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Physical Review A

Observation of modulated Hanle resonances in phase-diffusing optical fields

Gautam Vemuri, Michael H. Anderson, J. Cooper, Stuart James Smith

Published Dec 1, 1991• 6 citations
Publisher

Abstract

We report an experiment on the observation of Hanle resonances in weakly sinusoidally amplitude modulated, nonsaturating, phase-diffusing optical fields. The $^{1}$${\mathit{S}}_{0}$ to $^{1}$${\mathit{P}}_{1}$ transition (J=0 to 1) in an atomic beam of $^{138}\mathrm{Ba}$ was probed with a frequency stabilized dye laser, whose bandwidth and band shape 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 (\ensuremath{\Delta}E) of the excited ${\mathit{m}}_{\mathit{j}}$=\ifmmode\pm\else\textpm\fi{}1 sublevels equaled the modulation frequency (\ensuremath{\Omega}) for monochromatic light, and at a \ensuremath{\Delta}E of \ensuremath{\Omega}/2 for broadband light. The results are in good agreement with the theoretical predictions of Saxena and Agarwal [Phys. Rev. A 25, 2123 (1982)].

DOI
10.1103/physreva.44.7635

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We report an experiment on the observation of Hanle resonances in weakly sinusoidally amplitude modulated, nonsaturating, phase-diffusing optical fields. The ^{1}$${\mathit{S}}_{0} to ^{1}$${\mathit{P}}_{1} transition (J=0 to 1) in an atomic beam of 138Ba^{138}\mathrm{Ba}138Ba was probed with a frequency stabilized dye laser, whose bandwidth and band shape 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 (\ensuremath{\Delta}E) of the excited mj{\mathit{m}}_{\mathit{j}}mj​=\ifmmode\pm\else\textpm\fi{}1 sublevels equaled the modulation frequency (\ensuremath{\Omega}) for monochromatic light, and at a \ensuremath{\Delta}E of \ensuremath{\Omega}/2 for broadband light. The results are in good agreement with the theoretical predictions of Saxena and Agarwal [Phys. Rev. A 25, 2123 (1982)].
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
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.
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.
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 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.
The dependence of the optical Autler-Townes effect on laser field phase fluctuations describable by a two-dimensional Markovian process (phase-diffusion model with a non-Lorentzian line shape) is investigated experimentally in atomic sodium. An intense cw laser tuned to near resonance with the 3S1/2{3\mathrm{S}}_{1/2}3S1/2​(F=2,MF{\mathrm{M}}_{\mathrm{F}}MF​=2)\ensuremath{\rightarrow}3P3/2{3\mathrm{P}}_{3/2}3P3/2​(F=3,MF{\mathrm{M}}_{\mathrm{F}}MF​=3) transition was frequency modulated to generate phase fluctuations with the desired characteristics. A weak probe laser coupling to the 4D5/2{D}_{5/2}D5/2​(F=4, MF{M}_{F}MF​=4) state was used to obtain the double-peaked optical Autler-Townes absorption profiles in order to investigate their dependencies on band shape, bandwidth, intensity, and detuning of the intense fluctuating laser field. Reversals of the peak-height asymmetry and reversion to normal asymmetry with increased detuning were observed and measured. Quantitative studies of the spectral widths and splittings of the peaks are also reported. Measured absorption profiles are in excellent agreement with calculations based on previous theoretical work.