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.
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