Non-Markovian stochastic jump processes. I. Input field analysis

Abstract

A non-Markovian model of correlated phase jumps is introduced for phase fluctuations of an electromagnetic field. This generalized jump model (GJM) treats phase jumps of arbitrary size, occurring at random times; but in contrast to previous work, the jumps are allowed to be fully correlated, partially correlated, or uncorrelated. The degree of correlation is defined by a single parameter derived from the theory. The familiar phase-diffusion model, telegraph-noise model, Burshtein model, and Brownian-notion-like model are all obtained from the GJM in the proper limits. The standard way of characterizing the spectrum of a laser has been the assignment of a single parameterthe linewidth. However, in experiments where the details of the fluctuations are important, or where exact line shapes are measured, this single-parameter characterization might be insufficient. This GJM describes most cases by a set of three stochastic parameters: the degree of correlation between the jumps, the characteristic jump size, and the mean time between jumps. In this paper expressions are derived for the correlation function and the spectrum of a stochastic field in terms of these three stochastic parameters. In addition to analytical work, detailed numerical simulations are presented for the various limiting cases of the model, and the agreement between theory and simulation is excellent. Since the stochastic parameters are not a priori known, a procedure is described for extracting the stochastic parameters from measurable quantities such as the field correlation function or spectrum. Since correlated Auctuations are very common in optics (any stabilization feedback procedure involves anticorrelation), the questions of relevance of the present model to problems of current interest in optical communication and nonlinear optics are also discussed. Lorentzian line shapes is to consider a Gaussian-Markovian (Ornstein-Uhlenbeck, Brownian motion) process of frequency fluctuations. This model for an intensity stabilized laser field, which is similar to the Anderson-Kubo treatment of material energy level fluc-41 6434

Zeno and anti-Zeno effects for photon polarization dephasing

“…Equation (8) can be reduced to an equation for the average quantityQ(t), involving an expansion in cumulants of Cµ(t) [25,26]. As shown in Appendix A, by truncating the cumulant expansion at the second order, one obtains the following non-Markovian differential master equations for the average polarization probabilities,…”

“…where γ (−1 ≤ γ ≤ 1) is the correlation degree between two successive jumps [26]: ∆ϕ n ≈ ∆ϕ n+1 for γ ≈ 1, ∆ϕ n and ∆ϕ n+1 tend to have opposite signs for γ < 0 and are statistically independent for γ = 0. In this case [Eq.…”

“…(8) we use the cumulant expansion technique [25,26]. Assuming, for simplicity, that the odd moments of µ(t) vanish, we obtain…”

Three-dimensional telegrapher's equation and its fractional generalization

“…Let us incidentally mention that this model is similar in spirit to that of Gordon in his study of rotational relaxation in fluids wherein the angular momentum is randomized at each collision [52]. In the context of fluctuations in laser fields the model corresponds to the so-called Burshtein model [53,54].…”

Relaxation dynamics of the fastest channel in multichannel parallel relaxation mechanism

“…It is of importance in the study of a variety of problems from condensed matter physics [1,2], nuclear physics [3], spectroscopy [4,5], rheology [6], seismology [7], physical chemistry [8], molecular biophysics [9,10], cell and population dynamics [11,12], etc. Among dierent relaxation functions suggested in the literature [1] the Kohlrausch±Williams±Watts (KWW) function…”