Controllable single-photon scattering via coupling of a driven <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math> system with a topological waveguide

We investigate the coherent single-photon scattering process in a topological waveguide coupled to a driven three-level atom in the $\mathrm{\ensuremath{\Lambda}}$ configuration, where two ground states are optically connected to a common excited state. We derive an analytical expression for transmittance by using the scattering formalism for three distinct sublattice sites (A, B, and AB) of the waveguide, which couple to the $\mathrm{\ensuremath{\Lambda}}$ system. We have demonstrated that the system's response is topology independent for A and B sublattice-site coupling and becomes topology dependent for AB sublattice-site coupling. In a weak-control-field regime, the system behaves as a perfect mirror in all of these configurations. Upon control field strength enhancement, the transmission spectrum evolves from Electromagnetically Induced Transparency to Autler-Townes splitting in A and B sublattice-site coupling. The manipulation of transmission from opaque to transparent holds the key mechanism of a single-photon switching. Furthermore, the topology-dependent AB sublattice configuration allows multiple-path-induced quantum interference, resulting in the sharper Fano line shape. This topological Fano resonance is not only tunable via the control field and coupling strength but also robust against structural imperfections due to the topological nature of the waveguide. These combined features such as tunability and topological protection make our system a promising platform for next-generation quantum technologies, including single-photon switches, high-resolution sensors, and communication devices.