Resonant Processes and Exciton Propagation in a Frozen Gas

We present numerical simulations of a theory of resonant processes in a frozen gas of excited atoms interacting via dipole-dipole potentials that vary as $r^{-3}$, where $r$ is the interatomic separation. The simulations calculate time-dependent averages of transition amplitudes and transition probabilities for a single atom in a given state interacting resonantly with a uniformly distributed random gas of atoms in a different state. The averages are over spatial configurations of the gas atoms, which are held fixed while the resonant interaction creates a Frenkel exciton that can travel from atom to atom. We check that the simulations reproduce previously known exact results when the exciton is not allowed to propagate [Phys. Rev. A 59, 4358 (1999)]. Further, we develop an approximation for the average transition amplitude that compares well with the numerical results for a wide range of values of the system parameters.