  Ordinarily, photons do not interact with one another. However, atoms can be
used to mediate photonic interactions, raising the prospect of forming
synthetic materials and quantum information systems from photons. One promising
approach uses electromagnetically-induced transparency with highly-excited
Rydberg atoms to generate strong photonic interactions. Adding an optical
cavity shapes the available modes and forms strongly-interacting polaritons
with enhanced light-matter coupling. However, since every atom of the same
species is identical, the atomic transitions available are only those
prescribed by nature. This inflexibility severely limits their utility for
mediating the formation of photonic materials in cavities, as the resonator
mode spectrum is typically poorly matched to the atomic spectrum. Here we use
Floquet engineering to redesign the spectrum of Rubidium and make it compatible
with the spectrum of a cavity, in order to explore strongly interacting
polaritons in a customized space. We show that periodically modulating the
energy of an atomic level redistributes its spectral weight into
lifetime-limited bands separated by multiples of the modulation frequency.
Simultaneously generating bands resonant with two chosen spatial modes of an
optical cavity supports "Floquet polaritons" in both modes. In the presence of
Rydberg dressing, we find that these polaritons interact strongly. Floquet
polaritons thus provide a promising new path to quantum information
technologies such as multimode photon-by-photon switching, as well as to
ordered states of strongly-correlated photons, including crystals and
topological fluids.
