Modeling Rydberg gases using random sequential adsorption on random graphs

The statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a random sequential adsorption (RSA) process on a random geometric graph (RGG), and a RSA process on a decomposed random intersection graph (DRIG). The last model refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the first two models, it lends itself for a rigorous mathematical analysis, and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time evolution of the number of Rydberg atoms and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graph (ERRG). Finally, we also develop a heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long timescales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short timescales the statistics are affected most by quantum mechanical effects.