Topological amplification and frequency conversion in a photonic lattice with a two-photon driving

We investigate the topological amplification and frequency conversion of a one-dimensional photonic lattice subjected to a two-photon driving, considering both the thermodynamic- and dynamical-stability regimes. Our analysis shows that the critical behavior of their stability occurring at zero energy depends on the width of the lower band, as well as the strength and phase of the two-photon driving. We demonstrate that the presence of the driving field does not alter the band topology of the system under both regimes, and we showcase the nontrivial band topology. The occurrence of edge modes within the band gap is also demonstrated by employing a Bogoliubov (squeezed-state) representation. Furthermore, we show the amplification and frequency conversion of edge modes using the input-output formalism, relying on the mechanism of an optical parametric amplifier. These phenomena exhibit an absence-to-presence transition at a critical value of hopping strength, signifying a topological phase transition. Additionally, we demonstrate the robustness of these phenomena against symmetry-preserving disorders. Our findings have potential applications in engineering fault-tolerant quantum-optics devices based on the topological protection.