Many-body dynamics of Rydberg excitation using the Ω expansion

We investigate the excitation dynamics of Rydberg atoms in ultracold atomic samples by expanding the excitation probability and the correlation function between excited atoms in powers of the isolated atom Rabi frequency Ω. In the Heisenberg picture, we give recurrence relations to calculate any order of these expansions, which are expected to be well behaved for arbitrarily strong interactions. For large homogeneous samples, the explicit terms of the expansions, up to Ω4, are given and then averaged over all possible random spatial distributions of atoms for the most important cases of excitation pulses and interactions.