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Journal of Physics B Atomic Molecular and Optical Physics

Theory of the inelastic lineshape for two-step sub-Doppler excitation

W -K Liu, A S Dickinson

Published Mar 28, 1991• 8 citations
Publisher

Abstract

The inelastic lineshape expression for two-step sub-Doppler excitation of a four-level atomic or molecular system in the presence of a low-pressure perturber is derived by treating the collisional effects to first order. The degeneracies of the rotational levels and the tensorial coupling due to the laser fields and collisions have been fully taken into account by working with the irreducible tensorial components of the density matrix. Explicit expressions for the collision cross sections affecting the lineshape are derived using the Liouville-space formalism.

DOI
10.1088/0953-4075/24/6/015

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The inelastic lineshape expression for two-step sub-Doppler excitation of a four-level atomic or molecular system in the presence of a low-pressure perturber is derived by treating the collisional effects to first order. The degeneracies of the rotational levels and the tensorial coupling due to the laser fields and collisions have been fully taken into account by working with the irreducible tensorial components of the density matrix. Explicit expressions for the collision cross sections affecting the lineshape are derived using the Liouville-space formalism.
Line-shape expressions for two-step resonant excitation in the presence of low-pressure perturbers are derived. These expressions allow for velocity-dependent broadening rates and are well suited for studying the line-shape contributions due to coherent excitation, incoherent excitation (due to dephasing collisions), interference between coherent and incoherent excitation, and velocity-changing collisions. Using the sodium 3S-3P-4D system perturbed by noble-gas atoms as an example and assuming C6{C}_{6}C6​ interaction potentials, line shapes at a variety of laser detunings (Doppler selected atomic velocities) are numerically calculated to study variation in the line shape due to competing excitation paths and speed-dependent collision rates. In order to deal with both the line core and the line wing a composite collision kernel consisting of a Lorentzian kernel for small \ensuremath{\Delta}vz{v}_{z}vz​ collisions and a Keilson-Storer kernel for large \ensuremath{\Delta}vz{v}_{z}vz​ collisions is used. It is also demonstrated how the inelastic (3P3/2{P}_{3/2}P3/2​-3P1/2{P}_{1/2}P1/2​) velocity-changing collision kernel may be obtained from experimental data at low pressures. Theoretical line shapes and broadening rates are compared with experiment.
The theory of the collisional and radiative relaxation of an atom perturbed by foreign-gas atoms in the presence of an intense monochromatic laser beam is developed. The spatial degeneracy of the levels and the reorientation effects of the collisions are included. The impact approximation is not made, but the Doppler effect is neglected. The spectrum of the fluorescent and scattered light is obtained from an eigenvalue problem in Liouville space using the dressed-atom approach and rotationally irreducible tensors. Minimal symmetry-adapted bases in Liouville space are constructed for linear, circular and elliptical incident polarisation. The method is illustrated for S 0 -P 1 transitions.
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
Abstract The treatment of transient excitation of transitions between two atomic or molecular energy levels by monochromatic laser radiation under conditions in which phase-destroying collisions predominate is treated by means of numerical and approximate analytical solutions of the equations giving the rate of change of a modified density matrix. The analytical solutions are then used to obtain the transient values of the population of the excited state when Doppler broadening is taken into account. Numerical calculations are carried out for the ratio γ'/γ = 10, where 2γ is the damping constant for diagonal and γ' for non-diagonal elements of the density matrix. This extends the treatm ent given in two previous papers for the situation when γ' = γ, corresponding to very low pressures for which phase-destroying collisions are relatively unimportant.
A theory of collisionally aided radiative excitation (CARE) for three-level systems in the weak-field limit is presented. Cross sections for the excitation of three-level atoms by two off-resonant pulsed radiation fields in the presence of collisions with structureless perturbers are calculated. Analytic expressions for the cross sections as functions of atom-field detunings are obtained under usual classical-trajectory and rotating-wave approximations using perturbation theory for various regions of detunings. Examples for the resulting excitation line shapes are given mostly for van der Waals potentials. A dressed-atom picture of the CARE processes is discussed. Emphasis is put on an interesting effect arising from the interference between the "stepwise" and the "direct" channel of excitation. Such an interference effect manifests itself as modulations in the total excitation cross section as a function of relative interatomic speed in some cases.
Based on the quantum Liouville formalism, a theory of the two-color, triply resonant four-wave mixing is developed for molecules with isotropically oriented angular momenta. The approach allows to strictly incorporate the relaxation matrices Γ((r)) (r = 0, 1, 2) into the third-order susceptibility χ((3)) whose expression acquires therewith the form of a scalar product in the line space. Thanks to this representation, isolation of all resonance terms from χ((3)) becomes a routine task. Some of these terms correspond to the case when a molecule initially interacts with two pump photons of the same frequency. Such interactions give rise to the grating line-space vectors which have the same (zero) eigenfrequency. Due to this degeneracy, the latter are easily mixed by rotationally inelastic collisions which shows up in a state-resolved coherence transfer. The satellite signals induced thereby provide a great scope to study the state-to-state inelastic rates in situ by purely optical means. If the diagonal form of Γ is assumed, the satellites become forbidden and our results reduce to conventional expressions for the main resonances. Polarization configurations are designed for direct measurements of the population (r = 0), orientation (r = 1), and alignment (r = 2) contributions to χ((3)). Finally, depending on the photon-molecule interaction sequence, the resonance terms of χ((3)) are shown to be differently affected by velocity averaging, the effect which conspicuously manifests itself when Doppler broadening becomes paramount.