  In this work we investigate the nonperturbative decay dynamics of a quantum
emitter coupled to a composite right/left handed transmission line (CRLH-TL).
Our theory captures the contributions from the different spectral features of
the waveguide, providing an accurate prediction beyond the weak coupling
regime, and illustrating the multiple possibilities offered by the nontrivial
dispersion of metamaterial waveguides. We show that the waveguide is
characterized by a band-gap with two asymmetric edges: (i) a mu-near-zero (MNZ)
band edge, where spontaneous emission is inhibited and an unstable pole is
smoothly transformed into a bound state, and (ii) an epsilon-near-zero (ENZ)
band edge, where the decay rate diverges and unstable and real (bound state)
poles coexist. In both cases, branch cut singularities contribute with
fractional decay dynamics whose nature depend on the properties of the
band-edges.
