Can quantum materials be built out of light? In the hope of doing just that, we have developed a system for turning optical photons into cavity Rydberg polaritons: quasiparticles which inherit their spatial waveforms from the modes of an optical cavity and gain strong interactions from Rydberg excitations of an atomic gas. In a single cavity mode, the strong interactions between polaritons manifest as transport blockade, in which an individual photon in the cavity prevents any other photons from entering. To go beyond blockade, we use Floquet engineering to enable the polaritons to move around and self-organize among multiple transverse modes of the cavity. Finally, I will discuss our preliminary experiments on building photonic Laughlin states, the ground states of a fractional quantum Hall system.