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Abstract

The present thesis treats the interacting ultracold Rydberg gas with special emphasison the statistical footprint of the phase transition between unordered and crystalline phase, which can be understood as a consequence of the dipole blockade effect. After mapping the system onto an effective spin-1/2-model, exact diagonalization of the effective Hamiltonian is used to obtain the many-body ground state. Repeated application of this procedure on random realizations reveals the underlying probability distribution of the number of Rydberg atoms, allowing to calculate its statistical moments. In the regimes of weak and strong interaction these observables have power law character. The critical interaction strength is estimated by extrapolating these power laws up to their intersection point. The same procedure is applied to interacting excitons in bilayer heterostructures uncovering a phase transition here as well. Furthermore, new methods are introduced to handle the effects of finite detection efficiency and parameter fluctuations to establish a better connection between experimental and theoretical results. Finally, new models are introduced to include dynamics or additional Rydberg states. The last model is of purely statistical nature and its results may be used as a tool for detecting a potential clustering of Rydberg atoms.