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Annual Meeting Optical Society of America

Observation of modulated Hanle resonances in stochastic laser fields

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

Published Jan 1, 1991• 0 citations
Publisher

Abstract

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

DOI
10.1364/oam.1991.we1

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Observation of modulated Hanle resonances in phase-diffusing optical fields

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AC Stark effect and fluorescence in modulated laser beams. II. Two-photon stepwise excitations

Ragini Saxena, G. S. Agarwal

For pt.I see ibid., vol.12, no.12, p.1939 (1979). The modulated fluorescence in an optical double resonance experiment, i.e. when an atom interacts with two intense monochromatic laser beams each of which is weakly modulated in amplitude at frequency Omega , is studied in detail. The modulation rate corresponding to the steady-state population of the highest atomic level exhibits resonant structures at the modulation frequencies Omega =0, +or- alpha 0 , +or-2 alpha 0 due to the dynamical Stark splitting of the various energy levels; alpha 0 being related to the Rabi frequencies alpha 1 and alpha 2 associated with the two lasers and the two atomic transitions, by alpha 0 2 = mod alpha 1 mod 2 + mod alpha 2 mod 2 . The peak at Omega =+or- alpha 0 is the most intense and exhibits splitting when the applied fields are detuned from exact resonance with the respective atomic frequencies. Resonances are observed in the modulation rate of the intermediate-level population at the modulation frequencies Omega =0, +or-2 alpha 0 . In the presence of detunings of the applied fields, additional resonant structures occur at Omega =+or- alpha 0 which increase in magnitude with higher detunings and split. Explicit expressions for the positions, widths and weights of these resonant components are obtained analytically for intense, resonant fields. Numerical results for general values of the laser field strengths and the detunings are given.

Journal of Physics B Atomic and Molecular Physics•Feb 14, 1980•21 citations
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 report an experiment on the observation of Hanle resonances in weakly sinusoidally amplitude modulated, nonsaturating, phase-diffusing optical fields. The 1^{1}1S0{\mathit{S}}_{0}S0​ to 1^{1}1P1{\mathit{P}}_{1}P1​ 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)].
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.
A novel phase-modulation technique which permits subkilohertz-laser stability and new levels of precision in laser spectroscopy was reported recently. For spectroscopy, the basic arrangement consists of a combination of an optical pump and a probe field which is phase modulated. The pump prepares the atomic sample by burning a narrow hole within the atom's inhomogeneous line shape, and the probe beam samples the prepared hole when its modulation sidebands are swept into resonance. Off resonance, the probe is balanced as pairs of sidebands produce heterodyne beat signals of opposite phase which just cancel. On resonance, the balance is upset and yields a nonvanishing beat signal with a Lorentzian absorption or dispersion line shape and with residual noise approaching the shot noise limit. Here we investigate the theory of phase-modulation spectroscopy. We treat the nonlinear response of an atomic two-level quantum system subject to an intense pump and a weak copropagating or counterpropagating phase-modulated probe beam. The density-matrix equations of motion are solved by a Laplace-transform method and by the novel use of a translation operator which allows the infinite hierarchy of coupled equations to close. A solution equivalent to the rate-equation result is developed and coherence corrections are found which predict new resonances that have just been detected in this laboratory. The delayed pump-probe technique encountered in solid-state laser spectroscopy is analyzed in this context for two- and three-level quantum systems. The response of a Fabry-Perot cavity to a phase-modulated light wave is examined also and reveals an unexpected absorption feature.
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.
For pt.I see ibid., vol.12, no.12, p.1939 (1979). The modulated fluorescence in an optical double resonance experiment, i.e. when an atom interacts with two intense monochromatic laser beams each of which is weakly modulated in amplitude at frequency Omega , is studied in detail. The modulation rate corresponding to the steady-state population of the highest atomic level exhibits resonant structures at the modulation frequencies Omega =0, +or- alpha 0 , +or-2 alpha 0 due to the dynamical Stark splitting of the various energy levels; alpha 0 being related to the Rabi frequencies alpha 1 and alpha 2 associated with the two lasers and the two atomic transitions, by alpha 0 2 = mod alpha 1 mod 2 + mod alpha 2 mod 2 . The peak at Omega =+or- alpha 0 is the most intense and exhibits splitting when the applied fields are detuned from exact resonance with the respective atomic frequencies. Resonances are observed in the modulation rate of the intermediate-level population at the modulation frequencies Omega =0, +or-2 alpha 0 . In the presence of detunings of the applied fields, additional resonant structures occur at Omega =+or- alpha 0 which increase in magnitude with higher detunings and split. Explicit expressions for the positions, widths and weights of these resonant components are obtained analytically for intense, resonant fields. Numerical results for general values of the laser field strengths and the detunings are given.