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Physical Review A

Theory of resonance fluorescence in a fluctuating laser field

W. Vogel, D.‐G. Welsch, K. Wódkiewicz

Published Sep 1, 1983• 20 citations
Publisher

Abstract

A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.

DOI
10.1103/physreva.28.1546

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Physical Review A•Oct 1, 1978•211 citations
A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.
The effects of amplitude and phase fluctuations of the pump laser on the resonance fluorescence under two-photon resonant excitation are studied. The averaging of the stochastic Bloch equations is carried out within the approximation scheme that has recently been applied to the case of one-photon resonant excitation. In the case of two-photon resonant excitation the stochastic pump field gives rise to multiplicative nonlinear Gaussian fluctuation effects. The phase fluctuations are treated in the phase-diffusion model, and the averaging over the amplitude fluctuations is performed by means of generating-function methods. Both the intensity and the intensity correlation of the fluorescent light are studied. When the amplitude fluctuations are weak, the cases of one- and two-photon resonant excitations are similar. The difference between both cases consists of a modification of the Rabi frequency and a faster damping of the Rabi oscillations in the two-photon case. For sufficiently fast correlation decay of the amplitude fluctuations their effect can simply be described by a rate. In the general case of n-photon resonant excitation this rate is larger by a factor of n2{n}^{2}n2 than that in the case of one-photon resonant excitation. When the amplitude fluctuations are sufficiently strong, the Rabi oscillations are ``washed out.'' In the case of one-photon resonant excitation both the intensity and the intensity correlation function of the scattered light develop into step functions over time. In the two-photon case the dynamics of both quantities is determined by a new time constant which acts as both the oscillation time and the damping time. Consequently an overshoot peak occurs in both the intensity and the intensity correlation instead of the tendency to become step functions.
The relaxation process and resonance fluorescence of a two-level atom strongly driven by stochastically fluctuating laser light is studied. A laser with small intensity fluctuations about a constant mean value and a diffusing phase is assumed. Depending on the relative time scales of the intensity fluctuations and the atomic response to these fluctuations, the time decay of the atomic population is found to be non-exponential. The broadening of the sidebands in the triplet spectrum of resonance fluorescence is shown to be non-Lorentzian.
The effects of different laser line shapes on the intensity and spectrum of resonance fluorescence from a two-level atom are investigated. The laser model considered is that of an ideal single-mode laser operating high above threshold, with constant amplitude and undergoing phase-frequency fluctuations analogous to Brownian motion. As the correlation time of the frequency fluctuations increases from zero to infinity, the laser line shape changes from Lorentzian to Gaussian in a continuous way. It is shown that for intermediate and strong fields, the average intensity of fluorescence in the case of a resonant broadband Lorentzian line shape is higher than that in the case of a Gaussian line shape with the same bandwidth and total power. This is in contrast to the weak-field case where the higher peak power of the Gaussian line shape makes it more effective than the Lorentzian line shape. It is also shown that in the case of a nonzero frequency correlation time (non-Lorentzian line shape) the intensity of fluorescence undergoes non-Markovian fluctuations. In relation to the spectrum of resonance fluorescence, it is shown that as the line shape is varied from Lorentzian to Gaussian the following changes take place: In the case of off-resonance excitation, the asymmetry of the spectrum decreases. In the case of resonant excitation, the center-peak to side-peak height ratio for the triplet structure increases. Moreover, the recently predicted center-line dip, which develops in the spectrum in the case of broadband excitation when the Rabi frequency and the bandwidth are nearly equal, becomes increasingly deeper.
A systematic method is presented for averaging the integral equations of motion for the atomic density matrix operator and its quantum correlation functions over the fluctuations of Markovian exciting fields with arbitrary bandwidth. The theory takes into account the statistics of the field to all orders. The method is applied to the investigation of resonance fluorescence in the presence of three different Markovian fields: (i) a phase-diffusion field, (ii) a chaotic field, and (iii) a Gaussian-amplitude field. It is shown that in the case of a resonant, intense phase-diffusion field a center line dip develops in the spectrum of resonance fluorescence when the Rabi frequency is approximately equal to the bandwidth of the field. In the case of amplitude fluctuations the sidebands of the on-resonance spectrum tend to reproduce the probability distribution for the amplitude of the exciting field.
The influence of the laser fluctuations on resonance fluorescence is treated exactly within the framework of multiplicative stochastic processes. For the phase diffusion model of the laser, I find that, in the limit of strong fields, the ratios of the width and height of the central peak to the side peak are, respectively, 23x\frac{2}{3x}3x2​ and 3x3x3x, where $x=\frac{(\ensuremath{\gamma}+{\ensuremath{\gamma}}_{c})}{(\ensuremath{\gamma}+2{\ensuremath{\gamma}}_{c})}$. The envelope of the intensity correlations is found to decay at the rate $\frac{(3\ensuremath{\gamma}+{\ensuremath{\gamma}}_{c})}{2}$.
A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio 3x3x3x (23x\frac{2}{3x}3x2​), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.