Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Time-dependent fluctuations in optical bistability

Andrzej Zardecki

Published Oct 1, 1980• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.22.1664

Related papers

Time-dependent fluctuations and phase hysteresis in dispersive bistability

Jacek Chrostowski, Andrzej Zardecki, Claude A. Delisle

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.

Physical Review A•Jul 1, 1981•11 citations

Fluctuations of optically bistable systems in the bad-cavity limit

Andrzej Zardecki

In the limit of large field-relaxation rates as compared to the rates of collective and atomic relaxation, we model the stochastic evolution of a bistable system with the aid of the Langevin equations for atomic variables. This semiclassical approach leads to a Fokker-Planck equation for the joint probability distribution of the atomic polarization and inversion. The Monte Carlo integration scheme, consisting in construction of approximate random trajectories, enables us to evaluate numerically the adiabatic values of the output field, intensity, and intensity fluctuations as functions of time. The final model used in numerical work is equivalent to ignoring microscopic fluctuations and considering delta-correlated external fluctuations impressed on the driving field.

Physical Review A•Mar 1, 1981•12 citations

Quantum-statistical treatment of the transient in absorptive optical bistability—local relaxation

Luigi A. Lugiato, James D. Farina, Lorenzo M. Narducci

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.

Physical Review A•Jul 1, 1980•25 citations

Effect of laser amplitude and phase fluctuations on optical bistability

C. R. Willis

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-$Q$ 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.

Physical Review A•Feb 1, 1984•7 citations

Acousto-optic bistability with fluctuations

Michael F. Wehner, Jacek Chrostowski, W. J. Mielniczuk

A newly developed numerical method of solving nonlinear Fokker-Planck equations is applied to study an acousto-optic bistability with additive noise. The time evolution of the probability distribution for the transmitted light is calculated for a double-well potential. The time-dependent moments are calculated.

Physical Review A•Jun 1, 1984•8 citations

Absorptive Optical Bistability with Laser Amplitude Fluctuations

Marek Kuś, K. Wódkiewicz, Jason A. C. Gallas

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.

Topical Meeting on Optical Bistability (OB)•Jan 1, 1983•0 citations
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.
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.
In the limit of large field-relaxation rates as compared to the rates of collective and atomic relaxation, we model the stochastic evolution of a bistable system with the aid of the Langevin equations for atomic variables. This semiclassical approach leads to a Fokker-Planck equation for the joint probability distribution of the atomic polarization and inversion. The Monte Carlo integration scheme, consisting in construction of approximate random trajectories, enables us to evaluate numerically the adiabatic values of the output field, intensity, and intensity fluctuations as functions of time. The final model used in numerical work is equivalent to ignoring microscopic fluctuations and considering delta-correlated external fluctuations impressed on the driving field.
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.
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.
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.