  We investigate the relaxation of a superconducting qubit for the case when
its detector, the Josephson bifurcation amplifier, remains latched in one of
its two (meta)stable states of forced vibrations. The qubit relaxation rates
are different in different states. They can display strong dependence on the
qubit frequency and resonant enhancement, which is due to quasienergy
resonances. Coupling to the driven oscillator changes the effective temperature
of the qubit.
