Fig. 6: Topological ‘phase diagram’ for the phase sensitive amplifier, Eqs. (34) and (35), and the bosonic Kitaev chain, Eq. (38).

Topological regimes for (a) the phase sensitive amplifier of Fig. 5c, and (b) the bosonic Kitaev chain of Fig. 5 (d). We sketch the scattering matrix ∣S(0)∣², whereby the blue rectangles indicate the dominant matrix elements. It is block diagonal for both systems (see inset), since their equations of motion decouple, with blocks S_x and S_p addressing x and p quadratures, respectively. Each block is obtained from Eq. (23). In a depending on the parameters Γ and λ, either the x quadratures are in a non-trivial regime, the p quadratures, both or none. The central white region is topologically trivial, so the gain is \({\mathcal{O}}(1)\). In b, either both x and p quadratures are in a non-trivial regime with ν_x = −ν_p, or ν_x = ν_p = 0. Since the winding numbers ν_x and ν_p have opposite sign, the amplification direction is the reverse for the two quadratures.