Fig. 2: Two-stage relaxation towards the single unique steady state.

a The reflection coefficients, S₂₂, corresponding to the two metastable branches (blue and red) form a closed loop, which converge to the unique steady-state solution with τ. The inset shows the convergence of the metastable branches at each fixed ξ. b The loop area, A, decays with τ and shows two distinct decay rates. The dashed lines show the exponential fits, \(A\propto \exp (-\eta \tau )\), of the decay rate at small and large τ, with fitted values η₁ = 0.74 μs⁻¹ and η₂ = 0.04 μs⁻¹, respectively. (c) The Liouvillian gap, δ₁, is approximately equal to the total energy dissipation rate, γ, at a sufficiently small or large ξ (dashed). However, it decreases by more than two orders of magnitude when approaching the critical point, ξ^*/2π = 1.51 MHz, and achieves a minimum value of 0.02 μs⁻¹ at ξ^* (See Supplementary Fig. 9). In all panels, the resonant frequency is fixed at ω_A/2π = 7.10 GHz. The error bars represent the standard deviation over 16 independent experiments, which are smaller than the size of the dots in (b) and (c). Source data are provided as a Source Data file.