Long-lived entanglement of two multilevel atoms in a waveguide

We study the presence of nontrivial bound states of two multilevel quantum emitters and the photons propagating in a linear waveguide, when the emitters can be modeled as harmonic oscillators. We characterize the conditions for the existence of such states and determine their general properties, focusing in particular on the entanglement between the two emitters, that increases with the number of excitations. Wediscuss the relevance of our results for entanglement preservation and generation by spontaneous relaxation processes. We finally introduce a perturbative method for determining the existence of long-lived states, for general level spacings and couplings.