  Color centers in hexagonal boron nitride have shown enormous promise as
single-photon sources, but a clear understanding of electron-phonon interaction
dynamics is critical to their development for quantum communications or quantum
simulations. We demonstrate photon antibunching in the filtered auto- and
cross-correlations $g^{(2)}_{lm}(\tau)$ between zero-, one- and two-phonon
replicas of defect luminescence. Moreover, we combine autocorrelation
measurements with a violation of the Cauchy-Schwarz inequality in the filtered
cross-correlation measurements to distinguish a low quantum-efficiency defect
from phonon replicas of a bright defect. With no background correction, we
observe single photon purity of $g^{(2)}(0)=0.20$ in a phonon replica and
cross-spectral correlations of $g^{(2)}_{lm}(0)=0.18$ between a phonon replica
and the zero phonon line. These results illustrate a coherent interface between
visible photons and mid-infrared phonons and provide a clear path toward
control of photon-phonon entanglement in 2D materials.
