  Continuous spectroscopic measurements of a strongly driven superconducting
qubit by means of a high-quality tank circuit (a linear detector) are under
study. Output functions of the detector, namely, a spectrum of voltage
fluctuations and an impedance, are expressed in terms of the qubit spectrum and
magnetic susceptibility. The nonequilibrium spectrum of the current
fluctuations in the qubit loop and the linear response function of the driven
qubit coupled to a heat bath are calculated with Bloch-Redfield and rotating
wave approximations. Backaction effects of the qubit on the tank and the tank
on the qubit are analyzed quantitatively. We show that the voltage spectrum of
the tank provides detailed information about a frequency and a decay rate of
Rabi oscillations in the qubit. It is found that both an efficiency of
spectroscopic measurement and measurement-induced decoherence of the qubit
demonstrate a resonant behaviour as the Rabi frequency approaches the resonant
frequency of the tank. We determine conditions when the spectroscopic
observation of the Rabi oscillations in the flux qubit with the tank circuit
can be considered as a weak continuous quantum measurement.
