Fig. 1: The pump-probe scheme and the experimental setup.
a An M-level transmon is pumped by a strong resonant field with Rabi (carrier) frequency Ωpump (ωpump). The Rabi frequency is related to the radio-frequency (RF) power through \(\Omega =k\sqrt{P}\), where k is a coupling constant and P is the RF power14. The transmon is pumped from \(\left\vert 0\right\rangle\) to \(\left\vert N\right\rangle\) by an N-photon absorption process. A weak probe with frequency ωp is applied to the system. The relaxation rate between adjacent states \(\left\vert N\right\rangle\) and \(\left\vert N-1\right\rangle\) is denoted by ΓN,N−1. b Energy diagram of the dressed states in the rotating frame of a pump frequency ωpump. Here Di (i = 0, 1, …, M−1) is the ith eigenstate (with energy \(\hslash {\omega }_{i}^{{\rm {D}}}\)) of the system with Hamiltonian \({H}_{{\rm {a}}}^{{\prime} }={H}_{{\rm {a}}}+{H}_{{\rm {d}}}\) (Ha and Hd are defined in Eqs. (4) and (5), respectively, in the “Methods” section) and F denotes the photon number. The dashed purple line represents the pump frequency ωpump and the solid blue double-sided arrow indicates the transition between dressed states (see Supplementary Information Note 6 for more details). c A simplified circuit diagram of the experimental setup where a probe field (green) and a pump field (purple) are combined by an RF combiner at room temperature with attenuation (red rectangle) and fed into the sample (optical micrograph inside the red dashed box). The micrograph shows an artificial atom (transmon), formed by a large cross-shaped island, capacitively coupled to the end of a semi-infinite transmission line, with a characteristic impedance of Z0 ≃ 50 Ω. The reflected output field is measured in a vector network analyzer (VNA). The position of the superconducting quantum interference device (SQUID) loop of the transmon is shown in the scanning electron micrograph on the right. The yellow dashed box shows the cryogenic environment of the dilution refrigerator. Further details on the experimental setup are given in Supplementary Information Note 1.
