Extended Data Fig. 3: Full Φ-dependence of spectrum and dispersive shifts.

a, Φ-dependence of Q over a full half flux quantum. The \(\left|\downarrow ,1\right\rangle\) and \(\left|\uparrow ,1\right\rangle\) distributions (traced with purple and pink splines respectively) remain below the bare resonator Q (black dotted line) over the full Φ range, indicating negative dispersive shifts which are inconsistent with χ resulting from coupling to the inverse inductance operator. The dispersive shift of \(\left|g\right\rangle\) (traced with the gray spline) is likely due to a pair transition with frequency above our measurement bandwidth. We also observe a small number of counts around Φ = 0 at positive Q, indicating a residual quasiparticle population in \(\left|\uparrow ,2\right\rangle\) and \(\left|\downarrow ,2\right\rangle\). Assuming the observed dispersive shift of \(\left|g\right\rangle\) is due only to the properties of the lower doublet, the dispersive shift of a quasiparticle in the upper doublet should be given by χ_s,2 = − χ_s,1 + χ₀. Based on this formula and the plotted splines, we estimated the Φ-dependence of the \(\left|\downarrow ,2\right\rangle\) and \(\left|\uparrow ,2\right\rangle\) distributions (dashed, teal, and yellow). The predictions track roughly with the residual counts in the vicinity of Φ = 0 before crossing the bare resonator Q. (b) Spectroscopy over the same flux range. The inter-doublet transitions have maximum frequency at Φ = − 0.5Φ₀, consistent with Fig. 1(d). We attribute the sign change in the measured \(\Delta \bar{Q}\) to the crossings of χ_s,1 with \({\it\chi}_{s_2}\) indicated in (a). (c) Attempted modeling of the transitions using the double-barrier model of Ref. ¹³. Here we extract a chemical potential (as measured from the bottom of a sub-band) of 0.65 meV and a Rashba coefficient of 43 meV*nm. The effective transparencies of the two barriers are t₁ = 0.32 and t₂ = 0.46.