Fig. 1. Experimental setup:　(a) Measurement system: signals from both the PNA and the PSG generator are combined and attenuated before entering the dilution refrigerator; the sample is placed in the sample chamber of the refrigerator whose working temperature is 20 mK; (b) the schematic of the coplanar-waveguide resonator, with a four-junction flux qubit located at the center of the resonator; yellow and blue curves are current distributions of the first and third modes of the resonator, respectively; (c) the scanning electron microscope images of the flux qubit.

Fig. 2. The transmission |S₂₁| as a function of the flux bias: (a) The transmission |S₂₁| of the third mode of the resonator as a function of the flux bias; (b) the transmission |S₂₁| of the first mode of the resonator as a function of the flux bias. Red dashed curves are the fitting results numerically obtained from the Hamiltonian (2) equation.

Fig. 3. The spectrum measured using the first mode of the resonator as a function of the flux bias. The red dashed curve is the numerical result for the qubit transition and the average photon number in the resonator is 4.7 × 10^–3.

Fig. 4. The qubit transition frequency as a function of the average photon number. The red solid curve denotes the simulation results when both the ac Stark and Bloch-Siegert shifts are included. The blue dashed curve denotes the simulation results when only the Bloch-Siegert shift is included.