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Physical Review A

Effect of correlated noises in an optical bistable system

Li Zhang, Cao Li, Wu Da-Jin

Published Jan 2, 2008• 20 citations
Publisher

Abstract

In this paper we focus on the state equation of absorptive optical bistable system with cross-correlated multiplicative noise $\ensuremath{\xi}(t)$ and additive noise $\ensuremath{\eta}(t)$. We find that the area of optical bistability with positive cross-correlation expands in comparison with the deterministic case. Furthermore, a very peculiar phenomenon is shown, namely when the cooperativity parameter $C<4$, noise still can induce optical bistability. The phenomenon of noise-induced optical bistability in this paper is just the example of noise-induced phase transition.

DOI
10.1103/physreva.77.015801

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In this paper we focus on the state equation of absorptive optical bistable system with cross-correlated multiplicative noise \ensuremathξ(t)\ensuremath{\xi}(t)\ensuremathξ(t) and additive noise \ensuremathη(t)\ensuremath{\eta}(t)\ensuremathη(t). We find that the area of optical bistability with positive cross-correlation expands in comparison with the deterministic case. Furthermore, a very peculiar phenomenon is shown, namely when the cooperativity parameter C<4C<4C<4, noise still can induce optical bistability. The phenomenon of noise-induced optical bistability in this paper is just the example of noise-induced phase transition.
The effect of correlated white noises on the optical bistability behavior of three-level atomic systems is theoretically investigated. Noise-induced bistability is demonstrated in an atomic optical bistable system consisting of three-level atoms in Λ-type configurations confined in an optical ring cavity. The output field as a result of enhanced Kerr nonlinearity induced by atomic coherence in the electromagnetically induced transparency system is variable due to the noises in laser parameters and the cooperativity parameter. Therefore, noise is mainly responsible for such induced bistability in the system. It is found that with the cross-correlation noises, the hysteresis loop of optical bistability can be clearly expanded in comparison with the deterministic case and the atomic system could show multistability behavior. This is specifically considerable for positive cross-correlated noises where it could expand the domain of the hysteresis loop. In the presented atomic system the numerical result shows that noise can induce bistability in the absence of spontaneously generated coherence. Furthermore, it is salient that the noise could induce bistability when the system is below its deterministic threshold for bistability.
Stochastic differential equations for absorptive optical bistability derived using the positive P representation are solved numerically. The stochastic model is formulated in ten dimensions, twice the dimensions of the space of physical observables. Extra dimensions are needed to include the quantum character of the noise. Ensemble averages are computed for the field amplitude, the incoherent intensity, and the second-order correlation function of the transmitted light. Comparisons with linearized theory are made and detailed results are given in the vicinity of the critical point where nonlinear effects are paramount. The scaling of fluctuations with system size at the critical point is investigated. Transitions between deterministic steady states are studied in a system just beyond the threshold for bistability. A potential difficulty with the use of the positive P representation for modeling nonlinear quantum-statistical systems is identified.
The relaxation time T of an optical bistable system with cross-correlated color noises and small time delay is investigated. Using the Novikov theorem and Fox approach, the steady probability distribution is obtained. The expression of T is derived using the Stratonovich decoupling ansatz formalism. It is found that the relaxation time T increases with the increasing cross-correlation time τ 0 between the two noises or the self-correlation time τ 1 of the multiplicative noise, but decreases with the increasing self-correlation time τ 2 of the additive noise. T decreases with the increasing correlation intensity λ or the multiplicative noise intensity Q , but increases with the increasing additive noise intensity D . There exists a peak in the curve of T versus delay time τ.
Investigated are the effects of the cross-correlation time τ or the self-correlation times τ1 and τ2 on the relaxation time T of an optical bistable system using the Mei's method(Mei D C, Xie C W, Zhang L 2003 Phys. Rev. E 68 051102). It is found that the relaxation time T increases with the increase of the cross-correlation time τ between the two noises or the self-correlation time τ1 of the multiplicative noise, and that a one-peak structure appears on the T－τ2 plot. T increases with the increase of correlation intensity λ but decreases with the increase of multiplicative noise intensity Q.
The transient properties of a three-level atomic optical bistable system in the presence of multiplicative and additive noises are investigated. The explicit expressions of the mean first-passage time (MFPT) of the transition from the high intracavity intensity state to the low one are obtained by numerical computations. The impacts of the intensities of the multiplicative noise D M and the additive noise D A , the intensity of correlation between two noises λ, and the intensity of the incident light y on the MFPT are discussed, respectively. Our results show: (i) for the case of no correlation between two noises (λ = 0.0), the increase in D M and D A can lead to an increase in the probability of the transition to the low intracavity intensity state, while the increase in y can lead to a retardation of the transition; and (ii) for the case of correlation between two noises (λ = 0.0), the increase in λ can cause an increase in the probability of the transition, and the increase in D A can cause a retardation of the transition firstly and then an increase in the probability of the transition, i.e., the noise-enhanced stability is observed for the case of correlation between two noises.
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.