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Journal of Physics B Atomic and Molecular Physics

Hanle resonances for a J=0 to J=1 transition excited by a fluctuating laser beam

Paul Avan, Claude N. Cohen-Tannoudji

Published Feb 1, 1977• 32 citations
Publisher

Abstract

The theoretical method developed in a previous paper (see ibid., vol.10, no.2, p.155 (1977)) is applied to the study of the sensitivity of Hanle resonances to the fluctuations of an intense quasi-monochromatic laser beam. The calculations are performed for the simplest possible transition J=0 to or from J=1. Quantitative expressions are derived for the level-crossing signals which appear to be quite perturbed by the fast phase fluctuations.

DOI
10.1088/0022-3700/10/2/007

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Journal of Physics B Atomic and Molecular Physics•Feb 1, 1977•113 citations
The theoretical method developed in a previous paper (see ibid., vol.10, no.2, p.155 (1977)) is applied to the study of the sensitivity of Hanle resonances to the fluctuations of an intense quasi-monochromatic laser beam. The calculations are performed for the simplest possible transition J=0 to or from J=1. Quantitative expressions are derived for the level-crossing signals which appear to be quite perturbed by the fast phase fluctuations.
The optical Hanle effect (light-shift-induced zero-field level crossing) has been investigated theoretically and experimentally. In the case of a $J=0\ensuremath{\rightarrow}1\ensuremath{\rightarrow}0$ three-level system and of a $J=0\ensuremath{\rightarrow}1$ two-level system, the resonance line shape has been obtained using a nonlinear-response-function method. Approximate expressions, valid under the conditions for the observation of light shifts, are obtained and compared with exact calculations as well as with the corresponding expression for the magnetic Hanle effect. As concerns the weak laser beam which produces the coherent excitation of atomic substates, we have calculated the linear response in the limiting case of monochromatic and of broad-band excitation. The experimental investigation has been performed for the Ba resonance line using two cw dye lasers at $\ensuremath{\lambda}=5535$ \AA{} and an atomic beam. The absorption- and dispersion-shaped level crossing resonances have been observed, for narrow- and broad-band excitation, by varying the power of the strong nonresonant laser beam. We have checked that the line shapes are only dependent upon the reduced variable $\frac{{P}_{L}}{\ensuremath{\delta}}$ (PL{P}_{L}PL​; beam power; $\ensuremath{\delta}$: frequency detuning). Using the theoretical expressions, we have tried to fit the experimental curves. Provided that the Gaussian distribution of the beams is taken into account, we have obtained a good agreement between theory and experiment for the whole set of experimental data.
We have carried out experimental and theoretical investigations of the Hanle effect in the 1S0^{1}\mathrm{S}_{0}1S0​${\ensuremath{\rightarrow}}^{3}$P1{\mathrm{P}}_{1}P1​ transition at \ensuremath{\lambda}=555.6 nm in 174Yb^{174}\mathrm{Yb}174Yb, emphasizing the dependence of the Hanle signal on the intensity and on the statistical parameters of a well-characterized phase-diffusing optical field. The phase and frequency fluctuations imposed on the laser beam by a system of modulators are described by two parameters corresponding to the bandwidth \ensuremath{\beta} and the spectral density b of the noise spectrum. The shape of the laser power spectrum is continuously variable between nearly Gaussian (b/\ensuremath{\beta}\ensuremath{\gg}1) and nearly Lorentzian (b/\ensuremath{\beta}\ensuremath{\ll}1). Measurements are made with a well-collimated Yb atomic beam crossed at right angles by a linearly polarized laser beam. In accord with theoretical predictions, the full width at half maximum (FWHM) of the Hanle effect coherence dip is found to increase as the square root of the laser intensity for monochromatic excitation. For a given intensity, the FWHM decreases as the laser bandwidth increases. With the same laser intensity and spectral width, the FWHM of the Hanle coherence dip increases as the laser spectral profile approaches the Gaussian limit, particularly when \ensuremath{\beta}/2\ensuremath{\pi} is within an order of magnitude of the natural width of the atomic line. The Hanle signal shapes agree well with theoretical calculations using a Brownian motion phase-diffusion model of the laser field. For large \ensuremath{\beta}, it suffices to apply substitution rules for simple phase diffusion to the density-matrix equations.
We report results of calculations on the Hanle effect in a J=0\ensuremath{\leftrightarrows}J=1 atomic transition with three types of model fluctuating light fields: (a) the Brownian-motion phase-diffusion field, as produced in recent experiments by Arnett et al. [Phys. Rev. A 41, 2580 (1990)]; (b) Gaussian amplitude fluctuations; and (c) the chaotic field model, in which real and imaginary parts of the electric-field amplitude fluctuate. For the stochastic density-matrix equations, we use methods developed by Zoller and co-workers [e.g., Dixit, Zoller, and Lambropoulos, Phys. Rev. A 21, 1289 (1980)] employing the Fokker-Planck operator and leading to matrix continued-fraction expansions. The Hanle effect is of interest as a prototype for multisublevel atomic transitions. The width of the Hanle dip at zero magnetic field reflects the tendency of the light field to preserve the coherence between excited-state sublevels. For monochromatic light, the Hanle dip width increases as the square root of light intensity. When the laser bandwidth increases, power broadening of the coherence dip normally decreases. However, with the Brownian-motion phase-diffusion model, if the laser spectral profile is nearly Gaussian, broadening the laser up to several times the natural width of the atomic line does not diminish the Hanle dip width. With amplitude fluctuations, even in the limit of monochromatic light, power broadening of the Hanle dip with intensity is reduced by one-third to one-half depending on the particular model.
Calculations on the Hanle effect in a J - J:i atomic transition are performed for laser fields with a) Brownian motion phase diffusion b) Gaussian amplitude fluctuations and c) fluctuating real and imaginary parts of the laser electric field amplitude (chaotic field model). We use the Fokker-Planck operator approach to stochastic differential equations. The zero-field level-crossing dip broadens at high laser intensity significantly less so as the laser bandwidth increases or if the amp]itu(le fluctuates.
The behaviour of a two-level atom saturated by a resonant fluctuating laser beam is studied. A classical description of the light wave is used, corresponding to a laser well above threshold. Two different approximations are introduced for treating the effects of slow and fast fluctuations; slow fluctuations are followed adiabatically by the atom and fast fluctuations appear as a relaxation mechanism for the population difference and for the optical dipole moment obeying the motional-narrowing condition. The perturbation of the spectral distribution of the fluorescence light is determined quantitatively and the great sensitivity of this spectrum to higher order correlation functions of the light wave is exhibited.