Power and temperature dependent model for High Q superconductors

Abstract: Measuring the internal quality factor of coplanar waveguide superconducting resonators is an established method of determining small losses in superconducting devices. Traditionally, the resonator losses are only attributed to two-level system (TLS) defects using a power dependent model for the quality factor. However, excess non-equilibrium quasiparticles can also limit the quality factor of the planar superconducting resonators used in circuit quantum electrodynamics. At millikelvin temperatures, quasiparticles can be generated by breaking Cooper pairs via a single high-energy or multiple sub-gap photons. Here a two-temperature, power and temperature dependent model is proposed to evaluate resonator losses for isolating TLS and quasiparticle loss simultaneously. The model combines the conventional TLS power and temperature dependence with an effective temperature non-equilibrium quasiparticle description of the superconducting loss. The quasiparticle description is based on the quasiparticle number density calculated using rate equations for an external quasiparticle generation source, recombination, and trapping. The number density is translated to an effective temperature using a thermal distribution that may be different from the bath. Experimental measurements of high-quality factor resonators fabricated from single crystal aluminum and titanium nitride thin films on silicon are interpreted with the presented model. This approach enables identification of quasiparticle and TLS loss, resulting in the determination that the TiN resonator has comparable TLS and quasiparticle loss at low power and low-temperature, while the low-temperature Al resonator behavior is dominated by non-equilibrium quasiparticle loss.