Fig. 5: Controllable Floquet states with the N = 10 device in the Kerr regime.

Through the figure, we set the drive input power to P_in = −128 dBm. a–d Both modulation strength ζ and frequency Ω are swept together to maintain a constant ratio ζ/Ω. This choice ensures that the effective drives in Eqs. (3) and (7) are kept constant. This allows enhancing the visibility of the transition between the strongly- and weakly-modulated cases. a Sketch of the results of Eq. (7) for the transition \(\left\vert 0\right\rangle \to \left\vert 1\right\rangle\) (purple, labeled \(\left\vert 1\right\rangle\)) and \(\left\vert 0\right\rangle \to \left\vert 2\right\rangle\) (green, labeled \(\left\vert 2\right\rangle\)). For \(\left\vert 1\right\rangle\), the pattern radiates from Δ = 0 with frequency modulation Ω. For the multiphoton transition to \(\left\vert 2\right\rangle\), the LZSM interference pattern is centered at Δ = χ and scales with Ω/2. b Simulation of ∣S₂₁∣ as a function of Δ and Ω with the full quantum model in Eq. (2) (see Supplementary Information for details on the simulation method), having fixed ζ/Ω = 0.86. The corresponding measurement is shown in (c) and perfectly overlaps with the results of the numerical simulation. The black arrows in (a–d) mark the position of two avoided crossings, where the “bare levels” in (a) interact and hybridize in (b–d). The crossings are further highlighted by the solid (associated with \(\left\vert 1\right\rangle\)) and dashed (\(\left\vert 2\right\rangle\)) lines in (b). The amplitude of the different avoided crossings can be controlled by modulating the Bessel functions \({J}_{\bar{m}}(n\zeta /\Omega )\) as shown in (d), where a larger ratio ζ/Ω is chosen. e As in panel (a), the sketch of the results of Eq. (7) for Ω = ∣χ∣ and as a function of Δ and ζ. In this “bare picture”, the two independent LZSM interference patterns scale with Ω and Ω/2 for \(\left\vert 1\right\rangle\) and \(\left\vert 2\right\rangle\), respectively. f Full quantum simulation and g corresponding measurement of ∣S₂₁∣ for Ω = ∣χ∣. The position of some LZSM resonances in the bare picture is superimposed in (f) as a guideline for the eye. h Repeating the measurement for Ω = 1.5∣χ∣, we observe line splittings, indicating a modulation of the coupling between different Floquet states.