  Nitrogen-vacancy (NV) centers in diamond have become an important tool for
quantum technologies. All of these applications rely on long coherence times of
electron and nuclear spins associated with these centers. Here, we study the
energy level anti-crossings of an NV center in diamond coupled to a first-shell
13C nuclear spin in a small static magnetic field. These level anti-crossings
occur for specific orientations of the static magnetic field due to the strong
non-secular components of the Hamiltonian. At these orientations we observe
decoherence-free subspaces, where the electron spin coherence times (T_2* ) are
5-7 times longer than those at other orientations. Another interesting property
at these level anti-crossings is that individual transition amplitudes are
dominated by a single component of the magnetic dipole moment. Accordingly,
this can be used for vector detection of microwave magnetic fields with a
single NV center. This is particularly important to precisely control the
center using numerical optimal control techniques.
