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Journal of the Optical Society of America B

Velocity distribution of laser-induced atomic polarization moments in antirelaxation-coated cell and magneto-optic rotation

A. I. Okunevich

Published Jan 1, 2005• 3 citations
Publisher

Abstract

It is shown with the use of the solution of the Boltzmann equation for laser pumping in a cell with antirelaxation coating that the velocity distribution of atomic polarization moments (PMs) is essentially dependent on the value of the magnetic field H. The z-velocity distribution of PMs in a low field, H approximately 10^−4 A/m, is a Maxwellian one with a small admixture of an almost monokinetic one. At larger field the same distribution remains for longitudinal alignment, but for transverse alignment the Maxwellian part of the distribution disappears (at H approximately 1 A/m). It appears that the radial velocity distribution is also dependent on the field H. A calculation accounting for wall Maxwellization in low field gives for the pumping power needed to saturate the magneto-optic rotation a value similar to the experimentally determined value. It is shown that the known semiphenomenological theory neglecting the Maxwellization gives an acceptable description of magneto-optic rotation only for high (approximately 1 A/m) magnetic field.

DOI
10.1364/josab.22.000029

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Optoelectronics Instrumentation and Data Processing•Jan 1, 2013•4 citations
It is shown with the use of the solution of the Boltzmann equation for laser pumping in a cell with antirelaxation coating that the velocity distribution of atomic polarization moments (PMs) is essentially dependent on the value of the magnetic field H. The z-velocity distribution of PMs in a low field, H approximately 10^−4 A/m, is a Maxwellian one with a small admixture of an almost monokinetic one. At larger field the same distribution remains for longitudinal alignment, but for transverse alignment the Maxwellian part of the distribution disappears (at H approximately 1 A/m). It appears that the radial velocity distribution is also dependent on the field H. A calculation accounting for wall Maxwellization in low field gives for the pumping power needed to saturate the magneto-optic rotation a value similar to the experimentally determined value. It is shown that the known semiphenomenological theory neglecting the Maxwellization gives an acceptable description of magneto-optic rotation only for high (approximately 1 A/m) magnetic field.
The general solution of the Boltzmann kinetic equations obtained previously by the author for the atomic density matrix in a long cylindrical cell with an antirelaxation coating of the walls under conditions of laser pumping in a magnetic field is used for the calculation of the magneto-optical rotation of the light polarization plane. It is shown that, in accordance with experimental data, the dependence of the rotation angle of the polarization plane of the pumping light on a magnetic field ranging from −1 to 1 Oe represents a sum of two antisymmetric curves: a narrow curve with a spacing between extrema of about 2 μOe and a broad curve with a spacing of about 0.03 Oe. It is found that the characteristics (the width and the amplitude) of these two curves depend quite differently on the parameters of the problem—the pumping power, the coating Q factor, the pumping beam radius, and the cell radius. This difference is explained by the fact that the two curves are associated with different physical processes: the narrow curve characterizes the distortion caused by the magnetic field in the atomic polarization accumulated during multiple passages through the pumping beam after almost nondepolarizing collisions with the coating, while the broad curve corresponds to the distortion of the polarization formed during a single passage through the beam.
A general solution of the Boltzmann kinetic equations for the atomic density matrix in a long cylindrical cell with an antirelaxation coating of the walls under conditions of laser pumping in a magnetic field is obtained. The theory involves only one relaxation parameter—the coefficient K refl of the diffusion reflection of polarized atoms from the cell walls. The solution consists of two terms. The first term is proportional to K refl and describes the contribution to the atomic polarization made by multiple passages of atoms through the pumping beam after their reflection from the walls with the antirelaxation coating. The second term does not depend on K refl and describes the polarization formed during a single passage of atoms through the pumping beam. This solution allows one to calculate the Stokes parameters for the pumping beam transmitted through the cell that are measured in experiments on magneto-optical rotation of the light polarization plane.
Larmor precession of laser-polarized atoms contained in antirelaxation-coated cells, detected via nonlinear magneto-optical rotation (NMOR), is a promising technique for a new generation of ultrasensitive atomic magnetometers. For magnetic fields directed along the light propagation direction, resonances in NMOR appear when linearly polarized light is frequency or amplitude modulated at twice the Larmor frequency. Because the frequency of these resonances depends on the magnitude but not the direction of the field, they are useful for scalar magnetometry. Additional NMOR resonances at the Larmor frequency appear when the magnetic field is tilted away from the light propagation direction in the plane defined by the light propagation and polarization vectors. These resonances, studied both experimentally and with a density matrix calculation in the present work, offer a convenient method of achieving additional information about a direction of the magnetic field.
Results of a fairly straightforward experiment on resonant magneto-optical rotation by rubidium-87 atoms revealed strong time-dependence of the polarization plane of light emerging from atomic vapors following a sudden irradiation with a laser beam. The rotation of the plane appears as a not direct consequence of the influence of the magnetic field on atoms. Reported measurements conducted using a vapor cell without any buffer gas or an anti-relaxation wall coating show that transmitted light has initially the same (linear) polarization as the incident one. Rotation of the polarization plane caused by an axial magnetic field develops in time scales similar to the pace of establishing the optical pumping/relaxation equilibrium in the atomic ensemble. The traditional passive Faraday rotation picture providing working description for the resonant magneto-optical effects in steady-state conditions does not explain the observed sequence of evolution of the polarization. The picture has to be augmented with analysis of time dependence in the spatial distribution of relevant quantum states within the atomic ensemble as well as longitudinal and transversal intensity profiles of the light beam. Isolation of external effects caused by the optical pumping and relaxation processes from consequences of internal atomic coherences should uncover more details of mechanisms of the latter in the presence of magnetic field. Plentitude of data available from time-dependent polarization measurements provides strong incentives for further development and tests of comprehensive models of coherent interactions of atomic ensembles with radiation and possibly a base for additional advance in understanding and applications of these phenomena.
Nonlinear magneto-optical Faraday rotation (NMOR) on the potassium D1 and D2 lines was used to study Zeeman relaxation rates in an antirelaxation paraffin-coated $3\text{\ensuremath{-}}\mathrm{cm}$-diameter potassium vapor cell. Intrinsic Zeeman relaxation rates of ${\ensuremath{\gamma}}^{NMOR}∕2\ensuremath{\pi}=2.0(6)\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ were observed. The relatively small hyperfine intervals in potassium lead to significant differences in NMOR in potassium compared to rubidium and cesium. Using laser optical pumping, widths and frequency shifts were also determined for transitions between ground-state hyperfine sublevels of 39K^{39}\mathrm{K}39K atoms contained in the same paraffin-coated cell. The intrinsic hyperfine relaxation rate of ${\ensuremath{\gamma}}_{\mathit{expt}}^{\mathit{hf}}∕2\ensuremath{\pi}=10.6(7)\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ and a shift of $\ensuremath{-}9.1(2)\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ were observed. These results show that adiabatic relaxation gives only a small contribution to the overall hyperfine relaxation in the case of potassium, and the relaxation is dominated by other mechanisms similar to those observed in previous studies with rubidium.
It is predicted theoretically that the combination of elliptically polarized radiation and an antirelaxation coating of the walls of a vacuum cell containing alkali metal vapor provides an asymmetric shape of the magneto-optical resonance. The cause of the asymmetry is the optical pumping of the ground-state hyperfine component which is not destroyed upon collision of the atom with the wall. The effect is most pronounced when the laser frequency is tuned to the resonance with the the F → F + 1 transition.