Fig. 4: Influence of Josephson harmonics on the charge dispersion.

a, Measured charge dispersion δf_0j (blue diamonds) of states j = 1, 2, 3 for the experiment in Köln, plotted as a function of the f₀₁ frequency. All transition frequencies are tuned, as the Josephson energy is suppressed by up to 35% by means of an in-plane magnetic field B_∥ swept to 0.4 T. The standard model in equation (3), shown in dashed grey lines, underestimates the charge dispersion by a factor of 2–7 (grey arrows), while the Josephson harmonics model in equation (4) plotted in solid blue overlaps the measured data. Note that both are computed with the same parameters used for Fig. 3; the Josephson energy is reduced with increasing magnetic field, and the other parameters such as the E_Jm/E_J1 ratios are kept constant. The blue arrow indicates f₀₁ = 5.079 GHz, corresponding to the dataset shown in Fig. 3. b, Evidence that Josephson harmonics can reduce the charge dispersion by an order of magnitude (grey arrows). The dashed grey lines represent the standard model predictions. In contrast, the green bars show results from all Josephson harmonics models. The data correspond to IBM qubits 0–2 (green bars in Fig. 3c) for the levels j = 1, 2, 3, 4; results for all other samples are shown in Supplementary Fig. 6. c, The values of E_J1/E_C change compared to the standard model E_J/E_C, which constitutes the main correction to the predicted charge dispersions in a and b. The bars represent the range of suitable ratios E_J1/E_C (Fig. 3c) for the successive CDs of the KIT sample (red bars), the ENS sample (yellow bar), the Köln sample (blue diamonds, using the same colour coding as in s) and the IBM Hanoi device (green bars). The dashed diagonal line indicates the case in which the ratios E_J1/E_C of the harmonics model and E_J/E_C of the standard model are equal. Inset, correction (ε^har − ε^std)/ε^std to the relative charge dispersion ε = δf_0j/f₀₁ for fixed \({E}_{{{{\rm{J}}}}}^{{{{\rm{std}}}}}/{E}_{{{{\rm{C}}}}}^{{{{\rm{std}}}}}={E}_{{{{\rm{J}}}}1}^{{{{\rm{har}}}}}/{E}_{{{{\rm{C}}}}}^{{{{\rm{har}}}}}\) for the Köln sample, where ε^std is given by the standard charge dispersion³⁴ and ε^har is computed using the Josephson harmonics model.