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Motivated by recent cold-atom realisations of matter-wave waveguide QED, westudy simple fermionic impurity models and discuss fermionic analogues ofseveral paradigmatic phenomena in quantum optics, including formation ofnon-trivial bound states, (matter-wave) emission dynamics, and collectivedissipation. For a single impurity, we highlight interesting ground-statefeatures, focusing in particular on real-space signatures of an emergent lengthscale associated with an impurity screening cloud. We also present novelnon-Markovian many-body effects in the quench dynamics of single- andmultiple-impurity systems, including fractional decay around the Fermi leveland multi-excitation population trapping due to bound states in the continuum.

Fermionic matter-wave quantum optics with cold-atom impurity models