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Physical Review A

Statistical model for the redistribution of resonantly scattered light

David L. Huber

Published Jul 1, 1986• 0 citations
Publisher

Abstract

We outline a statistical model for the cross section characterizing the integrated intensity of the redistributed radiation resonantly scattered from a localized scattering center coupled to a thermal bath. The theory exploits a formal analogy between the resonant scattering of light and nuclear scattering in the compound-nucleus regime. Using statistical models developed for the nuclear problem, we express the cross sections describing the quasielastic scattering and the redistributed Raman scattering in terms of parameters which can be extracted from measurements of the coherent-scattering cross sections.

DOI
10.1103/physreva.34.629

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We outline a statistical model for the cross section characterizing the integrated intensity of the redistributed radiation resonantly scattered from a localized scattering center coupled to a thermal bath. The theory exploits a formal analogy between the resonant scattering of light and nuclear scattering in the compound-nucleus regime. Using statistical models developed for the nuclear problem, we express the cross sections describing the quasielastic scattering and the redistributed Raman scattering in terms of parameters which can be extracted from measurements of the coherent-scattering cross sections.
An analysis is presented of the redistribution of scattered light in resonant electronic Raman scattering from localized scattering centers. Results are obtained for the integrated cross sections characterizing the coherently scattered and the redistributed radiation. A statistical SSS-matrix theory is outlined which relates the cross section for the redistributed radiation to the coherent-scattering cross section. For the theory to be applicable it is necessary that the absorption lines from the various initial and final states to the resonant intermediate state have identical shapes. The conditions for this to occur are discussed.
We investigate the resonant scattering of photons from an atom in the thermal equilibrium with a phonon bath. A two-level model of the atom is discussed in detail. In the model problem, the atom is coupled directly to the lattice through a diagonal interaction which is bilinear in the lattice quanta. Relaxation effects are introduced indirectly via correlation functions. In the limit in which the relaxation time is much shorter than the radiative lifetime, the frequency spectrum of the scattered radiation consists of three parts: (1) a coherent, elastic term weighted by the factor ${tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$, (2) an incoherent quasielastic term weighted by the factor [$1\ensuremath{-}{tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$], (3) a fluorescent term characterizing emission at the natural frequency ${\ensuremath{\omega}}_{0}$, which arises from an effective two-photon-two-phonon interaction. The extension of the analysis to the problem of resonant scattering from paramagnetic ions in crystals is outlined with particular emphasis on the influence of neighboring ionic levels and the relative importance of the one- and two-phonon processes.
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 consider light scattering from a multilevel system (e.g., a molecule) interacting with several heat baths. The Liouville space approach is employed and weak incident radiation intensity is assumed, whereas the light scattering process is described in the lowest appropriate order in the interaction between the molecule and the incident and emitted photons. The interaction with other radiation field modes as well as with the thermal baths is treated more fully, such as to account for the correct damping terms and their temperature dependence. The formalism is suited for light scattering from an impurity molecule in both gas and dense phases. The thermal baths induce relaxation within the manifold of excited states as well as transitions to nonradiative channels. Damping due to such transitions (as well as radiative damping) corresponds to interactions with zero temperature baths. The physical interpretation of such a bath depends on the nature of the particular damping process. Thermal relaxation within the excited manifold is induced by interaction with the environment represented by a bath of temperature T. Part of this interaction which is diagonal in the molecular space corresponds to the so-called ’’quasielastic collisions’’ and induces modulations of the molecular energy levels and broadening of the relaxed emission. The remainder, nondiagonal part induces actual transitions between the excited molecular energy levels. The formalism yields in a single calculation expressions for the cross section for both the coherent (Raman) scattering process and the relaxed luminescence and enables us to discuss the effects of different factors such as temperature and quenching processes on the relative yields for these processes. The formalism also allows for interference between different levels of the excited manifold due to coupling of several levels to the same bath, and makes it possible to study the effect of thermal relaxation on these interference phenomena.
A study has been made on the scattering of light by the hydrodynamical and statistical atom model. Bloch's treatment of the hydrodynamical equations of motion for this model is supplemented here by inclusion of the interaction with the electromagnetic field. We limited attention to oscillations of small amplitude. By correspondence principle arguments, general expressions were derived for the cross sections for absorption, coherent and incoherent scattering. The energy can be expressed—following Bloch—as the energy of a Thomas-Fermi atom plus a Hamiltonian which is associated with departures from the Thomas-Fermi distribution. Using Bloch's quantization of this Hamiltonian and applying the method of quantum field theory, we rederived the correspondence principle results for elementary cross sections. Then applying the correspondence theoretical argument to the matrix element, we rederived Heisenberg's result for the total intensity of the Compton scattering. We also apply the method of stationary phase to the hydrodynamical treatment and show that this method gives the same result as does the linear term in the momentum transfer in Heisenberg's expression, except for a numerical factor 3½/2—a point that was discussed by Bloch many years ago. Application of the general formulas given here for angular distribution of Rayleigh and Compton scattering will require electronic machine calculations of higher modes of oscillation of the gas model of the atom analogous to those made by Wheeler and Fireman for l=1.
The resonant scattering of photons from a perturbed two-level system is studied in detail. Adiabatic perturbations are approximated by a randomly fluctuating term in the level splitting. Off-diagonal perturbations are accounted for in the linewidth and the time dependence of the correlation functions. Explicit expressions for the cross section are obtained in the statistical and motional-narrowing limits. The detailed-balancing symmetry of the scattering is established. Minor modifications of the approximate expressions for the cross sections are proposed in order to preserve this symmetry. The effects of off-diagonal perturbations, which couple the atom to a thermal bath, are investigated at length for a model interaction.