  We use a density matrix formalism to study the equilibrium phases and
non-equilibrium dynamics of a system of dissipative Rydberg atoms in an optical
lattice within mean-field theory. We provide equations for the fixed points of
the density matrix evolution for atoms with infinite on-site repulsion and
analyze these equations to obtain their Mott insulator- superfluid (MI-SF)
phase boundary. A stability analysis around these fixed points provides us with
the excitation spectrum of the atoms both in the MI and SF phases. We study the
nature of the MI-SF critical point in the presence of finite dissipation of
Rydberg excitations, discuss the fate of the superfluidity of the atoms in the
presence of such dissipation in the weak-coupling limit using a coherent state
representation of the density matrix, and extend our analysis to Rydberg atoms
with finite on-site interaction via numerical solution of the density matrix
equations. Finally, we vary the boson (atom) hopping parameter $J$ and the
dissipation parameter $\Gamma$ according to a linear ramp protocol. We study
the evolution of entropy of the system following such a ramp and show that the
deviation of the entropy from its steady state value for the latter protocol
exhibits power-law behavior as a function of the ramp time. We discuss
experiments which can test our theory.
