  We study the dynamics of a quantum emitter coupled to a two-dimensional
photonic crystal featuring a finite bandwidth with sharp edges and a Van-Hove
singularity. We study the effect of strong system-bath coupling and
non-Markovianity of the photonic environment using a nonperturbative approach
based on the recently introduced NCA dynamical map for open quantum systems. We
show that several characteristic features of the dynamics near a photonic
band-edge such as the freezing of spontaneous emission and the maximum
light-matter entanglement, get strongly modified in presence of
counter-rotating terms in the system-bath coupling, beyond the rotating-wave
approximation. Furthermore, by computing the spectral function of the quantum
emitter we comment on the role played by atom-photon bound-state and show that
this acquires a much larger lifetime once the rotating-wave approximation is
relaxed.
