Superradiant Scattering from Nonlinear Mode-Coupling

The paper’s nonlinear scattering formula S = C + D F^{-1} D† + (1/s) D Σ_j ρ_j e^{iϕ_j} ⟨j| (their Eq. (9)) is consistent and leads, for the 2-ports/1-mode construction with C and D derived from the ideal target S*, to the closed-form Eq. (18). The correct coupling is D = √(γ h) (-g, 1)^T with h(σ, ε) = (σ + √(ε^2+1))/(g(ε)^2+1); using this yields the background term with numerators γ h (not γ h^2) and the amplitude-dependent term scaling as √(γ h)/s, exactly as reported in Eq. (18) when γ = 2ν and F = γ − iΔ, i.e., 2ν − iΔ . The candidate solution instead uses D = √γ h (missing the square root on h), producing an S whose background entries carry h^2 instead of h and whose nonlinear term scales as √γ h/s instead of √(γ h)/s; this mismatch is not a mere phasor convention but a coupling normalization error that also violates the consistency condition D†D = 2(γ − γ_i) and the inferred relation γ_i/γ = 1 − (σ + √(ε^2+1))/2 (their Eq. (17)) . The sign choice in F (±iΔ) is indeed a convention, but the h vs h^2 discrepancy is substantive. The superradiance condition α_j = 1 − |S_1j|^2 − |S_2j|^2 < 0 is correctly identified by the model and matches the paper’s definition .