We study the controllable single-photon scattering via a one-dimensional waveguide which is coupled to a two-level emitter and a single-mode cavity simultaneously. The emitter and the cavity are also coupled to each other and form a three-level system with cyclic transitions within the zero- and single-excitation subspaces. As a result, the phase of emitter-cavity coupling strength serves as a sensitive control parameter. When the emitter and cavity locate at the same point of the waveguide, we demonstrate the Rabi splitting and quasidark-state--induced perfect transmission for the incident photons. More interestingly, when they locate at different points of the waveguide, a controllable nonreciprocal transmission can be realized and the non-reciprocity is robust to the weak coupling between the system and environment. Furthermore, we demonstrate that our theoretical model is experimentally feasible with currently available technologies.
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