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Physical Review A

Theory of velocity-dependent, collision-broadened, Doppler-free line shapes

M. J. O’Callaghan, J. Cooper

Published Jun 1, 1989• 23 citations
Publisher

Abstract

Line-shape expressions for two-step resonant excitation in the presence of low-pressure perturbers are derived. These expressions allow for velocity-dependent broadening rates and are well suited for studying the line-shape contributions due to coherent excitation, incoherent excitation (due to dephasing collisions), interference between coherent and incoherent excitation, and velocity-changing collisions. Using the sodium 3S-3P-4D system perturbed by noble-gas atoms as an example and assuming ${C}_{6}$ interaction potentials, line shapes at a variety of laser detunings (Doppler selected atomic velocities) are numerically calculated to study variation in the line shape due to competing excitation paths and speed-dependent collision rates. In order to deal with both the line core and the line wing a composite collision kernel consisting of a Lorentzian kernel for small \ensuremath{\Delta}${v}_{z}$ collisions and a Keilson-Storer kernel for large \ensuremath{\Delta}${v}_{z}$ collisions is used. It is also demonstrated how the inelastic (3${P}_{3/2}$-3${P}_{1/2}$) velocity-changing collision kernel may be obtained from experimental data at low pressures. Theoretical line shapes and broadening rates are compared with experiment.

DOI
10.1103/physreva.39.6206

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Physical Review A•Aug 1, 1983•36 citations
Line-shape expressions for two-step resonant excitation in the presence of low-pressure perturbers are derived. These expressions allow for velocity-dependent broadening rates and are well suited for studying the line-shape contributions due to coherent excitation, incoherent excitation (due to dephasing collisions), interference between coherent and incoherent excitation, and velocity-changing collisions. Using the sodium 3S-3P-4D system perturbed by noble-gas atoms as an example and assuming C6{C}_{6}C6​ interaction potentials, line shapes at a variety of laser detunings (Doppler selected atomic velocities) are numerically calculated to study variation in the line shape due to competing excitation paths and speed-dependent collision rates. In order to deal with both the line core and the line wing a composite collision kernel consisting of a Lorentzian kernel for small \ensuremath{\Delta}vz{v}_{z}vz​ collisions and a Keilson-Storer kernel for large \ensuremath{\Delta}vz{v}_{z}vz​ collisions is used. It is also demonstrated how the inelastic (3P3/2{P}_{3/2}P3/2​-3P1/2{P}_{1/2}P1/2​) velocity-changing collision kernel may be obtained from experimental data at low pressures. Theoretical line shapes and broadening rates are compared with experiment.
Measurements of resonant, two-step excitation of sodium in the presence of very-low-pressure buffer gases (He, Ar, Xe, and N2{\mathrm{N}}_{2}N2​) are reported. These data are analyzed to isolate the effects of line broadening and velocity-changing (VC) collisions in the single-collision limit. Velocity dependence of the several line broadenings involved is obtained from this data analysis. The inelastic VC kernels are also obtained as a function of initial velocity.
We report the results of an experimental study of the effects of velocity-changing collisions on two-photon and stepwise-absorption line shapes. Excitation spectra for the $3{S}_{\frac{1}{2}}\ensuremath{\rightarrow}3{P}_{\frac{1}{2}}\ensuremath{\rightarrow}4{D}_{\frac{1}{2}}$ transitions of sodium atoms undergoing collisions with foreign gas perturbers are obtained. These spectra are obtained with two cw dye lasers. One laser, the pump laser, is tuned 1.6 GHz below the $3{S}_{\frac{1}{2}}\ensuremath{\rightarrow}3{P}_{\frac{1}{2}}$ transition frequency and excites a nonthermal longitudinal velocity distribution of excited 3P123{P}_{\frac{1}{2}}3P21​​ atoms in the vapor. Absorption of the second (probe) laser is used to monitor the steady-state excited-state distribution which is a result of collisions with rare gas atoms. The spectra are obtained for various pressures of He, Ne, and Kr gases and are fit to a theoretical model which utilizes either the phenomenological Keilson-St\"orer or the classical hardsphere collision kernel. The theoretical model includes the effects of collisionally aided excitation of the 3P123{P}_{\frac{1}{2}}3P21​​ state as well as effects due to fine-structure state-changing collisions. Although both kernels are found to predict line shapes which are in reasonable agreement with the experimental results, the hard-sphere kernel is found superior as it gives a better description of the effects of large-angle scattering for heavy perturbers. Neither kernel provides a fully adequate description over the entire line profile. The experimental data is used to extract effective hard-sphere collision cross sections for collisions between sodium 3P123{P}_{\frac{1}{2}}3P21​​ atoms and helium, neon, and krypton perturbers.
The Rautian-Sobelman approach describing the collision-Doppler shape of spectral line that takes into account both soft and hard velocity-changing collisions is extended to the case when correlation between Doppler and collision broadening as well as the dispersion line asymmetry should be considered. Numerical calculations show that speed-dependent effects can produce the additional line narrowing as well as broadening and also the line asymmetry. It is shown that speed-dependent profiles based on the soft and hard collision models differ much more one from another than those which omit the speed-dependent effects.
A theory of the influence of binary collisions on the line shapes associated with Doppler-free spectroscopy is presented. The specific calculation is for two-photon absorption through a real or virtual intermediate state, but the extension of the results to alternative level schemes is included. Collisions are treated quite generally at first, but, ultimately, a simple but reasonable collision model is adopted to properly account for both the "phase-interrupting" and "velocity-changing" aspects of collisions. It is shown that collisional processes can be used to distinguish between the "two-quantum" and "stepwise" contributions to the two-photon excitation rate. Moreover, it is demonstrated that systematic experimental line-shape investigations, in addition to providing tests of current collision theories, can lead to values for collision broadening and shift parameters, excitation transfer rates, magnetic substate relaxation rates, velocity-changing collision rates, and collision kernels. Consequently, Doppler-free spectroscopy can provide a new and important probe of collision effects in atomic and molecular systems.
We report a series of measurements of the nearly degenerate four-wave-mixing response in atomic sodium in the presence of a helium buffer gas. The results show the line shape associated with both the ground-state and excited-state dynamics. The line shape evolves as a function of pressure from a dip at a pressure of 0 Torr to a spike at a few Torr due to the presence of collisions. The width of the spike is determined by residual Doppler broadening. At higher pressure (\ensuremath{\ge}10 Torr), the width of the spike is further narrowed below the residual Doppler width. The additional reduction is due to a form of Dicke narrowing associated with the dynamics of the lower- and upper-level excitations. The results are in qualitative agreement with a perturbation solution of the density-matrix equations which include the effects of velocity-changing collisions and state-changing collisions.
We report on a quantitative experimental investigation of velocity-changing collisions by means of velocity-selective optical pumping (VSOP). We have calculated the VSOP line shape for an atom with hyperfine structure with the use of two phenomenological kernels for the collision effects: the Keilson-Storer kernel, and a two-term kernel consisting of a broad Keilson-Storer part and a narrower Gaussian component. Corrections were included to account for the finite absorption in the sample and the backward reflection of the pumping beam. The experiments were carried out in sodium vapor with neon as the perturber gas. The D1{D}_{1}D1​ line of sodium was used for optical pumping, and the orientation of the ground state was detected. Free parameters of the theory were determined by fitting the predicted line shapes to experimental curves. The Keilson-Storer kernel proved unsatisfactory, but the two-term kernel reproduced well the observed line shapes over the entire collision profiles in the neon pressure range 0-57 mtorr. In an independent experiment using rapidly modulated VSOP we also measured directly the cross section of velocity-changing collisions: $\ensuremath{\sigma}=(1.13\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ cm2{\mathrm{cm}}^{2}cm2. The large weight obtained for the narrow Gaussian from the fits, as well as the collision cross section which is three times as large as the cross section deduced from tabulated gas kinetic radii, may indicate the presence of collisions with relatively small velocity changes in addition to hard-sphere encounters.