L\'evy statistics of interacting Rydberg gases

A statistical analysis of the laser excitation of cold and randomly distributed atoms to Rydberg states is developed. We first demonstrate with a hard ball model that the distribution of energy level shifts in an interacting gas obeys L\'evy statistics, in any dimension $d$ and for any interaction $-C_p/R^p$ under the condition $d/p<1$. This result is confirmed with a Monte Carlo rate equations simulation of the actual laser excitation in the particular case $p=6$ and $d=3$. With this finding, we develop a statistical approach for the modeling of probe light transmission through a cold atom gas driven under conditions of electromagnetically induced transparency involving a Rydberg state. The simulated results are in good agreement with experiment.