Fig. 7: Multimodal-like behavior in the Duffing regime.

a–c Measurement of ∣S₂₁∣ for increasing drive power, fixed frequency modulation Ω/2π = 30 MHz and increasing ratios ζ/Ω ≈ 1.4 (a), ζ/Ω ≈ 2.6 (b), and ζ/Ω ≈ 3.8 (c). For low input powers the system behaves as a collection of noninteracting nonlinear modes, each one well separated from the others. For larger values of P_in, the system enters a phase characterized by a single broad response where the notion of isolated mode is lost. Such a response can be observed in multimode nonlinear systems and has been associated with a transition from integrability to dissipative quantum chaos⁴⁵. To show that this is indeed a dissipative quantum chaotic phase we plot in (d) the histogram of the probability density p(s) of the level spacings s obtained by diagonalizing the Floquet Liouvillian in the broad-response region indicated by the star in (a). Parameters are set to ζ/2π = 41.3 MHz, Ω/2π = 30 MHz, Δ = −1.1Ω and F/2π = 49.5 MHz (P_in ≈ −105 dBm). The cutoff in the Hilbert space is set to 90. The solid black (orange) curve represents the ideal Poisson (Ginibre) distribution given by Eqs.(15) and (16) associated with integrability (chaos).