Fig. 2: Josephson harmonics result from junction barrier inhomogeneity.

a, False-coloured scanning electron microscope image of a typical Al–AlO_x–Al JJ fabricated at KIT. The bottom and top electrodes are coloured blue and yellow, respectively. Inset, circuit symbol for a JJ with phase difference φ across the barrier. b, Cross-section schematic of the superconductor–insulator–superconductor JJ at the location indicated by the dash-dotted line in a. The supercurrent I_n(φ) of each conduction channel n = 1, …, N depends on its transparency T_n (equation (2)). We sketch a distribution of multiple low and a few high transparencies T₁, …, T_N in green and red, respectively. c, False-coloured high-angle annular dark field STEM image centred on the AlO_x tunnel barrier of a typical JJ fabricated at KIT, with average thickness d ≈ 2 nm as indicated by the white arrow. Individual columns of atoms of the Al grain in the top electrode are visible due to zone axis alignment, which is not the case for the bottom Al electrode (additional STEM images with thickness variations and structural defects such as grain boundaries are shown in Supplementary Fig. 27). d, Normalized Fourier coefficients c_m(T_n) of the JJ CφR (equation (2)) for a low (10⁻⁶, green) and high (10⁻², red) transparency channel. Note the alternating sign for even and odd order m and the fact that high-transparency channel coefficients (in red) remain relevant to higher order. e, Sketch of how the higher-order terms in the JJ Hamiltonian modulate the potential and shift the energy levels (red) of superconducting artificial atoms compared to a purely cosφ potential (grey). In this Article, we focus on transmon devices, which consist of a large capacitor in parallel to the JJ (refer to the circuit schematic inset). The discrepancy between the models generally increases at higher levels. f, The higher-order Josephson harmonics also influence the charge dispersion of the transmon levels versus offset charge n_g. The two branches per energy level correspond to a change between even and odd charge parity (that is, quasiparticle tunnelling^79,80; Supplementary Fig. 23 in Supplementary Section IIIC).