Decay of the rotary echoes for the spin of a nitrogen-vacancy center in diamond

We study dynamics of the electron spin of a nitrogen-vacancy (NV) center subjected to a strong driving field with periodically reversed direction (train of rotary echoes). We use analytical and numerical tools to analyze in detail the form and time scales of decay of the rotary echo train by modeling the decohering spin environment as a random magnetic field. We demonstrate that the problem can be exactly mapped onto a model of spin 1 coupled to a single bosonic mode with imaginary frequency. This mapping allows comprehensive analytical investigation beyond the standard Bloch-Redfield-type approaches. We explore the decay of the rotary echo train under assumption of strong driving and identify the most important regimes of the decay. The analytical results are compared with the direct numerical simulations to confirm quantitative accuracy of our study. We present the results for realistic environment of substitutional nitrogen atoms (P1 centers) and provide a simplified but accurate description for the decay of the rotary echo train of the NV center's spin. The approach presented here can also be used to study decoherence and longitudinal relaxation of other spin systems under conditions of strong driving.