Broadband and high-precision two-level system loss measurement using superconducting multi-wave resonators

Two-level systems (TLS) are known to be a dominant source of dissipation and decoherence in superconducting qubits. Superconducting resonators provide a convenient way to study TLS-induced loss due to easier design and fabrication in comparison to devices that include non-linear elements. However, accurately measuring TLS-induced loss in a resonator in the quantum regime is challenging due to low signal-to-noise ratio (SNR) and the temporal fluctuations of the TLS, leading to uncertainties of 30% or more. To address these limitations, we develop a multi-wave resonator device that extends the resonator length from a standard quarter-wave $\lambda/4$ to $N\lambda/4$ where $N = 37$ at 6GHz. This design provides two key advantages: the TLS-induced fluctuations are reduced by a factor of $\sqrt{N}$ due to spatial averaging over an increased number of independent TLS, and the measurement SNR for a given intra-resonator energy density improves by a factor of $\sqrt{N}$. The multi-wave resonator also has fundamental and harmonic resonances that allow one to study the frequency dependence of TLS-induced loss. In this work we fabricate both multi-wave and quarter-wave coplanar waveguide resonators formed from thin-film aluminum on a silicon substrate, and characterize their TLS properties at both 10mK and 200mK. Our results show that the power-dependent TLS-induced loss measured from both types of resonators agree well, with the multi-wave resonators achieving a five-fold reduction in measurement uncertainty due to TLS fluctuations, down to 5%. The $N\lambda/4$ resonator also provides a measure of the fully unsaturated TLS-induced loss due to the improved measurement SNR at low intra-resonator energy densities. Finally, measurements across seven harmonic resonances of the $N\lambda/4$ resonator between 4GHz - 6.5GHz reveals no frequency dependence in the TLS-induced loss over this range.