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Physical Review A

Role of collisional repopulation of atomic levels on coherent and redistributed components in resonant light scattering

J. Cooper, R. J. Ballagh

Published Sep 1, 1978• 29 citations
Publisher

Abstract

Resonant scattering (redistribution) of weak monochromatic light by atoms in the presence of collisions has been considered by Omont et al., who allowed for the broadening of both upper and lower levels of the transition. Mollow (who also has considered this problem for intense radiation) has shown that their results are applicable only if collisional repopulation of the levels (in contrast to collisional depopulation) is ignored. We here consider in detail the effect of repopulation. It leads, in the weak-field limit, to a true-coherent component ($\ensuremath{\delta}$-function spectrum) at the driving frequency, as well as a component at this frequency whose width is essentially the inelastic collisional rate of the lower state. An additional redistributed component due to repopulation of the upper level is also obtained.

DOI
10.1103/physreva.18.1302

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Resonant scattering (redistribution) of weak monochromatic light by atoms in the presence of collisions has been considered by Omont et al., who allowed for the broadening of both upper and lower levels of the transition. Mollow (who also has considered this problem for intense radiation) has shown that their results are applicable only if collisional repopulation of the levels (in contrast to collisional depopulation) is ignored. We here consider in detail the effect of repopulation. It leads, in the weak-field limit, to a true-coherent component (\ensuremathδ\ensuremath{\delta}\ensuremathδ-function spectrum) at the driving frequency, as well as a component at this frequency whose width is essentially the inelastic collisional rate of the lower state. An additional redistributed component due to repopulation of the upper level is also obtained.
The techniques of modern line-broadening theory are used to investigate the scattering of polarized radiation in the rest frame of an atom undergoing collisions. The formulation explicitly includes both elastic and inelastic (quenching) collisions. When the lower state has zero width, a form for the redistribution function similar to that of Zanstra is obtained, but with the redistribution in the neighborhood of the resonance line being caused solely by elastic collisions. In the limit of no collisions, but with both levels of finite lifetime, the result of `veisskopf and Woolley is obtained. The effect of level-degeneracy is also explicitly included; in this case the results are a function of the polarization of the light and the different relaxation rates for the multipolar components of the atomic states.
We extend the theory of light scattering from an atom undergoing collisions to the case of strong laser fields, where the expansion in powers of $\ensuremath{\Omega}{\ensuremath{\tau}}_{c}$ that was used formerly in a series of papers by Burnett et al. breaks down. In particular we consider in detail the conditions necessary to relate the observable quantities to intense-field collisional rate constants in a dressed-state basis (such as those calculated by Light and Sz\"oke). We also show (following Rabin and Ben-Reuven) that by studying the spectrum emitted by the atoms in the presence of a strong field one may measure the collisional rates for transfer between the dressed states of atom plus radiation field.
Light scattering of monochromatic radiation by atoms and dilute vapours is reviewed. The relative contributions of incoherent and coherent scattering are calculated for weak incident fields. It is shown that, to lowest order in the intensity of the driving field, there can be several inelastic components in the scattered spectrum. The positions and widths of these components are related to light shifts and optical pumping of the atoms in the field. The complications that one encounters in scattering by two atoms, moving atoms, and atomic ensembles are discussed in qualitative terms.
A theory of resonant light scattering from two level atoms in a gas is developed for an arbitrarily strong pump field. The Doppler effect and two body foreign gas collisions are considered. Elastic, inelastic (upward and downward), and the velocity changing collisions are retained in the general formulation. The impact approximation is not made, and the expressions for all (field dependent) collision rates are generated internally. Particular attention is paid to the lower state interaction, and a more complete parameterization of the theory is achieved. The general theory is non-Markovian and the quantum regression theorem is not used; it reduces to existing theories in special cases. It is shown that the coherent scattering is not a general property of the system if the lower state interaction is present; the "coherent" component acquires a small, but nonzero, width in the strong field region. This width could provide information about some new collision parameters.
Explicit expressions for the angular distribution and frequency dependence of radiation scattered by a spatially degenerate atom undergoing collisions are derived using a previously developed formalism of the quantum theory of line broadening for an atom in its rest frame. Results valid in the impact regime for resonant scattering are presented for the cases of linear incident and scattered polarization circular incident and linear scattered polarization, and unpolarized incident radiation with no monitoring of the scattered polarization. These results relate to scattering involving a j - j + or - 1 - j atomic transition, with lower-state interactions neglected; the expressions obtained take a relatively simple form involving atomic multipolar relaxation rates of the order of k = 0 and k = 2. The angular distributions for each case are combined with the frequency dependences to yield a three-component frequency redistribution function for an s-p-s transition, the components being a coherent term, a frequency-redistributed term with the same angular dependence as the coherent term, and an isotropic frequency-redistributed term.
We present a calculation of the scattering of monochromatic radiation by a degenerate atom in the binary-collision approximation. We limit ourselves to field strengths such that $\ensuremath{\Omega}{\ensuremath{\tau}}_{c}\ensuremath{\ll}1$, where $\ensuremath{\Omega}$ is the Rabi frequency for the atomic transition, and ${\ensuremath{\tau}}_{c}$ is the duration of a strong collision. Our calculation is not limited to the impact regime, $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\tau}}_{c}^{\ensuremath{-}1}$, or to the region where thermal correlations may be neglected, i.e., $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}\frac{\mathrm{kT}}{\ensuremath{\hbar}}$. (Here $\ensuremath{\Delta}\ensuremath{\omega}$ is the difference between either the incoming or outgoing photon's frequency and the atom's natural frequency.) To do this we derive an equation of motion for the correlation function, valid outside the quantum-regression regime, whose solution for practical cases is straightforward. We present solutions for the weak-field (linear-response) regime in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles created.