Fig. 6: Flux-dependent state dispersion for different gate voltages and schematic overview of the observed transitions.
a Two-tone Andreev bound state spectrum at a gate voltage of Vg = 0.251 V (blue dashed line in Fig. 4 a) as a function of the magnetic field induced phase difference φ across the junction. The red line is a fit to the pair transition driven by the pump tone using the short channel model with parameters τ = 0.96 ± 0.03, Δ = 105 ± 37 μeV. The purple lines correspond to a corresponding two photon process caused by the absorption or emission of an additional resonator photon. The dashed blue line indicates another two-particle transistion feature from another set of Andreev states. Dashed green lines highlight a set of single-quasiparticle transitions between spin polarized Andreev levels. b Single-tone measurement of cavity response at the same Vg as in (a). The resonance frequency is extracted and plotted in c alongside a fit to a Jaynes-Cummings Hamiltonian (solid black line). Fit parameters are γ = 9.7 ± 0.1 GHz, Z = 1.09 ± 0.02 × 10−6, foffset = −29 kHz, geff(π) = 78 ± 1 MHz, and κ = 0.1964 MHz. d Two-tone spectrum at a gate voltage of Vg = 0.4374 V (red dashed line in Fig. 4a). Again a prominent two particle transition is observed and fitted with a short channel model with τ = 0.998 ± 0.001, Δ = 132 ± 40 μeV (solid red line). Two photon transitions (solid purple lines) which utilize both cavity and pump photons are also observed. Additionally, a solid blue line indicates a two photon pair transition using two pump photons. The solid green line corresponds to a higher-order transition due to the combination of a pump-induced single resonator photon and the relaxation of an Andreev level. e Single-tone measurement of cavity response at the same Vg as in (d). The resonance frequency is extracted and plotted in f alongside a fit to a Jaynes–Cummings Hamiltonian (solid black line). Fit parameters are γ = 0.9 ± 0.2 GHz, Z = 4.8 ± 0.7 × 10−8, foffset = −8.4 kHz, geff(π) = 22 ± 3 MHz, and κ = 0.1964 MHz. g–k Schematics of the different transitions being discussed in a and d showing quasi-particle excitations from Andreev levels (two parallel lines) or the ground state (solid line) and the pump or probe photons that contribute to the process.
