  We investigate a one-dimensional atomic lattice laser-driven to a Rydberg
state, in which engineered dissipation channels lead to entanglement in the
many-body system. In particular, we demonstrate the efficient generation of
ground states of a frustration-free Hamiltonian, as well as $W$ states. We
discuss the realization of the required coherent and dissipative terms, and we
perform extensive numerical simulations characterizing the fidelity of the
state preparation procedure. We identify the optimum parameters for high
fidelity entanglement preparation and investigate the scaling with the size of
the system.
