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Although the main loss channel of planar microwave superconducting resonatorshas been identified to be related to an external coupling to a two-level system(TLS) bath, the behavior of such a cavity in the presence of an off-resonantpump has yet to be fully understood. Here a semi-classical model that predictsa power-dependent frequency shift besides the well-known TLS-induced damping isderived. The model is validated experimentally by performing a two-tonespectroscopy of several resonators fabricated on various substrates. Togetherwith the provided analytic formulas, the technique proposed here is a simple,yet powerful, tool to characterize various properties of the TLS bath, such astheir average dephasing rate.

Probing a two-level system bath via the frequency shift of an off-resonantly-driven cavity