Microwave dielectric properties of LiNbO$_{\mathbf{3}}$ and AlN at millikelvin temperatures and single-photon power
Electro-optic materials are paramount to achieving efficient and reliable microwave-optical signal transduction in the quantum regime. It is of great interest to investigate the performance of these materials at the quantum level and at cryogenic temperatures to assess their compatibility with superconducting circuit-based quantum systems. In this work, we present a detailed study of the microwave electric properties of single-crystal bulk LiNbO$_{3}$ and AlN at millikelvin temperatures and at the single-photon level. We characterize the materials' dielectric loss tangent throughout a wide range of electromagnetic power levels in a temperature range from a few tens of millikelvin to above 1~K. Our findings indicate that both materials' loss tangent behavior is consistent with the two-level system model up to a certain power threshold, beyond which it increases logarithmically with power. This suggests that two-level system loss mechanisms dominate at the single-photon level, while additional loss channels become relevant at higher powers and temperatures.