We study many-body correlations in the stationary state of a continuously driven, strongly interacting dissipative system. Specifically, we examine resonant optical excitations of Rydberg states of atoms interacting via long-range dipole-dipole and van der Waals potentials employing numerical and analytical techniques. Collection of atoms within a blockade distance form a ``superatom'' that can accommodate at most one Rydberg excitation. The superatom excitation probability saturates to
21 for coherently driven atoms but is significantly higher in the presence of dephasing, approaching unity as the number of atoms increases. Using the exact numerical solution of the density-matrix equations for a small system, we demonstrate that strong dephasing of the optically driven dipoles renders the many-body problem amenable to semiclassical Monte Carlo simulations. We employ the Monte Carlo algorithm for a large number of atoms and find that in the steady state of a uniformly driven, extended one-dimensional system, the saturation of superatoms leads to quasicrystallization of Rydberg excitations whose correlations exhibit damped spatial oscillations. We show that the behavior of the system under the van der Waals interaction potential can be approximated by a rate-equations model based on a ``hard-rod'' interatomic potential, and by solving it we obtain the period and correlation length for the density wave of Rydberg excitations.
We study resonant optical excitations of strongly interacting Rydberg states of atoms in the presence of relaxations. We employ the quantum stochastic (Monte Carlo) wavefunctions to simulate the dissipative dynamics of tens of atoms in two-dimensional lattices. We show that under typical experimental conditions involving the slow Rydberg state decay and sizable relaxation of atomic coherences, on the timescale of several μs the atomic ensemble approaches a stationary state in which much of the quantum correlations between the atoms have decayed away. The steady state, however, exhibits strong classical correlations of Rydberg excitation probabilities.