Table 1 Extracted and derived transmon parameters

E_C/h	E_J/h	E_J/E_C	ω₁₀/2π	ω₂₁/2π	ω₃₂/2π	ω₄₃/2π	Γ₁₀/2π	\({\Gamma }_{1}^{\phi }/2\pi\)	γ₁₀/2π
[MHz]	[GHz]	–	[GHz]	[GHz]	[GHz]	[GHz]	[MHz]	[MHz]	[MHz]
228	13.67	59.96	4.766	4.538	4.287	4.005	2.264	0.0317	1.164

We extract the transition frequency ω₁₀, the relaxation rate Γ₁₀, and the decoherence rate γ₁₀ by fitting the magnitude and phase data from single-tone scattering (see Supplementary Information Fig. 2b)³⁴. We calculate the pure dephasing rate \({\Gamma }_{1}^{\phi }\) from Γ₁₀ and γ₁₀, using γ₁₀ = \({\Gamma }_{10}/2+{\Gamma }_{1}^{\phi }\). From the four-tone spectroscopy (see Supplementary Information Fig. 2d), we extract ω₂₁, ω₃₂, and ω₄₃, and the anharmonicity between the \(\left\vert 0\right\rangle \leftrightarrow \left\vert 1\right\rangle\) transition and the \(\left\vert 1\right\rangle \leftrightarrow \left\vert 2\right\rangle\) transition. The anharmonicity approximately equals the charging energy²⁴E_C. We calculate the Josephson energy E_J and E_J/E_C from ω₁₀ and E_C, where \(\hslash {\omega }_{10}\simeq \sqrt{8{E}_{{\rm {J}}}{E}_{{\rm {C}}}}-{E}_{{\rm {C}}}\).

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