Microwave dielectric loss at single photon energies and millikelvin temperatures

The microwave performance of amorphous dielectric materials at very low temperatures and very low excitation strengths displays significant excess loss. Here, we present the loss tangents of some common amorphous and crystalline dielectrics, measured at low temperatures ( T < 100 mK ) with near single-photon excitation energies, E ∕ ℏ ω 0 ∼ 1 , using both coplanar waveguide and lumped L C resonators. The loss can be understood using a two-level state defect model. A circuit analysis of the half-wavelength resonators we used is outlined, and the energy dissipation of such a resonator on a multilayered dielectric substrate is theoretically considered.