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

Dispersive optical bistability for large photon numbers and low cavity damping

Karl Vogel, H. Risken

Published Jul 1, 1990• 17 citations
Publisher

Abstract

For a quantum-mechanical model of dispersive optical bistability we derive in the classical limit, i.e., for large photon numbers, a two-variable Fokker-Planck equation. For low cavity damping this Fokker-Planck equation can be approximated by a one-variable Fokker-Planck equation which is solved numerically. If, in addition, the number of thermal photons is small compared with the number of photons inside the cavity, an analytical result for the transition rate is obtained.

DOI
10.1103/physreva.42.627

Related papers

Fokker-Planck equation in the good-cavity limit and single-atom optical bistability

Changxin Wang, Reeta Vyas

A generalized Fokker-Planck equation is used to study the problem of single-atom optical bistability in the good-cavity limit. The effects of quantum fluctuations are investigated by linearizing fluctuations around the steady-state value. It is shown that in the good-cavity limit, quantum fluctuations are relatively small, and hence there exist both absorptive and dispersive single-atom bistability. By comparing our single-atom Fokker-Planck equation with the N-atom Fokker-Planck equation, we find that in the good-cavity limit, these two different Fokker-Planck equations reduce to identical equations. We therefore come to an important conclusion that the results obtained from the many-atom Fokker-Planck equation in the good-cavity limit are also valid for a single atom. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Nov 1, 1996•21 citations

Quantum-tunneling rates and stationary solutions in dispersive optical bistability

Karl Vogel, H. Risken

For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.

Physical Review A•Sep 1, 1988•55 citations

Thermal and parametric noise in dispersive optical bistability I General formalism

L. A. Lugiato, R. J. Horowicz

We formulate a running-wave, single-mode model for purely dispersive optical bistability that incorporates amplitude and frequency fluctuations in the incident-field, cavity-length fluctuations, and thermal noise in the radiation field and in the material. This model is given by a set of three Langevin-type equations for the real and the imaginary parts of the slowly varying electric field and for the material variable. In the white-noise limit, it is equivalent to a Fokker–Planck equation in three variables. Even if our model can also describe a Kerr medium or a two-level system in a suitable range of parameters, we are interested mainly in the case of miniaturized all-optical bistable devices that utilize semiconductor dispersive media. In this situation, we can adiabatically eliminate the field variables and therefore reduce the problem to a single stochastic differential equation in one variable, which contains several terms of additive and multiplicative noise. We find that noise in the imaginary part of the slowly varying electric field does not contribute to this equation. In the white-noise case, our stochastic equation is equivalent to a Fokker–Planck equation in one variable, whereas in the case of colored noise we obtain a one-dimensional Fokker–Planck equation only in the limit of short correlation times. Finally, we calculate the stationary solution of the resulting Fokker–Planck equation. We show that the presence of colored noise narrows the peaks of the probability distribution at steady state with respect to the white-noise case.

Journal of the Optical Society of America B•Jun 1, 1985•19 citations

Quantum theory of optical bistability. III. Atomic fluorescence in a low- Q cavity

H. J. Carmichael, Daniel F. Walls, P. D. Drummond +1 more

We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-$Q$ cavity. From a linearized Fokker-Planck equation derived in the positive $P$ representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.

Physical Review A•Jun 1, 1983•35 citations

Quantum theory of optical bistability. II. Atomic fluorescence in a high- Q cavity

P. D. Drummond, Daniel F. Walls

A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized $P$ representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.

Physical Review A•May 1, 1981•194 citations

Noise Effects in Dispersive Optical Bistability

L. A. Lugiato, A. Colombo, G. Broggi +1 more

In a recent paper (1) we formulated a running wave, singlemode model for purely dispersive optical bistability, which incorporates amplitude and frequency fluctuations in the incident field, cavity length fluctuations, thermal noise in the radiation field and in the material. In the bad-cavity case of miniaturized bistable devices which utilize semiconductor dispersive media, our model reduces to a single stochastic differential equation for the material variable, which contains several terms of additive and multiplicative noise. In the white-noise case, it is equivalent to a Fokker-Planck equation in one variable, whereas in the case of colored noise it is equivalent to a multidimensional Fokker-Planck equation. On the basis of this model, we examined both problem of practical (P) and of fundamental (F) interest.

Topical Meeting on Optical Bistability (OB)•Jan 1, 1985•0 citations
For a quantum-mechanical model of dispersive optical bistability we derive in the classical limit, i.e., for large photon numbers, a two-variable Fokker-Planck equation. For low cavity damping this Fokker-Planck equation can be approximated by a one-variable Fokker-Planck equation which is solved numerically. If, in addition, the number of thermal photons is small compared with the number of photons inside the cavity, an analytical result for the transition rate is obtained.
A generalized Fokker-Planck equation is used to study the problem of single-atom optical bistability in the good-cavity limit. The effects of quantum fluctuations are investigated by linearizing fluctuations around the steady-state value. It is shown that in the good-cavity limit, quantum fluctuations are relatively small, and hence there exist both absorptive and dispersive single-atom bistability. By comparing our single-atom Fokker-Planck equation with the N-atom Fokker-Planck equation, we find that in the good-cavity limit, these two different Fokker-Planck equations reduce to identical equations. We therefore come to an important conclusion that the results obtained from the many-atom Fokker-Planck equation in the good-cavity limit are also valid for a single atom. \textcopyright{} 1996 The American Physical Society.
For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.
We formulate a running-wave, single-mode model for purely dispersive optical bistability that incorporates amplitude and frequency fluctuations in the incident-field, cavity-length fluctuations, and thermal noise in the radiation field and in the material. This model is given by a set of three Langevin-type equations for the real and the imaginary parts of the slowly varying electric field and for the material variable. In the white-noise limit, it is equivalent to a Fokker–Planck equation in three variables. Even if our model can also describe a Kerr medium or a two-level system in a suitable range of parameters, we are interested mainly in the case of miniaturized all-optical bistable devices that utilize semiconductor dispersive media. In this situation, we can adiabatically eliminate the field variables and therefore reduce the problem to a single stochastic differential equation in one variable, which contains several terms of additive and multiplicative noise. We find that noise in the imaginary part of the slowly varying electric field does not contribute to this equation. In the white-noise case, our stochastic equation is equivalent to a Fokker–Planck equation in one variable, whereas in the case of colored noise we obtain a one-dimensional Fokker–Planck equation only in the limit of short correlation times. Finally, we calculate the stationary solution of the resulting Fokker–Planck equation. We show that the presence of colored noise narrows the peaks of the probability distribution at steady state with respect to the white-noise case.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
A quantum treatment of a coherently driven cavity with a medium of two-level atoms is given. The system displays both absorptive and dispersive bistability. Quantum fluctuations are included via a Fokker-Planck equation in a generalized PPP representation, after adiabatic elimination of the atomic variables. The transmitted spectrum and correlation function is calculated in a linearized (Gaussian) approximation. In the limit of large detuning and radiative damping, the quantum fluctuations are identical to those of a simple nonlinear polarizability theory. An analysis is given of the nonclassical statistical properties of the intensity-correlation spectrum.
In a recent paper (1) we formulated a running wave, singlemode model for purely dispersive optical bistability, which incorporates amplitude and frequency fluctuations in the incident field, cavity length fluctuations, thermal noise in the radiation field and in the material. In the bad-cavity case of miniaturized bistable devices which utilize semiconductor dispersive media, our model reduces to a single stochastic differential equation for the material variable, which contains several terms of additive and multiplicative noise. In the white-noise case, it is equivalent to a Fokker-Planck equation in one variable, whereas in the case of colored noise it is equivalent to a multidimensional Fokker-Planck equation. On the basis of this model, we examined both problem of practical (P) and of fundamental (F) interest.