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Physical Review A

Berry phases and profiles of line wings and rainbow satellites induced by optical collisions

R. Ciuryło, Józef Szudy, W. E. Baylis

Published Sep 8, 2015• 3 citations
Publisher

Abstract

The concept of Berry phase is included in an analysis of the intensity distribution in far wings of pressure-broadened spectral lines emitted or absorbed by atoms placed in an external cone-rotating electric field. Particular attention is focused on frequency regions where rainbow satellite bands appear. A classical-path treatment that employs the time-dependent Schr\"odinger equation is used to derive an expression for the line shape, and it uses a dipole transition moment calculated with quasimolecular wave functions given by the Berry version of the adiabatic approximation. It is found that in the presence of an external rotating electric field, the intensity distribution in far wings can be expressed in terms of the universal line shape function of the unified Franck-Condon theory once energy shifts due to Stark and Berry effects are taken into account. We show that the influence of Berry phase in the profiles of the far wings can be manifested either in the form of deviations of observed profiles from the quasistatic distribution or the appearance of additional features in the vicinity of the maximum of the rainbow satellite band. As an example, the modification of the rainbow satellite at 162.3 nm in the red wing of the self-broadened Lyman-$\ensuremath{\alpha}$ line of hydrogen, caused by an external rotating electric field, is considered.

DOI
10.1103/physreva.92.032503

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Stark-Effect Broadening of Hydrogen Lines

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The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.

Physical Review•Apr 15, 1939•20 citations
The concept of Berry phase is included in an analysis of the intensity distribution in far wings of pressure-broadened spectral lines emitted or absorbed by atoms placed in an external cone-rotating electric field. Particular attention is focused on frequency regions where rainbow satellite bands appear. A classical-path treatment that employs the time-dependent Schr"odinger equation is used to derive an expression for the line shape, and it uses a dipole transition moment calculated with quasimolecular wave functions given by the Berry version of the adiabatic approximation. It is found that in the presence of an external rotating electric field, the intensity distribution in far wings can be expressed in terms of the universal line shape function of the unified Franck-Condon theory once energy shifts due to Stark and Berry effects are taken into account. We show that the influence of Berry phase in the profiles of the far wings can be manifested either in the form of deviations of observed profiles from the quasistatic distribution or the appearance of additional features in the vicinity of the maximum of the rainbow satellite band. As an example, the modification of the rainbow satellite at 162.3 nm in the red wing of the self-broadened Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα line of hydrogen, caused by an external rotating electric field, is considered.
The intensity distribution in the core region of the Lyman-$\ensuremath{\alpha}$ line of hydrogen perturbed by neutral hydrogen atoms in the presence of an external cone-rotating electric field has been evaluated using an impact approximation. It was found that the line shape in this region can be described as the sum of four Lorentzian profiles representing contributions coming from transitions between various Stark and Berry sublevels of the resonance and ground states of the hydrogen atom. The effects due to Berry's phase are shown to produce a Lorentzian-shaped feature on each side of the Lyman-$\ensuremath{\alpha}$ line, with maxima located at frequencies linearly dependent on the angular frequency with which the electric field rotates about the laboratory fixed zzz axis, as well as on the cosine of the polar angle that the field makes with respect to this axis. The remaining two Lorentzian profiles are due to the Stark effect, and their maxima are located on both sides of the Lyman-$\ensuremath{\alpha}$ line at frequencies corresponding to the Stark shift.
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.
A quantum-mechanical unified theory of foreign-gas pressure broadening of atomic lines which includes the impact and statistical theories and duration-of-collision effects is developed. The model consists of an absorber atom with two internal states uuu (upper) and lll (lower) interacting with a structureless perturber gas via two spherical pairwise additive potentials Vu(r){V}_{u}(\mathcal{r})Vu​(r) and Vl(r){V}_{l}(\mathcal{r})Vl​(r), e.g., a simple model of an alkali-atom line perturbed by a rare gas. The dipolemoment correlation function is analyzed in the time domain using Liouville (tetradic) techniques. The correlation function reduces exactly to a two-body form $\ensuremath{\varphi}(t)$ containing the proper thermal weight factors. Detailed balance is obeyed and no $|\frac{\ensuremath{\hbar}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}{\mathrm{kT}}|<1$ restriction is required. The impact theory follows from a long-time analysis of $\ensuremath{\varphi}(t)$. Sum rules or spectral moments are used to study short-time behavior revealing conventional statistical-theory effects (including satellite bands) and the important duration-of-collision effects which link the statistical and impact regimes. The sum rules are expressed as quadratures involving potential functions and the quantum-mechanical radial distribution function. Unified methods for calculating the total line shapes are suggested: (1) Pad\'e approximants (ratios of polynomials) can be used to interpolate from the known short-time behavior to the known long-time beahvior; this requires the least computational effort; (2) exact results for this model may be obtained numerically from the overlap integrals of radial wave functions (Franck-Condon factors) for the two potentials Vu{V}_{u}Vu​ and Vl{V}_{l}Vl​; (3) classical phase (trajectory) expressions are given which are reasonable approximations in both the impact and statistical limits. A number of related topics are discussed including diabatic effects, collision-induced absorption, non-Lorentzian wings of molecular vibration-rotation bands, and the Condon approximation for the dipole-moment transition strength vs r\mathcal{r}r. The use of line shapes to probe atomic interactions, especially in excited states, is emphasized and the suggestion made that experiments be done over as wide a temperature range as possible with the measured line shapes in computer-readable form. The entire line-shape problem is described as the study of the spectrum of an atom-perturber "quasimolecule" (with primarily unbound states).
An analysis of near-resonant collisions between dissimilar atoms in the presence of a radiation field reveals their effect on the far wings of the absorption profile. Extending a model due to Cavalieri, Arimondo, and Matera, we take into account the collisional shifts of atomic levels, as well as the collisional coupling between atoms and the coupling with the radiation field. A general formula for the absorption profile is derived and expressed in terms of the universal line-shape function of the unified Franck-Condon treatment of pressure broadening. We show that in some cases collision-induced coherence can give rise to a strongly depressed red wing that exhibits transparency for some values of the laser detuning. \textcopyright{} 1996 The American Physical Society.
The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.