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

Longitudinal relaxation of a nitrogen-vacancy center in a spin bath by generalized cluster-correlation expansion method