Fig. 5: Nonlinearity-enhanced dynamical backaction with dissipative coupling.

a Schematic of the experiment. A pump tone (red arrow) with frequency ω_p and power P_p is applied around the red sideband of the HF mode. Detuning of pump from the red sideband is \(\delta^{\prime}={\omega }_{\mathrm{p}}-({\omega }_{0}^{{\prime} }-{\Omega }_{0}^{{\prime} })\). For each pair (ω_p, P_p), the LF mode reflection is probed directly around \({\Omega }_{0}^{{\prime} }\) using the LF feedline and a VNA (black arrow). b Color-map of the LF reflection ∣S₁₁∣ vs. pump frequency ω_p. When the pump is far red-detuned from the HF mode red-sideband, the resonance is nearly unmodified compared to the pump-free case. Just below \(\delta^{\prime}=0\) or ω_p/2π ≈ 8.0 GHz, the absorption gets wider and shallower and slightly shifts in frequency; both effects indicate dynamical backaction at work. Strikingly, for ω_p/2π ≈ 8.05 GHz, which is still around the cavity red-sideband, a parametric instability region appears, which is experimentally identified by a strongly deformed resonance feature in S₁₁, and the simultaneous disappearance of the HF mode in the HF reflection⁵⁴. The two pairs of arrows indicate the linescans shown in c. Top curve in c shows ∣S₁₁∣ at ω_p/2π ≈ 8.0 GHz, the point of maximum photon-pressure damping, bottom curve is manually offset by  − 0.1 for clarity and is for ω_p/2π ≈ 7.84 GHz. Symbols are data, lines are fits, from which we extract Ω_eff and Γ_eff. d, e Effective LF mode linewidth Γ_eff and effective resonance frequency Ω_eff vs. detuning \(\delta^{\prime}\) for three different P_p, color code for P_p identical to Fig. 4, cf. Supplementary Fig. 15 for the corresponding n_c. Symbols are data and error bars are standard errors obtained from the fit routine, lines are theory curves. Note the two clearest signatures for dissipative coupling: first, the maximum for the photon-pressure damping is at negative detunings from the red sideband. And second, for positive detunings from the red sideband the total damping rate is smaller than the intrinsic damping rate Γ_eff < Γ₀, where Γ₀ is shown as dashed line. This indicates negative backaction damping with a red-detuned pump, and is the precursor for the approaching parametric instability. To fully model the data, we included a cross-Kerr frequency shift to the LF frequency \({\Omega }_{0}^{{\prime} }={\Omega }_{0}+{{{\mathcal{K}}}}_{\mathrm{c}}{n}_{\mathrm{c}}\) with \({{{\mathcal{K}}}}_{\mathrm{c}}=2\pi \times -130.8\,{\mathrm{Hz}}\), and a small nonlinear cross-damping \({\Gamma }_{0}^{{\prime} }={\Gamma }_{0}+{\kappa }_{\mathrm{c}}{n}_{\mathrm{c}}\) with κ_c = 2π × 38.4 Hz. Dashed lines in e show \({\Omega }_{0}^{{\prime} }\).