Research On Cross-correlated Noise Problems And Correlated Effects Of Quantum Noise In A Laser

Over the last decades, amazing progress has been made in the study of stochastic dynamics, that has developed into a vigorous branch in statistical physics. That Fluctuations become the sources of order implies noise can play active or constructive role in stochastic dynamical systems. Stochastic methods are bridging between phenomeno-logical models and statistical mechanics. Noise-induced diverse effects attract more and more attentions in many fields of natural science and applied engineering. A lot of related noise problems, and cross-correlated noises especially emerge to be solved. In this paper, the basic theory and statistical properties in some stochastic systems driven by cross-correlated Gaussian white noises and O-U noises as well are investigated.Firstly, the statistical characteristics of Gaussian and O-U noise are described, respectively, and the cross-correlated forms between different noises are presented based on the reasonable idea that noises are cross-coupling if their fluctuations stem from a common bath. Computational physics not only supplement theoretical and experimental physics, but provides new ways of thinking physics also through directly computer simulations. In Chapter II, the numerical algorithms for generalized Langevin equation (stochastic differential equation) with cross-correlated Gaussian white noises are developed. Computer experiments show that the second order stochastic equivalent method (SOS) is more accurate and stable than the first order conventional approach (FOC), The SOS algorithm from the point of view of Fokker-Planck dynamics avoids the difficulty with cross integrals of stochastic terms encountered in the FOC method. Furthermore, the second-order numerical algorithm for the case of cross-correlated O-U noises is derived. The cross-integral of different noise terms is dealt with the concept of mean values, correcting the vital mistake presented in publications.More realistic cross-correlated colored-noise problems are investigated in Chapter III. The effective Markovian approximation for multi-colored noise problems are obtained through extending the unified colored noise theory. The extended unified colored noise approximation is applied to the very typical bistable system which is found many practicalapplications in optics and electronics. The multiplicative O-U noise stemming from external perturbations, and additive one arising from intrinsic fluctuations driving bistable system is analyzed. It is found either the strength or the correlation time of the multiplicative colored noise can induce first-order phase transitions. Numerical simulations from the derived algorithm confirm the predictions and the validity of approximative theory. Moreover, Considering the coupling of the multiplicative and additive colored noise, the stationary characteristics in the bistable system can be changed, the noise cross-correlation leads to symmetry broken of the bistable structures. When the environment fluctuates strongly, the bistable system undergoes an asymmetric evolution, unique most stable state appears. Particularly, in the limiting case of perfect cross-correlation, the probability density at the most probable state approaches to infinity, the stochastic bistable system turns into a locked deterministic state.As a typical nonlinear system far from equilibrium, a single-mode laser affected simultaneously by pump fluctuation and spontaneous emission is studied theoretically and numerically in details in Chapter IV. For the real and imaginary terms of the complex pump noise and the quantum noise come from common origins, respectively, thus may be cross-correlated. A generalized Laser intensity Langevin equation is first derived through phase locking procedure. It is found that, the cross-correlation between the components of the quantum noise plays a crucial role, but the coupling of the pump noise does not affect the laser intensity. Theoretical analyses show the existence of cross-correlation between quantum noise terms leads the phase to the appearance of bistable states, the two stable states will be definitely locked in the limiting case of perfect correlation, and a first-order-like phase transition with the change of cross-correlation strength is predicted. On the other hand, direct numerical simulations using derived algorithms for the evolution of both phase and intensity are performed, which corroborate the novel phenomena. The mechanism of phase locking is revealed through theoretical and numerical analysis.Stochastic resonance (SR) has become one of the striking topics in recent years. Many peculiar phenomena, such as aperiodic SR, adaptive SR, SR in Nonlinear monostable systems with Gaussian noise, SR in linear system with cross-correlated dichotomic noises etc. have been reported. In Chapter V, a biased periodic force driving overdamped linearsystem with more commonly cross-correlated Gaussian white noise, cross-correlated O-U noises, as well as Gaussian white noises with time modulated coupling, are investigated, respectively. It is found that, multiplicative O-U noise, or time modulated cross-correlated white noise can induce SR, differ from SR in nonlinear systems with Gaussian noise and SR in linear systems with dichotomic noise, whereas the biased component can be adjusted directions for strong cross-correlation, with the change of the strength of multiplicative white noise or the correlation time of multiplicative O-U noise.