Fig. 1

Sample and qubit dispersions. a Schematic of sample and measurement scheme. The signals at frequencies \({\nu }_{{\rm{p}}}\) (probe) and \({\nu }_{{\rm{d}}}\) (drive) are routed with circulators as indicated by arrows. The reflected signal \(I+\,\text{i}\,Q\) at \({\nu }_{{\rm{p}}}\) is measured. The sample (dashed line) contains four quantum systems with transition frequencies \({\nu }_{i}\): a coupling resonator that consists of an array of SQUID loops (\({\nu }_{{\rm{C}}}\), blue), an RX qubit (\({\nu }_{{\rm{RX}}}\), red), a transmon (\({\nu }_{{\rm{T}}}\), green) and a read-out resonator (\({\nu }_{{\rm{R}}}\), gray). Empty black double-rectangles indicate electron tunnel barriers separating the three quantum dots (red circles) as well as the source (S) and drain (D) electron reservoirs. A drive tone at frequency \({\nu }_{{\rm{dRX}}}\) can be applied to one of the dots. Crossed squares denote the Josephson junctions of SQUIDs. Yellow arrows indicate the coupling between the quantum systems with coupling strengths \({g}_{i}\). \({\it{\Phi} }_{{\rm{C}}}\) and \({\it{\Phi} }_{{\rm{T}}}\) denote coupling resonator and transmon flux, respectively. b Two-tone spectroscopy of the transmon, with the RX qubit energetically far detuned. We plot the complex amplitude change \(| {\boldsymbol{A}}-{{\boldsymbol{A}}}_{0}|\) (see main text) as a function of drive frequency \({\nu }_{{\rm{d}}}\) and \({\it{\Phi} }_{{\rm{T}}}/{\it{\Phi} }_{0}\). The dashed line indicates \({\nu }_{{\rm{T}}}\) as obtained from the system Hamiltonian. c Scanning electron micrograph of the TQD and quantum point contact (QPC) region of the sample. Unused gate lines are grayed out. The gate line extending to the coupling resonator is highlighted in blue. d TQD energy level diagram indicating the tunnel couplings \({t}_{{\rm{l}}}\) and \({t}_{{\rm{r}}}\) and the electrochemical potentials, parametrized by \({\it{\Delta}}\), of the relevant RX qubit states (\({N}_{{\rm{l}}}\),\({N}_{{\rm{m}}}\),\({N}_{{\rm{r}}}\)) with \({N}_{{\rm{l}}}\) electrons in the left, \({N}_{{\rm{m}}}\) electrons in the middle and \({N}_{{\rm{r}}}\) electrons in the right quantum dot. e Two-tone spectroscopy of the RX qubit, with the transmon energetically far detuned for \({\nu }_{\rm{p}}\simeq {\nu }_{{\rm{C}}}=4.84\ {\rm{GHz}}\) as a function of \({\it{\Delta}}\) and \({\nu }_{{\rm{dRX}}}\). The dashed line shows the expected qubit energy obtained from the Hamiltonian of the system