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Physical Review A

Quantum-electrodynamic calculation of hyperfine-state populations in atomic sodium

Peter M. Farrell, W. R. MacGillivray, M. C. Standage

Published Jun 1, 1988• 58 citations
Publisher

Abstract

A quantum-electrodynamic (QED) description of the resonant interaction of monochromatic light with the ${3}^{2}$${\mathit{S}}_{1/2}$(F'=2)\ensuremath{\rightarrow}${3}^{2}$${\mathit{P}}_{3/2}$(F=3,2,1) hyperfine transitions of the sodium ${D}_{2}$ 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 ${3}^{2}$${P}_{3/2}$ level after transversing a 1-mm-diam laser beam is 1.15 mW/${\mathrm{mm}}^{2}$ for \ensuremath{\pi} excitation and 1.43 mW/${\mathrm{mm}}^{2}$ for \ensuremath{\sigma} excitation.

DOI
10.1103/physreva.37.4240

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A quantum-electrodynamic (QED) description of the resonant interaction of monochromatic light with the {3}^{2}$${\mathit{S}}_{1/2}(F'=2)\ensuremath{\rightarrow}{3}^{2}$${\mathit{P}}_{3/2}(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 {3}^{2}$${P}_{3/2} 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 resonant excitation of the sodium D2 transition by laser radiation was investigated experimentally in this thesis. Line polarization measurement of the resonance fluorescence was employed to investigate the excitation process. The sodium atoms, which were collimated in a very narrow beam, were excited by a monochromatic laser beam with the frequency tuned to resonance with the 32S 112(F=2) - 32P312 hyperfine transition. The direction of the laser beam was perpendicular to the atomic beam and the fluorescence photons were detected along a direction perpendicular to both the laser and atomic beams. The experimental measurements were performed to determine the effect due to the variation of the laser intensity and frequency on the line polarization. A theoretical expression of the line polarization as a function of density matrix elements of the upper excited hyperfine sub -states was derived. A full quantum electrodynamic theory was used to derive an atomic operator equation of motion which was needed to calculate atomic operator elements. These elements then yielded the corresponding density matrix elements. The experimental results and the theoretical modelling were in a good agreement. The first experimental results for the line polarization of the highly excited states, 42D512 and 42D312, are also presented. The use of two lasers enabled these highly excited states to be excited. The line polarization of the fluorescence for transitions from the 4D level down to lower states is derived in terms of the density matrix elements of the upper excited states. Two limiting cases for the theoretical predictions of the line polarization are considered.
We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 2^{2}2S1/2{\mathit{S}}_{1/2}S1/2​--3 2^{2}2P1/2{\mathit{P}}_{1/2}P1/2​ electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 2^{2}2S1/2{\mathit{S}}_{1/2}S1/2​ and the 3 2^{2}2P1/2{\mathit{P}}_{1/2}P1/2​ levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.
The quantum electrodynamic description of circularly and linearly polarized, single mode radiation exciting atomic transitions in a hyperfine manifold is extended to the more general case of elliptic polarization. Calculations for the sodium D2 line are tested by observing the line polarization of the resonance flourescence scattered perpendicular to the plane containing laser and atomic beams as a function of the laser polarization. Very good agreement between the calculations and the measurements is achieved. It is demonstrated that predetermined populations and substate coherences can be engineered by optically pumping the transition with the appropriately polarized radiation.
Detailed analysis of spectral line broadening and variations in relative intensities of hyperfine spectral components due to optical pumping is presented. Hyperfine levels of sodium 3p1/23{p}_{1/2}3p1/2​ and 3p3/23{p}_{3/2}3p3/2​ levels are selectively excited in a supersonic beam at various laser intensities under the conditions when optical pumping time is shorter than transit time of atoms through the laser beam. The excitation spectra exhibit significant line broadening at laser intensities well below the saturation intensity, and redistribution of intensities of hyperfine spectral components is observed, which in some cases is contradicting with intuitive expectations. Theoretical analysis of the dynamics of optical pumping shows that spectral line broadening sensitively depends on the branching coefficient of the laser-driven transition. Analytical expressions for branching ratio dependent critical Rabi frequency and critical laser intensity are derived, which give the threshold for onset of noticeable line broadening by optical pumping. The critical laser intensity has its smallest value for transitions with branching coefficient equal to 0.5, and it can be much smaller than the saturation intensity. Transitions with larger and smaller branching coefficients are relatively less affected. The theoretical excitation spectra were calculated numerically by solving density matrix equations of motion using the split propagation technique, and they well-reproduce the observed effects of line broadening and peak intensity variations. The calculations also show that presence of dark (i.e., not laser coupled) Zeeman sublevels in the lower state results in effective branching coefficients which vary with laser intensity and differ from those implied by the sum rules, and this can lead to peculiar changes in peak ratios of hyperfine components of the spectra.
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 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 $J=1/2\ensuremath{\leftrightarrow}J=1/2$ approximation.