  Motivated by the recent progress of phonon-mediated control in quantum spin
devices, we propose a possibility of hybrid measurement using electron
paramagnetic resonance (EPR) and a surface acoustic wave (SAW). Considering
quadrupole-strain (QS) couplings suggested for silicon vacancies, we present a
minimum model of the two-level system to investigate a magnetoacoustic
resonance (MAR) coupled to various strain modes driven by the SAW. The
longitudinal and transverse QS couplings can be changed by rotating a magnetic
field, which depends on a combination of the strain modes. Using the Floquet
theory, we elucidate each coupling effect on the time-averaged transition
probability, especially focus on a single-phonon transition process. The
important result is that the longitudinal QS coupling brings about a sharp
photon-assisted resonance and leads to an abrupt change in the field-angle
dependent transition probability. Since this phonon transition process is
always accompanied by the photon transition, the field angle for the sharp
resonance peak can be detected by the EPR measurement. The hybrid EPR-MAR
measurement is useful to confirm the existence of quadrupole degrees of freedom
strongly coupled to elastic strains, and thus it is expected to be a
complementary probe for the precise evaluation of quadrupole properties.
