  We theoretically study a system composed by a waveguide and a moving quantum
emitter in the single excitation subspace, treating the emitter motional degree
of freedom quantum mechanically. We first characterize single-photon scattering
off a single moving quantum emitter, showing both nonreciprocal transmission
and recoil-induced reduction of the quantum emitter motional energy. We then
characterize the bound states within the bandgap, which display a
motion-induced asymmetric phase in real space. We also demonstrate how these
bound states form a continuous band with exotic dispersion relations. Finally,
we study the spontaneous emission of an initially excited quantum emitter with
various initial momentum distributions, finding strong deviations with respect
to the static emitter counterpart both in the occupation dynamics and in the
spatial distribution of the emitted photons. Our work extends the waveguide-QED
toolbox by including the quantum motional degree of freedom of emitters, whose
impact in the few-photon dynamics could be harnessed for quantum technologies.
