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Physical Review A

Semiclassical theory for the interaction dynamics of laser light and sodium atoms including the hyperfine structure

S. Dangel, R. Holzner

Published Nov 1, 1997• 7 citations
Publisher

Abstract

The conventional $J=1/2\ensuremath{\leftrightarrow}J=1/2$ model is successful for the qualitative description of the interaction of sodium atoms and laser light tuned close to the atomic ${D}_{1}$ transition. However, time constants for dynamic effects due to optical pumping predicted by the model are about an order of magnitude too short compared to experimental observations. The inclusion of hyperfine states in the quantum-mechanical description of the atom leads to a satisfactory model that also predicts the correct absorption due to atomic diffusion and allows for the description of effects related to population trapping. This hyperfine model is derived in detail and compared with the $J=1/2\ensuremath{\leftrightarrow}J=1/2$ approximation.

DOI
10.1103/physreva.56.3937

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The conventional J=1/2\ensuremath↔J=1/2J=1/2\ensuremath{\leftrightarrow}J=1/2J=1/2\ensuremath↔J=1/2 model is successful for the qualitative description of the interaction of sodium atoms and laser light tuned close to the atomic D1{D}_{1}D1​ transition. However, time constants for dynamic effects due to optical pumping predicted by the model are about an order of magnitude too short compared to experimental observations. The inclusion of hyperfine states in the quantum-mechanical description of the atom leads to a satisfactory model that also predicts the correct absorption due to atomic diffusion and allows for the description of effects related to population trapping. This hyperfine model is derived in detail and compared with the J=1/2\ensuremath↔J=1/2J=1/2\ensuremath{\leftrightarrow}J=1/2J=1/2\ensuremath↔J=1/2 approximation.
A quantum-electrodynamic (QED) description of the resonant interaction of monochromatic light with the 32{3}^{2}32S1/2{\mathit{S}}_{1/2}S1/2​(F'=2)\ensuremath{\rightarrow}32{3}^{2}32P3/2{\mathit{P}}_{3/2}P3/2​(F=3,2,1) hyperfine transitions of the sodium D2{D}_{2}D2​ line is formulated in terms of the Heisenberg atomic operator. Off-diagonal state coherences and all relaxation terms are included. It is found that the equations for the populations, optical coherences, and certain state coherences form a closed subset of the total system of equations. This subset is small enough to be computed numerically. The results of calculations with this model are compared with three previously developed semiclassical descriptions using the density operator. One of the semiclassical models was developed to describe the interaction of weak light with the atomic transition while another is suitable in the case of high-intensity light. For both \ensuremath{\pi} and \ensuremath{\sigma} excitation it is shown that the QED calculated values for the time-averaged, excited-level population probability converge to those of each of the semiclassical models in the appropriate limit of light intensity. For homogeneous broadening, results show that the optimum light intensity to obtain the largest fraction of atoms in the 32{3}^{2}32P3/2{P}_{3/2}P3/2​ level after transversing a 1-mm-diam laser beam is 1.15 mW/mm2{\mathrm{mm}}^{2}mm2 for \ensuremath{\pi} excitation and 1.43 mW/mm2{\mathrm{mm}}^{2}mm2 for \ensuremath{\sigma} excitation.
The atomic collision parameters ?, ?, ?, ?, P, |P| and |�| were determined from measurements of the pseudo Stokes parameters for superelastic scattering of 20 eV incident electrons at angles of 10� and 15� from the 32P312 excited state of sodium. Two depolarising factors K and K' for the pseudo Stokes parameters are defined for it and a laser excitation respectively to account for the depolarising influence due to optical pumping in the hyperfine structure of the 32P3t2 excited state. A theoretical model of the laser excitation of the sodium transition is developed to determine K and K'. The theory developed is a quantum -electrodynamic description of the resonant interaction of monochromatic light with the 32S in(F'=2) <-> 32P3a(F=3,2,1) hyperfine transitions of the sodium D2 line formualated in terms of the Heisenberg atomic operator. Off diagonal state coherences and all relaxation terms are included. It is found that the equations for the populations, optical coherences and certain state coherences form a closed subset of the total system of equations. This subset is sufficiently small to enable numerical computation. As well as K and K', the model is used to calculate the probability of occupation of the various atomic states and the polarisation of the sideways scattered fluorescence from the laser excited atoms. The results of this model are compared with the predictions of four other models, a rate equation model which includes only the F=3 excited state, a semiclassical model based on the density operator, a low light intensity approximation to the semiclassical model and a high intensity model which uses a transformed basis for the states. For both it and a laser excitation it is shown that the QED calculated values converge to the values predicted by the high and low intensity approximations in the appropriate limits.
The time-dependent population dynamics of hyperfine (HF) levels of n 2 p 3/2 states is examined for cyclic transitions in alkali atoms. We study a slow and cold atomic beam of Na (n = 3) and Cs (n = 6), taking into account the long interaction time of light with atoms (~200 μs) inside the resonant laser beam. Simple analytical expressions for the populations of the excited states and for the intensities of the absorption lines are derived for a three-level system model. We show that at moderate pump laser power the mixing of HF levels is sufficient to form a flow of population from a cyclic transition to partially open transitions. We discuss various phenomena associated with the evolution of optical pumping that cannot be explained by general analysis of two-level system model.
We have developed a model for calculating saturated absorption spectra for dipole transitions in multi-level atoms. Using a semiclassical density matrix formalism, we derive a set of coupled differential equations for the internal state of the atom in a standing wave light field. The equations are solved using standard integration techniques. The absorption at each laser detuning is found from an average of the absorption for a number of velocities along the laser field, thermally weighted. The method is relatively efficient computationally yet quantitatively predicts important details of saturated absorption spectra including saturation, crossover resonances, merging of absorption lines at high intensity and optical pumping between hyperfine levels. We have measured saturated absorption and fluorescence spectra of 85 Rb, and compare to our computational results for a 36-level model.
Optical pumping of the ground states of sodium can radically alter the shape of the laser-induced fluorescence excitation spectrum, complicating measurements of temperature, pressure, etc., which are based on these spectra. Modeling of the fluorescence using rate equations for the eight hyperfine states of the sodium D manifolds can be used to quantify the contribution to the ground state pumping of transitions among the hyperfine excited states induced by collisions with buffer gas atoms. This model is used here to determine, from the shape of experimental spectra, cross sections for $\ensuremath{\Delta}F$ transitions of the P3/2{P}_{3/2}P3/2​ state induced by collisions with helium and argon atoms, for a range of values assumed for the ${P}_{1/2}, \ensuremath{\Delta}F$ cross sections. The hyperfine cross sections measured using this method, which to our knowledge is novel, are compared with cross sections for transitions involving polarized magnetic substates mF{m}_{F}mF​ measured previously using polarization sensitive absorption. Also, fine-structure transition $(\ensuremath{\Delta}J)$ cross sections were measured in the pumped vapor, giving agreement with previous measurements made in the absence of pumping.