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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlation functions and spectra for fields with highly anticorrelated phase jumps

Abraham G. Kofman

Published May 1, 1995• 1 citation
Publisher

Abstract

A non-Markovian jump model of phase-fluctuating electromagnetic fields is considered, which is a generalization of the random phase telegraph model. According to the present model, the phase jumps are highly anticorrelated Markovian jump process, i.e., they have alternating signs and approximately equal magnitudes. Closed analytical results for the field correlation function and spectrum have been obtained under rather general assumptions. The parameter \ensuremath{\gamma}, which is a measure of correlation of successive jumps, has been expressed in a general form through the parameters of the model. Two regimes have been found, depending on the values of the rms phase jump B and the correlation parameter \ensuremath{\gamma}. For highly anticorrelated and not very large jumps (1+\ensuremath{\gamma}\ensuremath{\ll}1,${\mathit{B}}^{\mathrm{\ensuremath{-}}2}$) the correlation function (spectrum) consists of two components with significantly different damping rates (widths). In the opposite case each phase jump destroys the coherence of the field and the decay rate of the correlation function is determined by the frequency of jumps. The possibility of extracting parameters of the model from measurements of the laser field correlation function and/or spectrum is discussed.

DOI
10.1103/physreve.51.4011

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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-motion-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 parameter---the 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 fluctuations 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.
A new non-Markovian, correlated-jump model for phase fluctuations of an electromagnetic field is introduced. The phase-diffusion, generalized telegraph, Burshtein, and Brownian-motion-like models are all obtained from this model in the proper limits. The spectrum and correlation functions of the stochastic field are predicted, and detailed line shapes of the Mollow triplet of resonance fluorescence are given and shown to depend on the field intensity, a purely non-Markovian effect. Two different experimental procedures are proposed for the determination of the stochastic parameters of a laser field.
A new non Markovian, correlated jump model for phase fluctuations of an electromagnetic field is introduced'. The phase diffusion, generalized telegraph, Burshtein and Brownian-motion-like models are all obtained from this model in the proper limits. The spectrum and correlation functions of the stochastic field are predicted, and detailed lineshapes of the Mollow triplet of resonance fluorescence are given and shown to depend on the field intensity, a purely non Markovian effect. Two different experimental procedures are proposed for the determination of the stochastic parameters of a laser field.
Strong laser fields, when tuned on resonance to an atomic (molecular) transition, give rise to the well-known triplet of resonance fluorescence. The laser field is normally assumed to be strong and monochromatic, even though the spectrum of real lasers is never a \ensuremath{\delta} function. A standard way to treat laser fluctuations has been to assume a well-defined nominal frequency and a randomly fluctuating phase. Recently we introduced the generalized jump model (GJM) for correlated phase fluctuations, where correlated phase jumps are considered, leading to Markovian as well as non-Markovian stochastic behavior. In the GJM, each phase jump may be correlated to the previous jump, and the degree of correlation, the typical jump size, and the mean time between jumps are the three parameters defining the stochastic character of the laser field. In this paper we apply the GJM model to the case of resonance fluorescence from an atom excited by a stochastic field. The ``standard'' Mollow triplet is obtained in the monochromatic limit of small jumps, and several new predictions are made in the different limits of the stochastic parameters. In the small-jump limit, a new ``field effect'' is predicted. The spectral line shape and the triplet separation deviate from the Mollow predictions for a monochromatic field in a substantial way, giving rise to observable differences.In the highly correlated small-jump limit, which is approximated by the Kubo oscillator model, the previous results are corrected to include nonzero line shifts and extended to the case of non-Lorentzian line shapes. In all regions, even input fields of the same line shape (which would be considered identical in the one-parameter linewidth description) give rise to very different output spectra, justifying the more complete stochastic description. Effective relaxation constants ${\mathrm{\ensuremath{\Gamma}}}_{1}^{\mathrm{*}}$ and ${\mathrm{\ensuremath{\Gamma}}}_{2}^{\mathrm{*}}$ that simplify the description of the spectrum are calculated in all appropriate cases, and ${\mathrm{\ensuremath{\Gamma}}}_{1}^{\mathrm{*}}$ is found to equal the input laser linewidth. In addition to predicting the resonance fluorescence spectra, the present analysis provides a way to extract the stochastic parameters from the nonlinear measurement in those cases where the linear characterization of the input field cannot do it. The analytic work is fully supported by extensive numerical simulations.
A new model describing laser noise in terms of non-Markovian stochastic phase jumps has been introduced by A.G. Kofman et al. (1988, 1990). An exact expression for the correlation function for the electric field is obtained. This expression, valid for all possible values of the parameters of the stochastic process, reduces to approximations reported by B.R. Mollow (1969). A procedure is described for the inverse problem of determining the stochastic parameters from measured data for real lasers.>
Stochastic fields with non-Gaussian Markovian phase or frequency jumplike noise are produced at microwave frequency and used to induce two-photon (TP) processes in a S=(1/2 spin system. The influence of the field statistics on the TP-induced second-order response is experimentally investigated in a system with very short relaxation times, in its stationary state and in the weak-field limit. In the case of frequency-telegraph noise the experimental spectra of the nonlinear response are found to reproduce roughly the spectral profiles of the corresponding driving fields but with a larger width. When the telegraph noise modulates the phase of the field, the relationship between input and output spectra depends on the phase-jump amplitude ${\ensuremath{\varphi}}_{0}$. For ${\ensuremath{\varphi}}_{0}$=\ensuremath{\pi}/4 the spectrum of the response is much broader than the input one, whereas it is much narrower for ${\ensuremath{\varphi}}_{0}$=\ensuremath{\pi}/2. The experimental results are interpreted using the calculation of the TP-induced second-order response worked out in the preceding paper [Boscaino and Mantegna, Phys. Rev. A 40, 5 (1989)].