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Physical Review

Calculation of Resonance Broadening

Haider R. Zaidi

Published Sep 5, 1968• 33 citations
Publisher

Abstract

The line-shape function is expressed in terms of three boundary functions related to the two-particle Green's function. The transverse as well as the longitudinal interaction among atoms is considered, using the photon propagator approach. The broadening of one of the radiative states through the resonance interaction with the ground state is considered in detail. The contributions of multiple transfer of excitation and of two-body scattering are calculated separately. Several validity conditions are also obtained which clarify the approximations in the existing results. The velocity dependence of the width parameter is investigated in some detail; our results differ considerably from the existing results because we have included higher-order contributions. The extreme low-density region [where (collision width) \ensuremath{\sim} (natural width)] is also considered; the linewidth is found to be nonlinear in the number density. The additional contribution of the coupling between the radiative states is also calculated in a special case.

DOI
10.1103/physrev.173.123

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Physical Review A•Dec 1, 1980•9 citations
The line-shape function is expressed in terms of three boundary functions related to the two-particle Green's function. The transverse as well as the longitudinal interaction among atoms is considered, using the photon propagator approach. The broadening of one of the radiative states through the resonance interaction with the ground state is considered in detail. The contributions of multiple transfer of excitation and of two-body scattering are calculated separately. Several validity conditions are also obtained which clarify the approximations in the existing results. The velocity dependence of the width parameter is investigated in some detail; our results differ considerably from the existing results because we have included higher-order contributions. The extreme low-density region [where (collision width) \ensuremath{\sim} (natural width)] is also considered; the linewidth is found to be nonlinear in the number density. The additional contribution of the coupling between the radiative states is also calculated in a special case.
The resonance broadening of a transition between two excited states of a radiator in a high density gas is investigated. The important many body shielding contributions are included to all orders, and the two body correlations for the close collisions are approximately taken into account. Corrections to the line width and shift at moderately high densities are calculated. The results agree with the experimental data for krypton. The density dependence of the width parameter is examined at very high densities also. It is found that this parameter, after reaching a peak value, decreases with further increase in the density. Moreover, the square root dependence of the line widths on the density is not valid over any extended range of gas densities.
The broadening of spectral lines due to dipole-dipole interactions with ground-state atoms of the same element is calculated with the impact and classical path approximations. Summation over all contributing intermediate states accounts for the multiplicity of the levels involved and results in a width proportional to the geometrical mean of absorption and emission oscillator strengths. Higher multipole interactions are shown to be almost always negligible, and for Lyman-$\ensuremath{\alpha}$ the iterated solution of the time-dependent Schr\"odinger equation describing the collisions is extended to fourth order. Estimated errors of its calculated width and of the widths of other lines are about 5% in the validity regime of the impact approximation, apart from any uncertainties in oscillator strengths and statistical weight factors.
The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these (frequency-dependent) imaginary parts must be included in both the resonant and antiresonant frequency denominators [P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 (2004)]; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schr\"odinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green's-function approach.
The excitation and decay probabilities for resonance scattering of radiation from an atom with two coupled excited states in an external static field are calculated as a function of time and frequency. Various oscillatory terms are found in the probabilities. These oscillations depend on the frequency of radiation, the external coupling and the energy difference between the excited states. Two special cases are investigated in detail. In the first case where one of the excited states does not decay a 'hole' appears in the emission line at a frequency equal to the frequency difference between the ground state and the unperturbed non-decaying excited state. In the second case where the two excited states decay with the same rate, one of the two lines in the emission spectrum is suppressed and the other is enhanced.
We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 \AA{}, respectively.
General expressions are derived for the spectral line shapes of resonance absorption and scattering of coherent radiation in collision-broadened gases, taking into account effects of coherent excitation of two or more atoms (or molecules), as steady-state solutions of a hierarchy of master equations described in a previous publication (paper I). Coupling between the coherent motions of the atoms, provided by a Bethe-Salpeter---type effective interaction, in the binary-collision approximation, forms the essential mechanism for introducing cooperative coherent effects into the steady-state spectra. Explicit expressions are given for the effects of two-atom coherence in the binary-collision approximation, in which the Bloch-type dressed-atom self-energy superoperator is modified by the presence of collisions in which both atoms retain memory of their coherent propagation before the collision. The self-energies include the effects of resonance exchange symmetrization in self-broadening, and are renormalized by the coincidence of radiative transitions during the collisions. The impact (near-resonance) and the quasistatic (line-wing) limits of the applied-frequency detunings are discussed. In the quasistatic limit, coherent many-atom excitations become irrelevant; however, interactions of both collision partners with the radiation during the collision accounts for such phenomena as collision-induced absorption or radiative collisions. In the impact limit, the inclusion of the Bethe-Salpeter interactions allows for the appearance of two-atom resonances. Magnitude estimates of these effects are discussed. Effects of higher-rank (many-body) coherences are formally discussed with the help of a diagrammatic method, leading into implicit bootstrap equations that can be solved by iterative or other procedures.