  Generating and detection coherent high-frequency heat-carrying phonons has
been a great topic of interest in recent years. While there have been
successful attempts in generating and observing coherent phonons, rigorous
techniques to characterize and detect these phonon coherence in a crystalline
material have been lagging compared to what has been achieved for photons. One
main challenge is a lack of detailed understanding of how detection signals for
phonons can be related to coherence. The quantum theory of photoelectric
detection has greatly advanced the ability to characterize photon coherence in
the last century and a similar theory for phonon detection is necessary. Here,
we re-examine the optical sideband fluorescence technique that has been used
detect high frequency phonons in materials with optically active defects. We
apply the quantum theory of photodetection to the sideband technique and
propose signatures in sideband photon-counting statistics and second-order
correlation measurement of sideband signals that indicates the degree of phonon
coherence. Our theory can be implemented in recently performed experiments to
bridge the gap of determining phonon coherence to be on par with that of
photons.
