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Synthetic quantum materials offer an exciting opportunity to explore quantummany-body physics and novel states of matter under controlled conditions. Inparticular, they provide an avenue to exchange the short length scales andlarge energy scales of the solid state for an engineered system with bettercontrol over the system Hamiltonian, more accurate state preparation, andhigher fidelity state readout. Here we propose a unique platform to studyquantum phases of strongly interacting photons. We introduce ideas forcontrolling the dynamics of individual photons by manipulating the geometry ofa multimode optical cavity, and combine them with recently establishedtechniques to mediate strong interactions between photons using Rydberg atoms.We demonstrate that this approach gives rise to crystalline- and fractionalquantum Hall- states of light, opening the door to studies of stronglycorrelated quantum many-body physics in a photonic material.

Quantum Crystals and Laughlin Droplets of Cavity Rydberg Polaritons