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Journal of Physics B Atomic and Molecular Physics

Collisional redistribution beyond the medium-coupling limit

Henk F. Arnoldus, Thomas F. George

Published May 28, 1987• 0 citations
Publisher

Abstract

The evaluation of the binary-collision operator Phi ( omega ), pertaining to collisional redistribution of strong laser radiation, involves the computation of a time-ordered exponential, which only reduces to a scalar exponential in the medium-coupling limit. In this approximation one neglects the alteration of the collisional dynamics due to the presence of the laser field. A two-state atom is considered, for which the authors expand Phi ( omega ) in the ratio of the Rabi frequency Omega to the detuning from resonance. This parameter vanishes in the medium-coupling case, and is small for common gas-phase experiments. It is shown that the correction to medium coupling can again be expressed in a scalar function, which represents the temporal correlation between the interaction potential and the full time evolution during a single collision. The decay constants for collisional transitions between dressed states, which are germane to the width of the spectral lines, appear to be sensitive to the deviation from medium coupling. Quite remarkably, a non-vanishing collisional contribution survives in the limit Omega to 0 of Phi ( omega ).

DOI
10.1088/0022-3700/20/10/013

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The evaluation of the binary-collision operator Phi ( omega ), pertaining to collisional redistribution of strong laser radiation, involves the computation of a time-ordered exponential, which only reduces to a scalar exponential in the medium-coupling limit. In this approximation one neglects the alteration of the collisional dynamics due to the presence of the laser field. A two-state atom is considered, for which the authors expand Phi ( omega ) in the ratio of the Rabi frequency Omega to the detuning from resonance. This parameter vanishes in the medium-coupling case, and is small for common gas-phase experiments. It is shown that the correction to medium coupling can again be expressed in a scalar function, which represents the temporal correlation between the interaction potential and the full time evolution during a single collision. The decay constants for collisional transitions between dressed states, which are germane to the width of the spectral lines, appear to be sensitive to the deviation from medium coupling. Quite remarkably, a non-vanishing collisional contribution survives in the limit Omega to 0 of Phi ( omega ).
The authors have studied resonance collisions amongst atoms dressed by a strong laser field. The equations of motion have been solved numerically to yield collisional transition rates for various values of the Rabi frequency (V) due to the laser and its detuning ( Delta ) from the atomic transition frequency ( omega 0 ). The atoms are assumed to follow classical rectilinear paths and only undergo binary collisions. The resonance interaction is taken to be the first-order dipole-dipole interaction acting within the ground and resonance levels of a pair of atoms. A qualitative explanation is offered for the results obtained by splitting the Hamiltonian into parts that represent the collision-free evolution of the system at the Rabi frequency, the 'normal' resonance interaction, and the 'dressed interaction' responsible for opening new scattering channels. The latter interaction depends upon the relative strengths of the resonance interaction (R), the generalized Rabi frequency ( Omega ) and the ratio V/ Delta .
Perturbation theory for the interaction of light with a binary system is applied to collisional redistribution of radiation in the weak-field limit. Within the very large domain of validity of the binary model applied to collisions between non-hydrogenic atoms and atomic perturbers at low densities, redistribution can be viewed as collisional-induced absorption followed by spontaneous decay during a second collision. The formalism can be applied from the quasimolecular domain at large detunings Delta omega 1 or Delta omega 2 up to the impact regime, but emphasis is placed here on the non-impact region, at small Delta omega 1 or Delta omega 2, where non-adiabatic effects are important in collision-induced radiative absorption or emission rates as well as in the redistribution. The impact limit of the redistribution function is easily rederived in this way.
The authors incorporate in the theory of collisional redistribution of resonance radiation the effect of a finite bandwidth of the intense incident radiation. The phase of the radiation is treated as a process with independent increments, which contains the phase-diffusion model as a special case. These phase fluctuations give rise to effective decay operators affecting the evolution of the density matrix and the regression of the correlation function. For a two-level atom suffering binary collisions the combined effect of collisions and phase fluctuations on the fluorescence spectrum is evaluated in several limiting cases of practical interest.
We present a calculation of the scattering of monochromatic radiation by a degenerate atom in the binary-collision approximation. We limit ourselves to field strengths such that $\ensuremath{\Omega}{\ensuremath{\tau}}_{c}\ensuremath{\ll}1$, where $\ensuremath{\Omega}$ is the Rabi frequency for the atomic transition, and ${\ensuremath{\tau}}_{c}$ is the duration of a strong collision. Our calculation is not limited to the impact regime, $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}{\ensuremath{\tau}}_{c}^{\ensuremath{-}1}$, or to the region where thermal correlations may be neglected, i.e., $\ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\ll}\frac{\mathrm{kT}}{\ensuremath{\hbar}}$. (Here $\ensuremath{\Delta}\ensuremath{\omega}$ is the difference between either the incoming or outgoing photon's frequency and the atom's natural frequency.) To do this we derive an equation of motion for the correlation function, valid outside the quantum-regression regime, whose solution for practical cases is straightforward. We present solutions for the weak-field (linear-response) regime in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles created.
We extend the theory of light scattering from an atom undergoing collisions to the case of strong laser fields, where the expansion in powers of $\ensuremath{\Omega}{\ensuremath{\tau}}_{c}$ that was used formerly in a series of papers by Burnett et al. breaks down. In particular we consider in detail the conditions necessary to relate the observable quantities to intense-field collisional rate constants in a dressed-state basis (such as those calculated by Light and Sz\"oke). We also show (following Rabin and Ben-Reuven) that by studying the spectrum emitted by the atoms in the presence of a strong field one may measure the collisional rates for transfer between the dressed states of atom plus radiation field.
The authors investigate collisional redistribution in a laser field with arbitrary lineshape. The phase diffusion model cannot be correct in general due to its inherent assumption of zero correlation times. The authors give general expressions for the state of the atom, the dipole correlation function and the steady-state fluorescence spectrum in the presence of collisions and phase fluctuations with finite correlation times. It turns out that collisional effects decouple from the free-atom stochastic problem to a good approximation and that modifications due to the laser bandwidth can be quite different from collisional effects, although they are additive on a microscopic level. They evaluate the fluorescence spectrum explicitly in the limit where the three lines are well separated and show that the spectrum is determined by simple field correlation functions. The arbitrary phase fluctuations do not affect the line strengths but only the lineshapes. Collisions, on the other hand, can transfer intensity between the lines, due to the very small collision time. For free atoms and strong driving fields, the fluorescence spectrum becomes a convolution of the normalised laser spectrum with the fluorescence spectrum at monochromatic irradiation, but this does not hold any more when collisions are present.