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Physical Review A

Modulated Hanle signals in partially coherent fields

Ragini Saxena, G. S. Agarwal

Published Apr 1, 1982• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.25.2123

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Observation of modulated Hanle resonances in phase-diffusing optical fields

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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 $^{138}\mathrm{Ba}$ 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)].

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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.

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

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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.

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Optical Hanle effect in fields of arbitrary strength and bandwidth

P. Anantha Lakshmi, G. S. Agarwal

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.

Physical Review A•May 1, 1981•10 citations
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 \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ. 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 s=\ifmmode±\else\textpm\fi\ensuremathΩ2(0,\ifmmode±\else\textpm\fi\ensuremathΩ)\mathcal{s}=\frac{\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega}}{2}(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\Omega})s=2\ifmmode±\else\textpm\fi\ensuremathΩ​(0,\ifmmode±\else\textpm\fi\ensuremathΩ) 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 \ensuremathΩ=0,\ifmmode±\else\textpm\fi\ensuremathα0\ensuremath{\Omega}=0,\ifmmode\pm\else\textpm\fi{}{\ensuremath{\alpha}}_{0}\ensuremathΩ=0,\ifmmode±\else\textpm\fi\ensuremathα0​, \ifmmode±\else\textpm\fi2\ensuremathα0(\ensuremathα02=s2+2\ensuremathα2)\ifmmode\pm\else\textpm\fi{}2{\ensuremath{\alpha}}_{0}({\ensuremath{\alpha}}_{0}^{2}={\mathcal{s}}^{2}+2{\ensuremath{\alpha}}^{2})\ifmmode±\else\textpm\fi2\ensuremathα0​(\ensuremathα02​=s2+2\ensuremathα2) due to the dynamical Stark splitting of the various energy levels, with the strength of the resonances depending on the direction of detection.
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)].
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 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.
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.
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.
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.