  We study coherent and radiative interactions induced among two or more
quantum units, by coupling them to two-dimensional lattices acting as
structured environments. This model can be representative of atoms trapped near
photonic crystal slabs, trapped ions in Coulomb crystals or to surface acoustic
waves on piezoelectric materials, cold atoms on state-dependent optical
lattices, or even circuit QED architectures, to name a few. We compare coherent
and radiative contributions for the isotropic and directional regimes of
emission into the lattice, for infinite and finite lattices, highlighting their
differences and existing pitfalls, e.g. related to long-time or large-lattice
limits. We relate the phenomenon of directionality of emission with
linear-shaped isofrequency manifolds in the dispersion relation, showing a
simple way to disrupt it. For finite lattices, we study further details as the
scaling of resonant number of lattice modes for the isotropic and directional
regimes, and relate this behavior with known van Hove singularities in the
infinite lattice limit. Further we export the understanding of emission
dynamics with the decay of entanglement for two quantum, atomic or bosonic,
units coupled to the 2D lattice. We analyze in some detail completely
subradiant configurations of more than two atoms, which can occur in the finite
lattice scenario, in contrast with the infinite lattice case. Finally we
demonstrate that induced coherent interactions for dark states are zero for the
finite lattice.
