  Atom-photon bound states arise from the coupling of quantum emitters to the
band edge of dispersion-engineered waveguides. Thanks to their tunable-range
interactions, they are promising building blocks for quantum simulators. Here,
we study the dynamics of an atom-photon bound state emerging from coupling a
frequency-tunable quantum emitter - a transmon-type superconducting circuit -
to the band edge of a microwave metamaterial. Employing precise temporal
control over the frequency detuning of the emitter from the band edge, we
examine the transition from adiabatic to non-adiabatic behavior in the
formation of the bound state and its melting into the propagating modes of the
metamaterial. Moreover, we experimentally observe multi-mode emission from the
bound state, triggered by a fast change of the emitter's frequency. Our study
offers insight into the dynamic preparation of atom-photon bound states and
provides a method to characterize their photonic content, with implications in
quantum optics and quantum simulation.
