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Physical Review A

Collisional redistribution of radiation. I. The density matrix

Keith R. Burnett, J. Cooper, R. J. Ballagh, E. W. Smith

Published Nov 1, 1980• 96 citations
Publisher

Abstract

As a first step in deriving accurate, calculable expressions for collisional redistribution of radiation, valid outside the impact regime, we present a method for calculating an accurate binary-collision density operator for an atom in the presence of a driving field and perturbers. We use projection-operator techniques to establish first the precise validity of the Markoff approximation for radiative relaxation. Using the same techniques, we show how the collisional-relaxation problem may be analyzed in a practical manner outside the Markoff approximation. The effect of correlations between radiative and collisional events is included in a consistent way, and the physical implications of these correlation effects are demonstrated for a simple example, the two-level atom with nondegenerate levels.

DOI
10.1103/physreva.22.2005

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As a first step in deriving accurate, calculable expressions for collisional redistribution of radiation, valid outside the impact regime, we present a method for calculating an accurate binary-collision density operator for an atom in the presence of a driving field and perturbers. We use projection-operator techniques to establish first the precise validity of the Markoff approximation for radiative relaxation. Using the same techniques, we show how the collisional-relaxation problem may be analyzed in a practical manner outside the Markoff approximation. The effect of correlations between radiative and collisional events is included in a consistent way, and the physical implications of these correlation effects are demonstrated for a simple example, the two-level atom with nondegenerate levels.
We apply the analysis of paper I to degenerate atomic systems and calculate the effect of correlations between an atom (the absorber) and perturbers in the binary-collision approximation. The result of our calculation is a generalized absorption profile that specifies the final state of the atom an absorption event. This profile is then related to the total intensities of Rayleigh scattering and fluorescence (redistributed radiation) from the atom. The profile is expressed in terms of collision operators that depend on the index of the multipole of the atom that is created. Some of the operators represent correlations between radiative and collisional events and cannot, except in certain limits, be expressed in terms of the usual line-broadening operators. This, of course, implies that one needs extra dynamical information, on top of that obtainable from ordinary absorption experiments, to be able to describe redistribution. Conversely, redistribution offers a probe of collisional dynamics, not given by absorption experiments, via the polarization of fluorescent light. We give practical expressions for the calculation of these operators and discuss their implications.
The effect of correlations between an absorber atom and perturbers in the binary-collision approximation are applied to degenerate atomic systems. A generalized absorption profile which specifies the final state of the atom after an absorption event is related to the total intensities of Rayleigh scattering and fluorescence from the atom. It is suggested that additional dynamical information to that obtainable from ordinary absorption experiments is required in order to describe redistributed atomic radiation. The scattering of monochromatic radiation by a degenerate atom is computed in a binary-collision approximation; an equation of motion is derived for the correlation function which is valid outside the quantum-regression regime. Solutions are given for the weak-field conditions in terms of generalized absorption and emission profiles that depend on the indices of the atomic multipoles.
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.
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 have derived reduced-density-matrix equations of motion for a pair of two identical atoms moving in the radiation field as the first step in establishing a theory of collisional redistribution of light from neutral-atom traps. We use the Zwanzig projection-operator technique to average over spontaneous field modes and establish the conditions under which Born and Markov approximations can be applied to the system of moving atoms. It follows from these considerations that when these conditions hold, the reduced-density-matrix equation for moving atoms has the same form as that for the stationary case: time dependence is introduced into the decay rates and interaction potentials by making the substitution R=R(t).