Fig. 3: Standard transmon model fails to describe the measured frequency spectra.

a, Differences between the frequencies \({f}_{0j}^{{{\,{\rm{model}}}}}\) predicted by the standard transmon model in equation (3) and the measured transitions \({f}_{0j}^{{{\,{\rm{experiment}}}}}\). The markers indicate the different experiments at KIT (red stars), ENS (yellow squares), Köln (blue diamonds) and IBM (green circles). For the KIT experiment, we show results for three successive CDs of the same sample (CD1–3, dark red to bright red, respectively). For the Köln experiment, we chose a set of measured transitions at a fixed magnetic field (blue arrow in Fig. 4a). For the IBM experiment, we show results for 20 qubits in the IBM Hanoi device, using different marker sizes and shades of green. Measurement imprecisions are on the order of 1 MHz and not visible in the figure. Note that the scale on the vertical axis is linear between ±100 MHz and logarithmic onward. Dashed and dotted lines are guides to the eye. b, Same as a, with \({f}_{0j}^{{{\,{\rm{model}}}}}\) given by the Josephson harmonics Hamiltonian in equation (4). Top, model truncated at E_J4. Bottom, mesoscopic model of tunnelling through an inhomogeneous AlO_x barrier, where \({E}_{{{{\rm{J}}}}m}(\bar{d},\sigma )\) is parameterized in terms of the average barrier thickness \(\bar{d}\) and the standard deviation σ (equation (5); the fit values are listed in the table inset). c, Ranges of the Josephson harmonics ratios ∣E_Jm/E_J1∣ that are consistent with the measured spectra. The ranges are represented by coloured vertical bars using the same colouring as in a. For the IBM Hanoi device, we show the ranges for qubits 0–2 from left to right (ranges for the other qubits are shown in Supplementary Section IC3). The shaded grey area highlights the region between two limiting cases: the fully open quantum point contact with unit transparency and a homogeneous barrier with T_n = 10⁻⁶ for all n. Turquoise markers on the vertical bars indicate the harmonics ratios calculated from the mesoscopic model, where the average thickness \(\bar{d}\) and the standard deviation σ are given in b. Inset, an Al–AlO_x–Al junction obtained from molecular dynamics simulations (Supplementary Fig. 25) with average barrier thickness \(\bar{d}=1.5\,{{{\rm{nm}}}}\) (Fig. 2c).