  Nitrogen-vacancy (NV) centers in diamond have been successfully coupled to
various optical structures to enhance their radiation by the Purcell effect.
The participation of many NV centers in these studies may naturally lead to
cooperative emission and superradiance, and our recent experimental study with
a diamond membrane in a fiber-based ultra-narrow optical cavity demonstrated
nonlinear radiation power and fast photon bunching which are signatures of such
collective effects. In this theoretical article, we go beyond the simple model
used in the previous study to address more phenomena, such as the appearance of
bunching shoulders in the second-order correlation function, Rabi splitting in
the steady-state spectrum, and population dynamics on excited Dicke states,
which for moderate pumping explains the observed collective effects. Overall,
our results can guide further experiments with NV centers, and they are also
relevant for other solid-state color centers, such as silicon-vacancy centers
in diamond and silicon carbide, boron-vacancy centers and carbon-related
centers in hexagonal boron-nitride.
