Single-photon scattering in a dissipative superconducting-qubit--SSH lattice hybrid

We study single-photon scattering in a Su--Schrieffer--Heeger (SSH) photonic lattice locally coupled to a superconducting qubit with tunable loss or gain. Working in the single-excitation sector, we derive an explicit real-space scattering formulation for the full energy-dependent scattering matrix S(E) and identify how its eigenvalues encode coherent perfect absorption, amplification, and spectral singular behavior. The analytical results are benchmarked against time-domain wave-packet simulations, which reproduce the stationary scattering probabilities with high accuracy. We show that the SSH dimerization, the qubit-induced non-Hermitian self-energy, and the synthetic gauge phase cooperate to reshape the reflection and transmission spectra in a highly selective way. In particular, changing the dimerization can switch the system between transmission-dominated and reflection-dominated regimes, while the flux provides a direct handle on interference and symmetry-controlled response. We also find a robust loss--gain correspondence in the reflection landscape and show that the linewidth broadening is governed predominantly by the magnitude |γ| of the non-Hermitian coupling. These results establish a compact and experimentally relevant framework for topological scattering in superconducting quantum networks.