  The nitrogen-vacancy (NV) center in diamond is well known in quantum
metrology and quantum information for its favorable spin and optical
properties, which span a wide temperature range from near zero to over 600 K.
Despite its prominence, the NV center's photo-physics is incompletely
understood, especially at intermediate temperatures between 10-100 K where
phonons become activated. In this work, we present a rate model able to
describe the cross-over from the low-temperature to the high-temperature
regime. Key to the model is a phonon-driven hopping between the two orbital
branches in the excited state (ES), which accelerates spin relaxation via an
interplay with the ES spin precession. We extend our model to include magnetic
and electric fields as well as crystal strain, allowing us to simulate the
population dynamics over a wide range of experimental conditions. Our model
recovers existing descriptions for the low- and high-temperature limits, and
successfully explains various sets of literature data. Further, the model
allows us to predict experimental observables, in particular the
photoluminescence (PL) emission rate, spin contrast, and spin initialization
fidelity relevant for quantum applications. Lastly, our model allows probing
the electron-phonon interaction of the NV center and reveals a gap between the
current understanding and recent experimental findings.
