  Bulk-mode acoustic waves in a crystalline material exert lattice strain
through the thickness of the sample, which couples to the spin Hamiltonian of
defect-based qubits such as the nitrogen-vacancy (NV) center defect in diamond.
This mechanism has been previously harnessed for unconventional quantum spin
control, spin decoherence protection, and quantum sensing. Bulk-mode acoustic
wave devices are also important in the microelectronics industry as microwave
filters. A key challenge in both applications is a lack of appropriate operando
microscopy tools for quantifying and visualizing gigahertz-frequency dynamic
strain. In this work, we directly image acoustic strain within NV
center-coupled diamond thin-film bulk acoustic wave resonators using
stroboscopic scanning hard X-ray diffraction microscopy at the Advanced Photon
Source. The far-field scattering patterns of the nano-focused X-ray diffraction
encode strain information entirely through the illuminated thickness of the
resonator. These patterns have a real-space spatial variation that is
consistent with the bulk strain's expected modal distribution and a
momentum-space angular variation from which the strain amplitude can be
quantitatively deduced. We also perform optical measurements of strain-driven
Rabi precession of the NV center spin ensemble, providing an additional
quantitative measurement of the strain amplitude. As a result, we directly
measure the NV spin-stress coupling parameter $b = 2.73(2)$ MHz/GPa by
correlating these measurements at the same spatial position and applied
microwave power. Our results demonstrate a unique technique for directly
imaging AC lattice strain in micromechanical structures and provide a direct
measurement of a fundamental constant for the NV center defect spin
Hamiltonian.
