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Journal of the Optical Society of America B

Effects of hidden transient noise processes on multiphoton absorption

K. Wódkiewicz, Joseph H. Eberly

Published Apr 1, 1986• 16 citations
Publisher

Abstract

We consider a category of transient processes that is usually hidden from view in coherent-transient experiments. These are the very fast noise processes that give rise to relaxation effects. We present different Markov-type models of a class of these noise processes, namely, laser phase and frequency fluctuations. These models lead to different forms for the laser-field autocorrelation function, depending on which quantity, the phase or the frequency, is the primary source of noise. We show that it is possible to establish a regime such that all these kinds of noise will lead to a Lorentzian laser power spectrum. Despite this common feature, N-photon absorption exhibits observable differences depending on the source of fluctuations. For example, for diffusive noises the absorption linewidth increases as N2, but for jumplike or shotlike phase fluctuations it can be N independent.

DOI
10.1364/josab.3.000628

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We consider a category of transient processes that is usually hidden from view in coherent-transient experiments. These are the very fast noise processes that give rise to relaxation effects. We present different Markov-type models of a class of these noise processes, namely, laser phase and frequency fluctuations. These models lead to different forms for the laser-field autocorrelation function, depending on which quantity, the phase or the frequency, is the primary source of noise. We show that it is possible to establish a regime such that all these kinds of noise will lead to a Lorentzian laser power spectrum. Despite this common feature, N-photon absorption exhibits observable differences depending on the source of fluctuations. For example, for diffusive noises the absorption linewidth increases as N2, but for jumplike or shotlike phase fluctuations it can be N independent.
The effect of stochastic laser phase fluctuations on nonlinear optical phenomena are investigated within the phase-diffusion model. Averaged equations of motion are derived in the limit of infinitely short correlation times and the results are applied to multiphoton absorption and to four-wave mixing. The effect of the stochastic phase fluctuations is shown to be equivalent to a T2{T}_{2}T2​ dephasing process, and a procedure is described for the inclusion of this equivalence in many nonlinear-optical problems. When a given mode contributes n photons to a multiphoton absorption process, its contribution to the width of the multiphoton transition is n2{n}^{2}n2 times its own stochastic width. When correlated lasers are used as the excitation sources in four-wave-mixing experiments, (new) stochastic-fluctuation-induced extra resonances in four-wave mixing (SFIER4) will be observed, in complete analogy to the extra resonances induced by collisions [pressure-induced extra resonances in form-wave mixing (PIER4)].
The effect of amplitude and phase fluctuations on N-photon excitation from N independent lasers is investigated. Each laser is assumed to have a multi-lorentzian line shape. The Fokker-Planck equation is solved to yield atomic density matrix equations which incorporate the fluctuations. Most of the results discussed are for the lowest-order approximation. No enhancement in the Rabi frequency or transition probability appears, but the Stark shift is found to be enhanced by a factor which is independent of the degree of non-linearity of the process. A general expression for the transition probability in the rate equation regime is presented and its characteristic features discussed. The difference between N-photon excitation from N independent lasers and from a single laser is examined.
Experimental evidence of field-correlation effects on weak-field two-photon absorption in atomic sodium is presented. In the case of a nearly Lorentzian laser power spectrum the absorption profile has four times the spectral width of the exciting field, in agreement with predictions by Mollow. The measurement is carried out with counterpropagating laser beams to cancel Doppler broadening. The width of the two-photon absorption spectrum is decreased by partially decorrelating the counterpropagating laser beams. Increasing the correlation time of the frequency fluctuations, resulting in a nearly Gaussian laser power spectrum, has also been observed to decrease the width of the absorption spectrum. An extension of the time-dependent second-order perturbation theory to these additional cases yields good agreement.
A phase-diffusion stochastic field with frequency fluctuations obeying Gaussian Markovian statistics is produced at microwave frequency and used to drive a two-level spin system. The system is tuned to the two-photon (TP) resonance and its TP-induced second-order response is measured as a function of the characteristic parameters of the frequency noise. It is found that, when the statistics of the driving field are varied from the fast- to the slow-modulation limit, the power spectrum of the nonlinear response of the system varies in a continuous way from a Lorentzian form toward a Gaussian one, and its relative width (normalized to the input spectral width) varies from 4.0\ifmmode\pm\else\textpm\fi{}0.4 to 2.0\ifmmode\pm\else\textpm\fi{}0.2. The experimental results are interpreted on the basis of a semiclassical calculation which takes into account the stochastic nature of the field. The effect of the filter action of the cavity modes is discussed. The relationship with previous experiments and theoretical calculations of TP absorption is also examined.
The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of nthn\mathrm{th}nth-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the (n,n)th(n, n)\mathrm{th}(n,n)th normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is nnn! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in nnn-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.
We have measured the variance in fluorescence intensity and the spectrum of those fluctuations for two-level atoms in a phase-diffusing laser field. We compare our results with recent theoretical predictions that have been extended to include the effects of Doppler broadening and spatial variation of the laser intensity. Our study includes the effects of laser power, bandwidth, and shape (Lorentzian and non-Lorentzian) of the laser spectrum. At lower laser intensities, our measured variances versus detuning are in quantitative agreement with the theoretical predictions. At intensities above saturation, the variance is sensitive to inhomogeneities in the distribution of laser intensity in the interaction region, and only qualitative agreement is achieved. Asymmetries in the variance versus detuning resulting from correlated amplitude and phase noise have been observed. Spectra of the phase-noise-induced fluorescence-intensity fluctuations are shown to contain contributions from transients excited by the phase noise and are more sensitive to artifacts in the noise modulation process than the intensity spectrum of the laser field itself.