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arXiv:1504.04409 · Physical Review A

Quantum noise of free-carrier dispersion in semiconductor optical cavities

Ryan Hamerly, Hideo Mabuchi

Published Aug 12, 2015• 20 citations
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Abstract

This paper derives Langevin equations for an optical cavity where the dominant nonlinearity arises from free-carrier dispersion. We define a generalized Wigner function, compute a Fokker-Planck equation that approximates the master equation, and convert this to a system of stochastic differential equations. These equations are similar to the Wigner equations for an optical Kerr cavity, but have additional noise terms due to the incoherent carrier excitation and decay processes. We use these equations to simulate a phase-sensitive amplifier and latch and compare the results to a Kerr model.

DOI
10.1103/physreva.92.023819

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This paper derives Langevin equations for an optical cavity where the dominant nonlinearity arises from free-carrier dispersion. We define a generalized Wigner function, compute a Fokker-Planck equation that approximates the master equation, and convert this to a system of stochastic differential equations. These equations are similar to the Wigner equations for an optical Kerr cavity, but have additional noise terms due to the incoherent carrier excitation and decay processes. We use these equations to simulate a phase-sensitive amplifier and latch and compare the results to a Kerr model.
The Fokker-Planck approach is used to study the statistics of an optical pulse propagating in semiconductor optical amplifiers in which the amplified spontaneous emission noise dominates. The probability density functions (pdfs) of the pulse peak sample are calculated for different saturation levels. Analytical results are possible for an ideal linear amplifier and when the simplest first-order approximation for nonlinear amplification is used. In other cases, we solve numerically the Fokker-Planck equation and the appropriate Langevin equation. Multicanonical Monte Carlo simulations ensure efficient calculations of the pdfs whose high- and low-power tails deviate from noncentral chi-square statistics for moderate and deep saturation levels, thus implying that the electric field process is not Gaussian.
We formulate 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 (1). This model is given by a set of three Langevin-type equations for the real and the imaginary part 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 describe also a Kerr medium or a two-level system in a suitable range of parameters, we are mainly interested in the case of miniaturized all–optical bistable devices which 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 onedimensional Fokker-Planck equation only in the limit of short correlation times.
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.
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.
We present a systematic semiclassical model for the simulation of the dynamics of a single two-level atom strongly coupled to a driven high-finesse optical cavity.From the Fokker-Planck equation of the combined atom-field Wigner function we derive stochastic differential equations for the atomic motion and the cavity field.The corresponding noise sources exhibit strong correlations between the atomic momentum fluctuations and the noise in the phase quadrature of the cavity field.The model provides an effective tool to investigate localization effects as well as cooling and trapping times.In addition, we can continuously study the transition from a few-photon quantum field to the classical limit of a large coherent field amplitude.
We present a systematic semiclassical model for the simulation of the dynamics of a single two-level atom strongly coupled to a driven high-finesse optical cavity. From the Fokker-Planck equation of the combined atom-field Wigner function we derive stochastic differential equations for the atomic motion and the cavity field. The corresponding noise sources exhibit strong correlations between the atomic momentum fluctuations and the noise in the phase quadrature of the cavity field. The model provides an effective tool to investigate localisation effects as well as cooling and trapping times. In addition, we can continuously study the transition from a few photon quantum field to the classical limit of a large coherent field amplitude.