The Transient Properties Research Of The Optical Bistable System Driven By Cross-Correlated Noises

The fluctuation of various physical quantities in nature are usually described by noises. According to the different origins, noises could be divided into internal noise and external noise. Usually, the internal noises are caused by the statistical properties of the system, like quantum noise; while the external noises are produced by the external environment of the system, like pump noise. The research results show that the noise and its cross-correlation forms have great influence on the dynamic behaviors of the nonlinear system. The bistable system is a typical nonlinear system, the study of the dynamic behaviour for such a system has been attracting many researchers’extensive attention in recent years. In this paper, using the Novikov theory and Fox approxiation, we have investigated the statistical properties of the optical bistable system driven by white pump noise and white quantum noise modulated by periodic signal, and the main work as follows:Firstly, The analytical expression of the steady-state probability density function of the optical bistable system Pst(x) is calculated, and various moments of the state variable are obtained. The steady-state probability density function Pst(x) and moments of the optical bistable system driven by cross-correlation noises with signal modulation are researched, the influences of the signal and the noises on Pst(x)and the moments of the system are investigated. The research shows that, considering the noise modulated by the signal, it appears two peaks in the curve of the Pst(x) versus the state variable x. It can be found that, the two peaks locate x≈0.5and x≈11respectively, that is to say, the probability of the system at x≈0.5and at x≈11are much larger than at other locations, and the probability at x≈11is larger than at x≈0.5. When0<B<0.2, though the amplitude of the input signal B can change the Pst(x), the influence is not obvious. When0.2<B<1, the curve of the Pst(x) versus x greatly changes with different amplitude of the input signal B. The frequency of the input signal Ω hardly have influence on the Pst(x). It also can be found that, the Pst(x) slightly changes with the pump noise intensity Q, while the influence of the quantum noise intensity D on the Pst (x) is clear. From the curve of the mean of the state variable<x>versus the amplitude of the input signal B, we can find that, the amplitude of the input signal B is smaller, the mean of the state variable<x>is larger, as the increase of the amplitude of the input signal B, the mean of the state variable<x>is monotonous decreasing and gradually approaching to zero, the pump noise intensity Q and the quantum noise intensity D also have great influence on the mean of the state variable<x>. The curve of the secondary moment of the state variable<x2>and the third moment of the state variable<x3>versus the amplitude of the input signal B is similar to the curve of the mean of the state variable<x>versus the amplitude of the input signal B. There is a maximum both in the curve of the variance of the state variable κ2versus the pump noise intensity Q and the amplitude of the input signal B, the influences of the amplitude of the input signal B and the pump noise intensity Q on κ2are obvious, while the small frequency of the input signal Q has less influence on κ2. The influence of parameters on the skewness of the state variable κ3is similar to the influence on κ2.Secondly, the issue of the mean first passage time of the optical bistable system driven by cross-correlation noises with periodic signal modulation MFPTis researched. We have obtained the two opposite direction expressions of the MFPT using the analytic method, on the basis of the expressions, the influences of the bistable parameter, signal, noises and other parameters on MFPTare discussed. From the curves of T+(xs1â†'xs2)and T-(xs2â†'xv1)versus the amplitude of the input signal B, we can find that, with the increase of the amplitude of the input signal B, the curves of T+(xs1â†'xs2)and T-(xs2â†'xs1)both decline till to zero, then keep unchanged, and there are some discontinuous points, after that, the versus appear a maximum. With the augment of the bistable parameter c, the curve of T+(xs1â†'xs2)shifts down wholly, and the position of the peak moves toward the B increasing direction, while the curve of T-(xs2â†'xs1)moves up firstly and then moves down with the enhancement of the bistable parameter c, and the location of the peak is still. The curves of T+(xs1â†'xs2)and T-(xs2â†'xs2)both shift up wholly with the increase of the input laser mode,y0, the peak position of T+(xs1â†'xs2)curve move toward the B decreas-ing direction, and the peak position of T-(xs2â†'xs1)is immobile. Both of the curves of (T+(xs1â†'xs2),Q)and(T-(xslâ†'xs2),Q)increase with the augment of the quantum noise intensity D, but decrease with the enhancement of the pump noise intensity Q. The curves of T+(xs1â†'xs2)versus the quantum noise intensity D increase with the enhancement of D, and it appears a maximum in the curves of T-(xs2â†'xs1)versus the quantum noise intensity D. The curve of T+(xs1â†'xs2)versus the frequency of the input signal Ω exhibits periodic equal amplitude oscillations, and the curve of T(xx2â†'xsl)versus the frequency of the input signal Ω shows periodic peaks oscillations.