  We have performed spectroscopic measurements of a superconducting qubit
dispersively coupled to a nonlinear resonator driven by a pump microwave field.
Measurements of the qubit frequency shift provide a sensitive probe of the
intracavity field, yielding a precise characterization of the resonator
nonlinearity. The qubit linewidth has a complex dependence on the pump
frequency and amplitude, which is correlated with the gain of the nonlinear
resonator operated as a small-signal amplifier. The corresponding dephasing
rate is found to be close to the quantum limit in the low-gain limit of the
amplifier.
