  While dissipative Rydberg gases exhibit unique possibilities to tune
dissipation and interaction properties, very little is known about the quantum
many-body physics of such long-range interacting open quantum systems. We
theoretically analyze the steady state of a van der Waals interacting Rydberg
gas in an optical lattice based on a variational treatment that also includes
long-range correlations necessary to describe the physics of the Rydberg
blockade, i.e., the inhibition of neighboring Rydberg excitations by strong
interactions. In contrast to the ground state phase diagram, we find that the
steady state undergoes a single first order phase transition from a blockaded
Rydberg gas to a facilitation phase where the blockade is lifted. The first
order line terminates in a critical point when including sufficiently strong
dephasing, enabling a highly promising route to study dissipative criticality
in these systems. In some regimes, we also find good quantitative agreement
with effective short-range models despite the presence of the Rydberg blockade,
justifying the wide use of such phenomenological descriptions in the
literature.
