Fig. 1: Setup.

a Schematic of the device. A flux qubit embedded in a λ/2 coplanar waveguide resonator, working as a nonlinear coupler between two modes of the resonator. The dashed cyan lines represent the vacuum current distribution of the n = 1 (λ/2) and n = 2 (λ) modes of the resonator. κ_out = 3κ_in, where κ_in(out) is the loss rate from the input (output) port of the resonator. b Optical image of the flux qubit. c Effective coupling mechanism between the bare states \(\vert 2,0,g\rangle\) and \(\vert 0,1,g\rangle\) via virtual transitions involving intermediate states of the circuit-QED system. Here, the first two entries in the kets denote the number of photons in the first two modes of the resonator, and the third indicates the qubit state (\(\vert g\rangle\) is the ground state). d Scheme of the energy levels at the flux offset corresponding to the minimum gap of the \(\vert 2,0,g\rangle -\vert 0,1,g\rangle\) anticrossing, resulting in an effective coupling between these two states. A key feature of this configuration is that, at the minimum anticrossing gap, all the transitions shown by the arrows have comparable large efficiency, with transition matrix elements ∣X_1,0∣ ≃ ∣X_+,1∣ ≃ ∣X_+,0∣ ~ 1 (see Supplementary Fig. 2). Moreover, the transition energy \({\tilde{\omega }}_{3}\) is almost equal to \(2{\tilde{\omega }}_{1}\), so that the second harmonic generation (with only two initial photons) and degenerate down conversion (with only one initial photon) can occur efficiently at sub-photon input levels.