. It is shown that the phase shift of the transmitted field in the purely dispersive case exhibits hysteresis and bistable behavior in a manner similar to that of the transmitted intensity. The Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted-field amplitude is obtained. The time-dependent moments of the transmitted light are calculated with the use of a Monte Carlo integration scheme.

DOI
10.1103/physreva.24.345

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Physical Review A•May 1, 1980•15 citations
Dispersive optical bistability with fluctuations is analyzed in the mean-field approximation. It is shown that the phase shift of the transmitted field in the purely dispersive case exhibits hysteresis and bistable behavior in a manner similar to that of the transmitted intensity. The Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted-field amplitude is obtained. The time-dependent moments of the transmitted light are calculated with the use of a Monte Carlo integration scheme.
On the basis of the absorptive bistability model, including the fluctuations, we rederive in the good-cavity limit the Fokker-Planck equation describing the time evolution of the quasiprobability distribution for the transmitted field amplitude. It is shown that on a semiclassical level different variants of the Fokker-Planck equation are possible. The equation of Bonifacio, Gronchi, and Lugiato is solved in terms of a path integral which involves a generalized Wiener measure. In order to calculate the averages of transmitted field moments as functions of time, a Monte Carlo integration scheme is used. This scheme consists in constructing random trajectories which result from the Langevin equation statistically equivalent to the derived Fokker-Planck equation. Numerical results in the bistable domain are presented for large values of the fluctuation parameter.
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.
Systems exhibiting dispersive optical bistability are considered as examples of nonequilibrium steady states lacking detailed balance. Methods for describing such states are developed. The Fokker-Planck equation for the steady-state distribution function of the transmitted electromagnetic field is derived. In the limit of small fluctuations the Fokker-Planck equation is reduced to the form of a Hamilton-Jacobi equation for a "nonequilibrium thermodynamic potential." This equation is solved in various approximations. The nonequilibrium thermodynamic potential acts like a free-energy for the first-order-type transition far from thermodynamic equilibrium lacking detailed balance. The potential entirely determines the steady-state distribution, the coexistence curve in the bistable domain, a Lyapunoff function of the deterministic equations of motion, and allows us to cast the deterministic equations of motion into the standard form of nonequilibrium thermodynamics.
A generalized Fokker-Planck equation is derived for a system undergoing optical bistability when the dominant source of fluctuations is due to fluctuations in the incident field rather than spontaneous emission. The fluctuations are treated as a time-dependent Gaussian process whose properties are determined by the individual laser's characteristics. Both amplitude and phase fluctuations are included on the incident laser field. The effect of fluctuations of the incident laser on optical bistability are different for amplitude and phase fluctuations and are different for absorptive and dispersive optical bistability. The combination of incident laser phase fluctuations and dispersive optical bistability leads to large amplitude fluctuations near the turning points of the optical bistability curve and causes the system to make a transition from one branch of the optical bistability curve to the other before reaching the mean-field turning points. We take the high-QQQ cavity limit of our generalized Fokker-Planck equation, solve the linearized equations, and provide criteria for the magnitude of fluctuations in optical bistability in terms of laser and cavity parameters. The magnitude of incident laser fluctuations are typically orders of magnitude larger than fluctuations due to spontaneous emission and are therefore important for practical applications of optical bistability devices.
The purpose of this work is to study the behavior of atomic fluctuations and the spectrum of transmitted light by a nonlinear medium with absorption and dispersion. The authors make use of the system-size expansion and apply it to the quantum-mechanical Langevin's equations for the atomic operators and obtain the characteristic curves for bistable behavior, macroscopic atomic expectation values, and the spectrum of the atomic fluctuations. The calculated spectrum of transmitted light exhibits line narrowing near bistable thresholds, discontinuous formation of bands along the high-transmission bistable branch, and hysteresis effects, etc., for detuning $|\ensuremath{\delta}|$ less than a critical value ${\ensuremath{\delta}}_{c}$. For $|\ensuremath{\delta}|>{\ensuremath{\delta}}_{c}$, the discontinuous character of all the physical variables of the system vanishes; the system no longer exhibits bistable behavior. The system, however, still shows cooperative and single-atom behavior separated by a critical value of the intensity of the incident coherent pump at which a critical slowing down of the medium occurs, exhibiting a phase transition.