  We theoretically study the longitudinal relaxation of a nitrogen-vacancy (NV)
center surrounded by a 13C nuclear spin bath in diamond. By incorporating
electron spin in the cluster, we generalize the cluster-correlation expansion
(CCE) to theoretically simulate the population dynamics of electron spin of NV
center. By means of the generalized CCE, we numerically demonstrate the decay
process of electronic state induced by cross relaxation at the ambient
temperature. It is shown that the CCE method is not only capable of describing
pure-dephasing effect at large-detuning regime, but it can also simulate the
quantum dynamics of populations in the nearly-resonant regime.
