Theory of weak continuous measurements in a strongly driven quantum bit

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. Back-action 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 the frequency and 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 behavior 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.