  We study numerically the onset of higher-level excitations and resonance
frequency shifts in the generalized multilevel Rabi model with dispersive
coupling under strong driving. The response to a weak probe is calculated using
the Floquet method, which allows us to calculate the probe spectrum and extract
the resonance frequency. We test our predictions using a superconducting
circuit consisting of transmon coupled capacitively to a coplanar waveguide
resonator. This system is monitored by a weak probe field, and at the same time
driven at various powers by a stronger microwave tone. We show that the
transition from the quantum to the classical regime is accompanied by a rapid
increase of the transmon occupation and, consequently that the qubit
approximation is valid only in the extreme quantum limit.
