Fig. 4: Phase space picture of apparent damping.

Here phase space operators are defined by \(\hat{a}=\left(\hat{x}+i\hat{p}\right)/\sqrt{2}\), see Supplementary Note 5A for further details. a Wigner distribution of the steady-state in absence of Kerr nonlinearity (driven at ω_r with P_in = − 124 dBm). The balance between quantum noise, damping and drive is shown by vectors corresponding to Wigner currents. b Growth of phase uncertainty of a coherent state under Kerr nonlinearity. The amplitude-dependent resonance frequency (Kerr effect) translates to a radius-dependent rotation around the origin. The center of mass of the distribution rotates at a frequency Kα². In a frame rotating at that frequency, the effect of the Kerr nonlinearity is to increase the uncertainty in phase (this is the frame adopted in (c)). The larger the uncertainty in phase (the extreme case being a ring around the origin), the closer the center of mass of the distribution gets to the origin (i.e., \(| \langle \hat{a}\rangle | \to 0\)). This is the first contribution (effect A) to a reduced resonant amplitude’s magnitude \(| \langle \hat{a}\rangle |\). c Wigner distribution of the steady-state with Kerr nonlinearity (at minimum ∣S₂₁∣ with P_in = − 124 dBm). The Kerr effect is eventually balanced by the damping, quantum noise and drive. Since the drive now opposes both damping and Kerr effect, it is less effective at opposing the damping and driving the state away from the origin (compared to (a)). This brings the distribution closer to the origin, and constitutes the second contribution (effect B) to a reduced resonant amplitude’s magnitude \(| \langle \hat{a}\rangle |\). The center of mass (\(\langle \hat{a}\rangle\)) (white dot) is compared to the classical steady-state (white cross). Since the Wigner current of the Kerr effect grows with the amplitude squared \(| \langle \hat{a}\rangle {| }^{2}\propto {\epsilon }^{2}\) and the drive and dissipation currents grow with ϵ and \(| \langle \hat{a}\rangle |\) respectively, the reduction in \(| \langle \hat{a}\rangle |\) does not linearly follow the driving strength ϵ (see Supplementary Note 5B).