Exact Breit-Rabi formulae for the nitrogen-vacancy center in diamond

The hyperfine structure that the nitrogen-vacancy (NV) center in diamond has with its nitrogen nucleus is of immense importance for quantum sensing and quantum computing. Current methods of calculating the NV hyperfine energy eigenvalues and eigenstates with respect to an axial magnetic field - i.e., calculating the NV Breit-Rabi formula - resort to numerical methods and analytic approximations. Here we provide an exact, closed-form expression to the NV hyperfine energy eigenvalues and eigenstates for both the nitrogen-14 and -15 cases.