  The polariton, a quasiparticle formed by strong coupling of a photon to a
matter excitation, is a fundamental ingredient of emergent photonic quantum
systems ranging from semiconductor nanophotonics to circuit quantum
electrodynamics. Exploiting the interaction between polaritons has led to the
realization of superfluids of light as well as of strongly correlated phases in
the microwave domain, with similar efforts underway for microcavity
exciton-polaritons. Here, we develop an ultracold-atom analogue of an
exciton-polariton system in which interacting polaritonic phases can be studied
with full tunability and without dissipation. In our optical-lattice system,
the exciton is replaced by an atomic excitation, while an atomic matter wave is
substituted for the photon under a strong dynamical coupling. We access the
band structure of the matter-wave polariton spectroscopically by coupling the
upper and lower polariton branches, and explore polaritonic many-body transport
in the superfluid and Mott-insulating regimes, finding quantitative agreement
with our theoretical expectations. Our work opens up novel possibilities for
studies of polaritonic quantum matter.
