  Hybrid quantum systems, which combine quantum-mechanical systems with
macroscopic mechanical oscillators, have attracted increasing interest as they
are well suited as high-performance sensors or transducers in quantum
computers. A promising candidate is based on diamond cantilevers, whose motion
is coupled to embedded Nitrogen-Vacancy (NV) centers through crystal
deformation. Even though this type of coupling has been investigated
intensively in the past, several inconsistencies exist in available literature,
and no complete and consistent theoretical description has been given thus far.
To clarify and resolve these issues, we here develop a complete and consistent
formalism to describe the coupling between the NV spin degree of freedom and
crystal deformation in terms of stress, defined in the crystal coordinate
system XYZ, and strain, defined in the four individual NV reference frames. We
find that the stress-based approach is straightforward, yields compact
expressions for stress-induced level shifts and therefore constitutes the
preferred approach to be used in future advances in the field. In contrast, the
strain-based formalism is much more complicated and requires extra care when
transforming into the employed NV reference frames. Furthermore, we illustrate
how the developed formalism can be employed to extract values for the
spin-stress and spin-strain coupling constants from data published by Teissier
et al..
