Amplification and attenuation of a probe signal by doubly-dressed states

We analyse a system composed of a qubit coupled to electromagnetic fields of two high quality quantum oscillators. Particular realization of such a system is the superconducting qubit coupled to a transmission-line resonator driven by two signals with frequencies close to the resonator's harmonics. One strong signal is used for exciting the system to a high energetic state while the second weak signal is applied for probing effective eigenstates of the system. We demonstrate that a description of the system dynamics as doubly dressed qubit is applicable. Experiments show that in the case of high quality resonators the energy levels and the resonance conditions can be probed even for high driving amplitudes. The interaction of the qubit with photons of two harmonics has prospects to be used as a quantum amplifier or an attenuator.