  The characteristics of spontaneous emission can be strongly modified by the
mode structure of the vacuum. In waveguide quantum-electrodynamics based on
photonic crystals, this modification is exploited to engineer atom-photon
interactions near a band edge, but the physics of coupling to an entire band
has not yet been explored in experiments. Using ultracold atoms in an optical
lattice, we study the decay dynamics of matter-wave quantum emitters coupled to
a single band of an effective photonic crystal waveguide structure with tunable
characteristics. Depending on the ratio between vacuum coupling and bandwidth,
we observe a transition from irreversible decay to fully oscillatory dynamics
linked to the interplay of matter-wave bound states near the band edges, whose
spatial structure we characterize. Our results shed light on the emergence of
coherence in an open quantum system in a controllable environment, and are of
relevance for the understanding of vacuum-induced decay phenomena in photonic
systems.
