  We consider a quantum emitter ("atom") radiating in a one-dimensional (1D)
photonic waveguide in the presence of a single mirror, resulting in a delay
differential equation for the atomic amplitude. We carry out a systematic
analysis of the non-Markovian (NM) character of the atomic dynamics in terms of
refined, recently developed notions of quantum non-Markovianity such as
indivisibility and information back-flow. NM effects are quantified as a
function of the round-trip time and phase shift associated with the atom-mirror
optical path. We find, in particular, that unless an atom-photon bound state is
formed a finite time delay is always required in order for NM effects to be
exhibited. This identifies a finite threshold in the parameter space, which
separates the Markovian and non-Markovian regimes.
