Probing a Two-Level System Bath via the Frequency Shift of an Off-Resonantly Driven Cavity

Although the main loss channel of planar microwave superconducting resonators has been identified to be related to an external coupling to a two-level system (TLS) bath, the behavior of such resonators in the presence of an off-resonant pump has yet to be fully understood. Alongside the well-known power-dependent damping, we observe a frequency shift with a conspicuous maximum for intermediate pump power that is attributed to a spectrally asymmetric saturation of the TLSs. We derive a semiclassical model that describes both of these effects quantitatively. The model is validated experimentally by performing a two-tone spectroscopy of several resonators fabricated on various substrates. Together with the provided analytic formulas, the technique proposed here is a simple yet powerful tool to unambiguously identify the presence of a limiting TLS bath, and to characterize various properties thereof, such as its average dephasing rate.