Frequency redistribution and step-size distribution of light scattered by atomic vapor: Applications to Lévy flight random walk

The propagation of light that undergoes multiple-scattering by resonant atomic vapor can be described as a Lévy flight. Lévy flight is a random walk with heavy tailed step-size (r) distribution, decaying asymptotically as $P\left(r\right)\sim r^{-1-\alpha }$, with $\alpha <2$. The large steps, typical of Lévy flights, have its origins in frequency redistribution of the light scattered by the vapor. We calculate the frequency redistribution function and the step-size distribution for light diffusion in atomic vapor. From the step-size distribution we extract a Lévy parameter $\alpha$ that depends on the step’s size. We investigate how the frequency redistribution function and step-size distribution are influenced by the finite size of the vapor and the many-level structure typical for alkali vapors. Finite size of the vapor introduces cutoff on the light scattered spectrum and thus in the size of steps. Multi-level structure introduces oscillations in $P\left(r\right)$ slope. Both effects might have an impact on measurables related to the Lévy flight random walk.