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Physical Review A

Fluorescence fluctuations from a multilevel atom in a nonstationary phase-diffusion field: Deterministic frequency modulation

James C. Camparo

Published Jan 8, 2004• 4 citations
Publisher

Abstract

It is well known that a field's quantum noise can have a profound effect on the fluctuations of laser-induced fluorescence (LIF). However, though previous studies have led to a good understanding of this process in the case of stationary fields, many of the important applications of LIF employ some form of laser modulation, yielding a field of nonstationary stochastic character. Here, we discuss the results from numerical experiments that examine the influence of a nonstationary field on the LIF noise. Specifically, we consider a phase-diffusion field and LIF from a beam of alkali-metal-like atoms with ground and excited-state Zeeman splitting when the field undergoes deterministic frequency modulation. Our computational results show that deterministic modulation at high Fourier frequencies (i.e., 10 MHz) can significantly increase the LIF noise at low Fourier frequencies (e.g., 1 Hz), and that the amount of LIF noise amplification depends on a complicated interplay among the laser's linewidth, the modulation frequency, the modulation index, and the multilevel atomic system's energy-level spacing. Interestingly, we find that certain values of the atom's Zeeman splitting significantly decrease the magnitude of LIF noise.

DOI
10.1103/physreva.69.013802

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It is well known that a field's quantum noise can have a profound effect on the fluctuations of laser-induced fluorescence (LIF). However, though previous studies have led to a good understanding of this process in the case of stationary fields, many of the important applications of LIF employ some form of laser modulation, yielding a field of nonstationary stochastic character. Here, we discuss the results from numerical experiments that examine the influence of a nonstationary field on the LIF noise. Specifically, we consider a phase-diffusion field and LIF from a beam of alkali-metal-like atoms with ground and excited-state Zeeman splitting when the field undergoes deterministic frequency modulation. Our computational results show that deterministic modulation at high Fourier frequencies (i.e., 10 MHz) can significantly increase the LIF noise at low Fourier frequencies (e.g., 1 Hz), and that the amount of LIF noise amplification depends on a complicated interplay among the laser's linewidth, the modulation frequency, the modulation index, and the multilevel atomic system's energy-level spacing. Interestingly, we find that certain values of the atom's Zeeman splitting significantly decrease the magnitude of LIF noise.
The fluorescence emitted by atoms whose excited states have Zeeman structure is studied for the case of excitation by partially coherent optical sources whose amplitudes are weakly modulated at a low frequency $\ensuremath{\Omega}$. The temporal fluctuations in lasers are taken into account exactly by using the phase-diffusion model of the laser. Analytical results for the modulated fluorescence are presented for the case of weak fields. It is shown that the shape of the fluorescent signal is extremely sensitive to the bandwidth of the field; for example, one gets resonances at a Larmor frequency equal to $\mathcal{s}=\frac{\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}}{2}(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega})$ for broadband (monochromatic) fields. Various limiting cases are treated, and in the limit of very broadband fields, results reduce to those of Corney and Series. The fluorescence is studied both as a function of the modulation frequency and as a function of the magnetic field. Numerical studies of the modulated Hanle signals in intense fields reveal resonant structures in the modulation-frequency scan at $\ensuremath{\Omega}=0,\ifmmode\pm\else\textpm\fi{}{\ensuremath{\alpha}}_{0}$, $\ifmmode\pm\else\textpm\fi{}2{\ensuremath{\alpha}}_{0}({\ensuremath{\alpha}}_{0}^{2}={\mathcal{s}}^{2}+2{\ensuremath{\alpha}}^{2})$ due to the dynamical Stark splitting of the various energy levels, with the strength of the resonances depending on the direction of detection.
The influence of relaxation parameters, as well as temporal fluctuations in the laser field on the modulation rate and phaseshift of the fluorescence from a two-level atom excited by a weakly modulated source is studied in detail. It is shown that, for intense resonant laser fields, the resonance in the modulation rate occurs at the modulation frequency Omega =2 due to the dynamical Stark splitting and has a width equal to 1 / 2 (T 1 -1 +T 2 -1 ). When temporal fluctuations of the exciting laser field are taken into account, the phaseshift of the scattered light is found to be strongly dependent on the bandwidth of the exciting source, as well as the frequency of modulation; hence experimental results on the determination of the lifetime by the phaseshift method have to be interpreted carefully. Numerical results for the modulation rates and phaseshift at different field strengths and various values of T 1 -1 , T 2 -1 , Delta laser bandwidth and Doppler broadening are presented.
We report an experiment on the observation of Hanle resonances in sinusoidally amplitude modulated, weak, phase diffusing optical fields. The J = 0 to J = 1 transition in an atomic beam of barium was probed with a frequency stabilized dye laser, whose bandwidth and bandshape were precisely controlled with an extra-cavity noise modulation system. Phase sensitive detection of the Hanle fluorescence signal at the amplitude modulation frequency revealed resonances in the in-phase and in-quadrature parts, whose position, shape, and strength were sensitive to the laser bandwidth. These resonances, which were Lorentzian or dispersion shaped or a combination of the two, occurred when the Zeeman splitting of the exited sublevels equaled the modulation frequency for monochromatic light and at splitting of half the modulation frequency for broadband light. The results are in good agreement with the theoretical predictions of Saxena and Agarwal.1
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.
Significant fluctuations, above the shot-noise limit, have been observed in the intensity of fluorescent light scattered from atoms excited by an intense noisy laser, where a central role is played by the nonlinearity of the atom-field interaction. By considering the variance of the atomic populations, we show that ``noise spectroscopy'' is sensitive to the field statistics of the laser. We specifically consider the cases of resonance fluorescence (analogous in the weak-field limit to analysis by a Fabry-P\'erot interferometer), two-photon excitation, and double optical resonance for variable laser intensity and bandwidth. We show that large differences in the ``noise'' spectra can occur between lasers characterized by a phase-diffusion or a phase-jump model, although these models would give the same values for the mean atomic populations. We believe that observation of population fluctuations can become a useful method for characterizing laser noise.
Stochastic fluctuations of coherent and incoherent components of the resonance fluorescence intensity induced by Wiener-Levy laser phase noise are investigated. Statistical properties of the atomic dipole moment for different values of the laser linewidth and different strength of the driving field are calculated. The stochastic Mollow spectra of atomic dipole fluctuations are derived. It is shown that these spectra exhibit a triplet structure which is purely classical and entirely laser-noise dependent.