  We investigate the possibility of realizing supersolid quantum phases in
bosonic Rydberg-excited quantum lattice gases in the presence of non-unitary
processes, by simulating the dynamical evolution starting from initial
preparation in non-dissipative equilibrium states. Within Gutzwiller theory, we
first analyze the many-body ground-state of a bosonic Rydberg-excited quantum
gas in a two dimensional optical lattice for variable atomic hopping rates and
Rabi detunings. Furthermore, we perform time evolution of different supersolid
phases using the Lindblad-master equation. With the inclusion of two different
non-unitary processes, namely spontaneous decay from a Rydberg state to the
ground state and dephasing of the addressed Rydberg state, we study the effect
of non-unitary processes on those quantum phases and observe long-lived states
in the presence of decay and dephasing. We find that long-lived supersolid
quantum phases are observable within a range of realistic decay and dephasing
rates, while high rates cause any initial configuration to homogenize quickly,
preventing possible supersolid formation.
