  Superradiance and subradiance are collective effects that emerge from
coherent interactions between quantum emitters. Due to their many-body nature,
theoretical studies of extended samples with length larger than the atomic
transition wavelength are usually restricted to their early time behavior or to
the few-excitation limit. We use herein a mean-field approach to reduce the
complex many-body system to an effective two-atom master equation that includes
all correlations up to second order and that can be numerically propagated in
time. We find that three-dimensional and two-dimensional inverted atomic arrays
sustain superradiance below a critical lattice spacing and quantify the scaling
of the superradiant peak for both dimensionalities. Finally, we study the
late-time dynamics of the system and demonstrate that a subradiant phase
appears before the system finally relaxes.
