In this dissertation I describe measurements of dielectric loss at microwave frequencies due to two level systems (TLS) using superconducting resonators. Most measurements were performed in a dilution refrigerator at temperatures between 30 and 200 mK and all resonators discussed were fabricated with thin-film superconducting aluminum.

I derive the transmission through a non-ideal (mismatched) resonant circuit and find that in general the resonance line-shape is asymmetric. I describe an analysis method for extracting the internal quality factor (Q_i), the diameter correction method (DCM), and compare it to a commonly used phenomenological method, the φ rotation method (φRM). I analytically find that the φRM deterministically overestimates Q_i when the asymmetry of the resonance line-shape is high.

Four coplanar resonator geometries were studied, with frequencies spanning 5-7 GHz. They were all superconducting aluminum fabricated on sapphire and silicon substrates. These include a quasi-lumped element resonator, a coplanar strip transmission line resonator, and two hybrid designs that contain both a coplanar strip and a quasi-lumped element. Measured Q_i's were as high as 2 × 105 for single photon excitations and there was no systematic variation in loss between quasi-lumped and coplanar strip resonance modes.

I also measured the microwave loss tangent of several atomic layer deposition (ALD) grown dielectrics and obtained secondary ion mass spectrometry (SIMS) measurements of the same films. I found that hydrogen defect concentrations were correlated with low temperature microwave loss. In amorphous films that showed excess hydrogen defects on the surface, two independent TLS distributions were required to fit the loss tangent, one for the surface and one for the bulk. In crystalline dielectrics where hydrogen contamination was uniform throughout the bulk, a single bulk TLS distribution was sufficient.

Finally, I measured the TLS loss in 250 nm thick HD-PECVD deposited silicon nitride (SiN_x) while sweeping an independent applied bias electric field across the capacitor. With a strong microwave field and an increasing bias rate, the loss tangent changed from a low value, where saturation occurs on resonance near the steady state, to a larger value approximately equal to the linear-response loss tangent, where saturation appears to be avoided. This increase was explained with a new theory in which TLSs can experience Landau-Zener transitions as they're swept, where the maximum excitation probability is 1/2 at resonance. Data is found to scale if plotted as a function of the dimensionless sweep rate. The functional form of this loss tangent agrees well with the theory, and is predicted to hold for any amorphous dielectric. By fitting the measured loss tangent as a function of bias sweep rate to the theory, I was able to extract an average TLS dipole moment of 7.9 D and a TLS spectral spatial density of P₀=4.9 × 1043 J-1 m-3.