Multiband and array effects in matter-wave-based waveguide QED

Recent experiments on spontaneous emission of atomic matter waves [Krinner, Nature (London) 559, 589 (2018)0028-083610.1038/s41586-018-0348-z; Stewart, Phys. Rev. Research 2, 043307 (2020)2643-156410.1103/PhysRevResearch.2.043307] open a window into the behavior of quantum emitters coupled to a waveguide. Here we develop an approach based on infinite products to study this system theoretically, without the need to approximate the band dispersion relation of the waveguide. We solve the system for a one-dimensional array of one, multiple, and an infinite number of quantum emitters and compare with the experiments. This leads to a detailed characterization of the decay spectrum, with a family of in-gap bound states, mechanisms for enhanced Markovian emission different from superradiance, and the emergence of matter-wave polaritons.