  The cooperative modification of spontaneous radiative decay is a paradigmatic
many-emitter effect in quantum optics. So far its experimental realization has
involved interactions mediated by rapidly escaping photons that do not play an
active role in the emitter dynamics. Here we explore cooperative dynamics of
quantum emitters in an optical lattice that interact by radiating atomic matter
waves. Using the ability to prepare weakly and strongly interacting many-body
phases of excitations in an array of matter-wave emitters, we demonstrate
directional super- and subradiance from a superfluid phase with tunable
radiative phase lags, and directly access the buildup of coherence imprinted by
the emitted radiation across a Mott insulator. We investigate the onset of
cooperative dynamics for slow wave propagation and observe a coupling to
collective bound states with radiation trapped at and between the emitters. Our
results in open-system quantum electrodynamics establish ultracold matter waves
as a versatile tool for studying many-body quantum optics in spatially extended
and ordered systems.
