A quantum-statistical theory of optical Hanle effect is developed for arbitrary intensities and bandwidths of the pump field. The excited- and ground-state level shifts, caused by another light field with appropriate polarization, are directly included in the theory by using the method of time averaging. The laser fluctuations are taken into account exactly by using the phase-diffusion model of the pump laser. The analytical and numerical results for the fluorescence signals are presented for a variety of excitations. Our results in the limiting case reduce to those of Kaftandjian, Klein, and Hanle. Finally, the case of a very broadband excitation under saturation conditions is analyzed and the results for signals in various directions and the polarizations are obtained.
A quantum-statistical theory of optical Hanle effect is developed for arbitrary intensities and bandwidths of the pump field. The excited- and ground-state level shifts, caused by another light field with appropriate polarization, are directly included in the theory by using the method of time averaging. The laser fluctuations are taken into account exactly by using the phase-diffusion model of the pump laser. The analytical and numerical results for the fluorescence signals are presented for a variety of excitations. Our results in the limiting case reduce to those of Kaftandjian, Klein, and Hanle. Finally, the case of a very broadband excitation under saturation conditions is analyzed and the results for signals in various directions and the polarizations are obtained.
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 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.
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 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${\ensuremath{\rightarrow}}^{3}$P1 transition at \ensuremath{\lambda}=555.6 nm in 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.
For pt.I see ibid., vol.9, p.357 (1976). A theory of the optical pumping of high-J molecule levels with a broad-band CW laser has previously been developed. The linewidth of the Hanle effect is analysed in the present paper for two particular cases in which an exact integration is possible. These cases are: (i) gT (where g is the Lande factor and T the lifetime) is the same for the two levels; and (ii) one of the Lande factors vanishes. In general, the Hanle resonance is power-broadened for high laser intensities. However, in the case of a zero Lande factor of the ground state, the theory predicts a radiative narrowing of the Hanle effect. It is shown that this most unusual feature is connected to the fact that, in the latter case, the physical observables of the resonance level may get the ground-state lifetime, when the transition is saturated.
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; 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.