  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 a cavity in the presence of an off-resonant
pump has yet to be fully understood. Here a semi-classical model that predicts
a power-dependent frequency shift besides the well-known TLS-induced damping is
derived. 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 characterize various properties of the TLS bath, such as
their average dephasing rate.
