  Spatial structures commonly emerge in interacting nonlinear systems. In this
study, we focus on the out-of-equilibrium dynamics of the recently-established
platform of Rydberg exciton-polaritons, fueled by their characteristic
long-range non-local interactions, in the presence of an external drive and
dissipation. Our work elucidates how modulational instability sets off
spontaneous density pattern formations in a Rydberg polariton system with
characteristic scales in the micron range. Under conditions of an unstable
flattop state, we track the evolution of the polariton ensemble, showing the
emergence of meta-stable patterns and their collapse in the long-time limit. We
trace this phenomenon back to the destructive interference between the
polariton state and the pump in a driven ensemble. Finally, we map out
conditions that allow stable patterns to form under incoherent pumping. These
findings provide new opportunities for exploring the emerging field of
long-range interacting gases through Rydberg exciton-polaritons.
