  We analyze the Purcell relaxation rate of a superconducting qubit coupled to
a resonator, which is coupled to a transmission line and pumped by an external
microwave drive. Considering the typical regime of the qubit measurement, we
focus on the case when the qubit frequency is significantly detuned from the
resonator frequency. Surprisingly, the Purcell rate decreases when the strength
of the microwave drive is increased. This suppression becomes significant in
the nonlinear regime. In the presence of the microwave drive, the loss of
photons to the transmission line also causes excitation of the qubit; however,
the excitation rate is typically much smaller than the relaxation rate. Our
analysis also applies to a more general case of a two-level quantum system
coupled to a cavity.
