  We study the backaction of a driven nonlinear resonator on a multi-level
superconducting qubit. Using unitary transformations on the multi-level
Jaynes-Cummings Hamiltonian and quantum optics master equation, we derive an
analytical model that goes beyond linear response theory. Within the limits of
validity of the model, we obtain quantitative agreement with experimental and
numerical data, both in the bifurcation and in the parametric amplification
regimes of the nonlinear resonator. We show in particular that the
measurement-induced dephasing rate of the qubit can be rather small at high
drive power. This is in contrast to measurement with a linear resonator where
this rate increases with the drive power. Finally, we show that, for typical
parameters of circuit quantum electrodynamics, correctly describing
measurement-induced dephasing requires a model going beyond linear response
theory, such as the one presented here.
