Estimation of vector static magnetic field by a nitrogen-vacancy center with a single first-shell ¹³C nuclear (NV-¹³C) spin in diamond

We suggest an experimental scheme that a single nitrogen-vacancy (NV) center coupled to a nearest neighbor ¹³C nucleus as a sensor in diamond can be used to detect a static vector magnetic field. By means of optical detection magnetic resonance (ODMR) technique, both the strength and the direction of the vector field could be determined by relevant resonance frequencies of continuous wave (CW) and Ramsey spectrums. In addition, we give a method that determines the unique one of eight possible hyperfine tensors for an (NV-¹³C) system. Finally, we propose an unambiguous method to exclude the symmetrical solution from eight possible vector fields, which correspond to nearly identical resonance frequencies due to their mirror symmetry about ¹⁴N-Vacancy-¹³C (¹⁴N-V-¹³C) plane.