  We theoretically investigate the transport properties of a weak coherent
input field scattered by an ensemble of $\Lambda$-type atoms coupled to a
one-dimensional photonic crystal waveguide. In our model, the atoms are
randomly located in the lattice along the crystal axis. We analyze the
transmission spectrum mediated by the tunable long-range atomic interactions,
and observe the highest-energy dip. The results show that the highest-energy
dip location is associated with the number of the atoms, which provides an
accurate measuring tool for the emitter-waveguide system. We also quantify the
influence of a Gaussian inhomogeneous broadening and the dephasing on the
transmission spectrum, concluding that the highest-energy dip is immune to both
the inhomogeneous broadening and the dephasing. Furthermore, we study
photon-photon correlations of the reflected field and observe quantum beats.
With tremendous progress in coupling atoms to photonic crystal waveguides, our
results may be experimentally realizable in the near future.
