Collisional redistribution of radiation. III. The equation of motion for the correlation function and the scattered spectrum

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 $\Omega {\tau }_c\ll 1$, where $\Omega$ is the Rabi frequency for the atomic transition, and ${\tau }_c$ is the duration of a strong collision. Our calculation is not limited to the impact regime, $\Delta \omega \ll {\tau }_c^{-1\mathrm{}}$, or to the region where thermal correlations may be neglected, i.e., $\Delta \omega \ll \frac{\mathrm{kT}}{\hslash }$. (Here $\Delta \omega$ 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.