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Physical Review A

Semiclassical Shape of Satellite Bands

Kenneth M. Sando, J. Wormhoudt

Published Jun 1, 1973• 130 citations
Publisher

Abstract

When an atom emits or absorbs radiation in the presence of perturbing atoms, satellite bands frequently appear in the spectrum. Such bands may occur when the active atom is subject to equal forces in the two states involved in the transition for some particular configuration of perturbers. In this work a semiclassical approach is used to derive an approximate expression for the one-perturber spectrum in the two-state adiabatic approximation. The expression, which involves an average over Airy functions, is readily adapted for rapid numerical evaluation. Also, an analytic expression for the rate of decline in intensity on the classically forbidden far side of the satellite is found. The semiclassical one-perturber line shape for satellite bands is compared with the classical and quantum-mechanical counterparts for the case of resonance broadening of the Lyman-$\ensuremath{\alpha}$ line in absorption. The semiclassical line shape is found to be much more realistic than the classical shape in which the satellite appears as a singularity. The semiclassical treatment provides an interpretation of the shape of a satellite band in terms of the intermolecular potentials, a possibility not evident in the exact quantum-mechanical treatment.

DOI
10.1103/physreva.7.1889

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IEEE Journal of Quantum Electronics•Sep 1, 1974•0 citations
When an atom emits or absorbs radiation in the presence of perturbing atoms, satellite bands frequently appear in the spectrum. Such bands may occur when the active atom is subject to equal forces in the two states involved in the transition for some particular configuration of perturbers. In this work a semiclassical approach is used to derive an approximate expression for the one-perturber spectrum in the two-state adiabatic approximation. The expression, which involves an average over Airy functions, is readily adapted for rapid numerical evaluation. Also, an analytic expression for the rate of decline in intensity on the classically forbidden far side of the satellite is found. The semiclassical one-perturber line shape for satellite bands is compared with the classical and quantum-mechanical counterparts for the case of resonance broadening of the Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα line in absorption. The semiclassical line shape is found to be much more realistic than the classical shape in which the satellite appears as a singularity. The semiclassical treatment provides an interpretation of the shape of a satellite band in terms of the intermolecular potentials, a possibility not evident in the exact quantum-mechanical treatment.
When two atoms are in proximity, their resonance lines are broadened because of the formation of molecular potentials, between which photon transitions can occur. The level of experimental refinement in measuring excimer and collisional far-wing spectra calls for interpretive theoretical methods that accurately treat the observed structure in line shapes. Of particular interest are satellites and undulations, for they can give much information on intermolecular potentials. We derive an expression for TTT-matrix elements, based on JWKB wave functions and stationary-phase techniques, that is universally applicable to situations with one, two, or more Condon points. Each transition point is treated individually, with effects of potential shape, wave-function phases, and interference separately highlighted. The expression is tested against quantum-mechanical line shapes of the red and blue wings of the Rb DDD lines broadened by Xe perturbers. Agreement is quite good for both the red-wing undulations and the blue-wing satellite-supernumerary structure.
Recent advances in experimental spectroscopy offer the promise of yielding much detailed information about molecular potentials and collisional interactions from a study of the light emitted or absorbed in the far wings of atomic lines. However, readily manipulable, accurate theoretical expressions and methods are needed to extract such knowledge. Straightforward formulas, based on the distorted-wave approximation, for the normalized emission intensity in the far wing of a pumped atomic line are given here. By normalizing theoretical spectra with respect to the strength of the atomic line, one obtains a convenient form for the analysis of experimental differential spectra that are reported in nonarbitrary units. Theoretical and experimental spectra are numerically consistent in the test case of the blue-wing satellite (\ensuremath{\sim}7610 A\r{}) of the Rb D2{D}_{2}D2​ line (7800 A\r{}) broadened by Xe perturbers.
Oscillatory or bandlike structure is predicted to occur in the wings of pressure-broadened atomic lines when the interaction potential between the emitting atom M* and the perturbing atom A is deeply attractive compared to kT and the interaction in the final state is repulsive. The oscillations in intensity arise from quantum effects and are associated with the nodal structure of the continuum wavefunction which describes the dissociated state of the diatomic molecule (A + M*). The number of maxima in the line shape is a measure of the number of bound states which can be supported in the potential while the amplitude and resolution of the oscillations depend on the long-range behavior of the potential. The bandlike structure, which may extend over thousands of cm−1 on the long-wavelength side of the atomic emission line, is resolvable as long as kT < ∼ 1 / 10 the vibrational quantum ℏωe of the AM* molecule. This is a generalization of similar bandlike structure which has been calculated and observed to occur in the collision-induced radiation of helium metastable atoms and which is merely a special case of pressure broadening involving forbidden lines. Specific calculations are presented for the broadening of the hydrogen Lyman-α line by H(1s), and the visible absorption spectra of H(1s) in the presence of free protons. The quantum theory of line shape which is employed here was first developed by Jablonski but then neglected due to difficulties involved in evaluating essentially the band strength for continuum–continuum transitions of diatomic molecules. Jablonski introduced the WKB and other approximations into his theory which reduced it to the classical statistical theory. These approximations are compared to exact quantum-mechanical calculations. The WKB approximation is shown to be quite valid in the special case of attractive–replusive transitions, but Jablonski then improperly averaged over those very quantum oscillations of interest here. This structure offers a unique and extremely sensitive method of studying interaction potentials involving excited states of atoms.
It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.
The effect of a driving or pump field on the emission and absorption line-shape functions for an atom is discussed. The pump field is assumed to oscillate at a frequency near the resonance frequency for transitions between a single pair of atomic states, and transitions between one of these states and any other state of the atom are analyzed. The pump field is treated classically, and atomic relaxation is treated in general terms. The emission and absorption line-shape function (the latter is defined as the rate of absorption of energy from a weak signal field, applied in addition to the pump field, as a function of the signal-field frequency) are found by evaluating the relevant two-time atomic correlation functions in the Markoff approximation. In the limit of high pump-field intensity, both the absorption and the emission spectra are doubly peaked at frequencies which differ from the usual resonance frequency by $\ifmmode\pm\else\textpm\fi{}\frac{1}{2} \ensuremath{\Omega}$, where $\ensuremath{\Omega}$ is the frequency of the pump-field-induced oscillations in the populations of the two strongly coupled states. The absorption and the emission spectra are represented by essentially the same function in the limit of high pump-field intensity, as they are also in the limit of vanishing pump-field intensity. For intermediate pump-field intensities, however, the two functions are quite different in form, and no simple proportionality exists between them.
A semiclassical theory of two-photon Lamb dips is presented, when one of the radiations is in the radio-frequency region and the other is in the optical region. Perturbation solutions as well as an exact recurrence formula are discussed. For the latter case numerical results are given. The resonance line shape of the two-photon Lamb dip shows a variety of changes according to the experimental conditions. When both the optical and the rf fields are weak a line is of a dispersion form. When the rf field is increased, the line changes to a Lorentzian form. When the optical field is increased, the line is shifted and broadened. If the radio frequency is close to or less than the Doppler width of the gas, a two-photon Lamb dip is accompanied with satellite dips which correspond to molecules with nonzero velocity components along the propagation of the laser radiation. The intensity of a two-photon Lamb dip is expected to decrease rapidly with increasing radio frequency. Additional effects when the sample cell is placed inside the laser cavity are also discussed. Those results agree qualitatively with recent observations by Fruend, R\omheld, and Oka.