  We study resonant optical excitations of atoms in a one-dimensional lattice
to the Rydberg states interacting via the van der Waals potential which
suppresses simultaneous excitation of neighboring atoms. Considering two- and
three-level excitation schemes, we analyze the dynamics and stationary state of
the continuously-driven, dissipative many-body system employing time-dependent
density-matrix renormalization group (t-DMRG) simulations. We show that
two-level atoms can exhibit only nearest neighbor correlations, while
three-level atoms under dark-state resonant driving can develop finite-range
crystalline order of Rydberg excitations. We present an approximate rate
equation model whose analytic solution yields qualitative understanding of the
numerical results.
