No abstract

Several recent experiments have established by measuring the Mandel Q parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly-excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel Q parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment.

We identify a relation between the dynamics of ultracold Rydberg gases in which atoms experience a strong dipole blockade and spontaneous emission, and a stochastic process that models certain wireless random-access networks. We then transfer insights and techniques initially developed for these wireless networks to the realm of Rydberg gases, and explain how the Rydberg gas can be driven into crystal formations using our understanding of wireless networks. Finally, we propose a method to determine Rabi frequencies (laser intensities) such that particles in the Rydberg gas are excited with specified target excitation probabilities, providing control over mixed-state populations.

We develop many-server asymptotics in the Quality-and-Efficiency-Driven (QED) regime for models with admission control. The admission control, designed to reduce the incoming traffic in periods of congestion, scales with the size of the system. For a class of Markovian models with this scaled control, we identify the QED limits for two stationary performance measures. We also derive corrected QED approximations, generalizing earlier results for the Erlang B, C and A models. These results are useful for the dimensioning of large systems equipped with an active control policy. In particular, the corrected approximations can be leveraged to establish the optimality gaps related to square-root staffing and asymptotic dimensioning with admission control.

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.Comment: 25 pages, 3 figure