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Physical Review A

Theory of collision effects in Doppler-free spectroscopy

P. R. Berman

Published Jun 1, 1976• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.13.2191

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Effects of velocity-changing collisions on two-photon and stepwise-absorption spectroscopic line shapes

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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 $3{P}_{\frac{1}{2}}$ 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 $3{P}_{\frac{1}{2}}$ 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 $3{P}_{\frac{1}{2}}$ atoms and helium, neon, and krypton perturbers.

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Collision kernels and laser spectroscopy

P. R. Berman, THOMAS W. MOSSBERG, Sven R. Hartmann

Collisional processes occurring within an atomic vapor can be conveniently described in terms of collision kernels. The population kernel ${W}_{\mathrm{ii}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ gives the probability density per unit time that an "active" atom in state $i$ undergoes a collision with a perturber that changes the active atom's velocity from ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}$ to $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. For active atoms in a linear superposition of states $i$ and $j$, there is an analogous coherence kernel ${W}_{\mathrm{ij}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}) (i\ensuremath{\ne}j)$ reflecting the effects of collisions on the off-diagonal density-matrix element ${\ensuremath{\rho}}_{\mathrm{ij}}$. In this work, we discuss the general properties of the collision kernels which characterize a two-level active atom which, owing to the action of a radiation field, is in a linear superposition of its two levels. Using arguments based on the uncertainty principle, we show that collisions can be divided roughly into the following two categories: (1) collisions having impact parameters less than some characteristic radius which may be described classically and (2) collisions having impact parameters larger than this characteristic radius which give rise to diffractive scattering and must be treated using a quantum-mechanical theory. For the population kernels, collisions of type (1) can lead to a large-angle scattering component, while those of type (2) lead to a small-angle (diffractive) scattering component. For the coherence kernel, however, assuming that the collisional interaction for states $i$ and $j$ differ appreciably, only collisions of type (2) contribute, and the coherence kernel contains a small-angle scattering component only. The absence of a large-angle scattering component in the coherence kernel is linked to a collision-induced spatial separation of the trajectories associated with states $i$ and $j$. Interestingly enough, the width of the diffractive kernel, as measured in the laboratory frame, is found to be insensitive to the perturber to active-atom mass ratio. To illustrate these features, a specific calculation of the kernels is carried out using a hard-sphere model for the scattering. The relationship of the present description of collisions to that of traditional pressure-broadening theory in which trajectory separation effects are ignored is discussed. It is explained why traditional pressure-broadening theory correctly describes collision effects in linear spectroscopy, but fails to provide an adequate description of some saturation spectroscopy and photonecho experiments in which velocity-changing collisions associated with the coherence kernel play a significant role. An expression for the collisionally modified photon-echo amplitude is derived which clearly displays the role played by velocity-changing collisions associated with the coherence kernel.

Physical Review A•May 1, 1982•91 citations
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 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.
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.
A theory of saturation spectroscopy in three-level gas-vapor systems, including collisional effects, is presented. Using a model of collisions in which they are phase interrupting in their effect on level coherences and velocity changing in their effect on level population densities, the authors calculate the absorption profile of a weak probe field on a given transition when an arbitrarily strong pump field acts on a coupled transition. Line shapes are derived for an arbitrary collision kernel describing the velocity-changing collisions, and these line shapes are evaluated for the collision kernel proposed by Keilson and Storer. Various limiting forms for the line shapes are derived and representative line shapes are displayed. Several new features of the line shapes, including collisionally induced increased probe absorption are discussed. The line shapes are seen to reflect the various collisional processes that may occur in atomic or molecular systems.
A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.
Non-stationary two-photon absorption coherent spectroscopy free of Doppler broadening in an atomic gas has been considered in detail. Correct account has been taken of the level degeneracy and depolarizing collisions. This two-photon Doppler-free spectroscopy has been developed for determining the relaxation rates of the atomic multipole moments in gas. A criterion to test the adopted model of the depolarizing collisions has been developed. A possible method for the identification of the atomic transitions in the angular momentum has been formulated. A technique of precisely determining the separation between the energy levels of the two-photon transition has been proposed. The conditions under which the Stokes wave is conjugated with respect to the probe pulse in amplitude, phase and polarization have been determined for the atomic gas in the presence of depolarizing collisions.
Collisional processes occurring within an atomic vapor can be conveniently described in terms of collision kernels. The population kernel ${W}_{\mathrm{ii}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ gives the probability density per unit time that an "active" atom in state iii undergoes a collision with a perturber that changes the active atom's velocity from ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}$ to $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. For active atoms in a linear superposition of states iii and jjj, there is an analogous coherence kernel ${W}_{\mathrm{ij}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}^{\ensuremath{'}}\ensuremath{\rightarrow}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}) (i\ensuremath{\ne}j)$ reflecting the effects of collisions on the off-diagonal density-matrix element ${\ensuremath{\rho}}_{\mathrm{ij}}$. In this work, we discuss the general properties of the collision kernels which characterize a two-level active atom which, owing to the action of a radiation field, is in a linear superposition of its two levels. Using arguments based on the uncertainty principle, we show that collisions can be divided roughly into the following two categories: (1) collisions having impact parameters less than some characteristic radius which may be described classically and (2) collisions having impact parameters larger than this characteristic radius which give rise to diffractive scattering and must be treated using a quantum-mechanical theory. For the population kernels, collisions of type (1) can lead to a large-angle scattering component, while those of type (2) lead to a small-angle (diffractive) scattering component. For the coherence kernel, however, assuming that the collisional interaction for states iii and jjj differ appreciably, only collisions of type (2) contribute, and the coherence kernel contains a small-angle scattering component only. The absence of a large-angle scattering component in the coherence kernel is linked to a collision-induced spatial separation of the trajectories associated with states iii and jjj. Interestingly enough, the width of the diffractive kernel, as measured in the laboratory frame, is found to be insensitive to the perturber to active-atom mass ratio. To illustrate these features, a specific calculation of the kernels is carried out using a hard-sphere model for the scattering. The relationship of the present description of collisions to that of traditional pressure-broadening theory in which trajectory separation effects are ignored is discussed. It is explained why traditional pressure-broadening theory correctly describes collision effects in linear spectroscopy, but fails to provide an adequate description of some saturation spectroscopy and photonecho experiments in which velocity-changing collisions associated with the coherence kernel play a significant role. An expression for the collisionally modified photon-echo amplitude is derived which clearly displays the role played by velocity-changing collisions associated with the coherence kernel.