Single-photon coherent transparency in a coupled atoms-waveguide system

We investigate the single-photon propagation in the one-dimensional waveguide coupled to $N$ two-level atoms. For a waveguide coupled to $N$ distant atoms, the transparency can be induced by coherent interaction at resonance for an $even$ $number$ of $N$, while for an $odd$ $number$ of $N$, the photon can be fully reflected. We then can switch the photon transport by controlling the atomic number. For a waveguide coupled to a small atomic ensemble of $N$ atoms in the same position, the transparency disappears, and the photonic transport property displays a jump behavior near the resonance point.