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

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 the electron spin of the NV center. By means of the generalized CCE, we numerically demonstrate the decay process of the electronic state induced by cross relaxation at ambient temperature. It is shown that the CCE method is not only capable of describing pure-dephasing effects in the large-detuning regime, but it can also simulate the quantum dynamics of populations in the nearly-resonant regime.