Fig. 2: Nonequilibrium steady-state phase diagram.

Steady-state properties of the last resonator in the chain, ℓ = L, as functions of the chain length L and drive strength F. a Photon number n_L. b Photon-number fluctuation δn_L, defined in Eq. (5). c Saturation value of the steady-state phase OTOC, D_1,L(τ → ∞), defined in Eq. (6). The three distinct regimes labeled I, II, and III are discussed in throughout the “Results" section. d–f Cuts of a–c, respectively, at fixed chain length L = 10. Results are computed by averaging over N_traj = 10³ independent Wigner trajectories. Statistics are further improved by averaging over a time window Δτ after reaching the NESS: Δτ = 25 for D_1,L(τ → ∞), and Δτ = 2 × 10³ for n_L and δn_L. Throughout the manuscript, the dissipation rate γ sets the unit of energy, and the other parameters are fixed to Δ = 2.5, J = 2, and U = 0.1.