Microwave dielectric properties of LiNbO$_{\mathbf{3}}$ and AlN at millikelvin temperatures and single-photon power
Efficient bidirectional microwave-optical photon conversion is a key capability for scaling superconducting quantum processors into distributed networks. However, achieving the necessary conversion efficiency requires filling a critical knowledge gap in understanding the loss mechanisms of electro-optic materials. Here, we characterize the microwave properties of single-crystal bulk LiNbO3 and AlN over a broad range of powers, down to single-photon levels, and spanning from millikelvin temperatures to above 1 K. We demonstrate that both materials exhibit two-level systems (TLS) behavior, while piezoelectric-related losses are excluded. We show that TLS-induced dissipation is predominantly localized on the surface rather than being an intrinsic bulk property, a result further corroborated by room-temperature 3D XPS and time-of-flight SIMS analyses. These findings provide useful insights to engineer hybrid architectures that integrate bulk electro-optic crystals within superconducting cavities, proving that microwave quality factors compatible with high-efficiency microwave-optical transduction are within reach.