  We explore special features of quantum light-matter interactions inside
structured waveguides due to their finite bandwidth, band edges, and
non-trivial topological properties. We model the waveguides as either a
tight-binding (TB) chain or a Su-Schrieffer-Heeger (SSH) chain. For
unstructured waveguides with infinite bandwidth, the transmission and
reflection amplitude of a side-coupled two-level emitter (2LE) are the same as
the reflection and transmission amplitude of a direct-coupled 2LE. We show that
this analogy breaks down for structured waveguides with finite bandwidth due to
the appearance of Lamb shift only for the direct-coupled 2LE. We further
predict a robust light-emitter coupling at zero collective decay width of a
single giant 2LE (with two couplings at different points) near the band edges
of the structured waveguides where topological features can be beneficial.
Finally, we study single-photon dynamics in a heterojunction of a long TB and
short SSH waveguide connected to a 2LE at the SSH end. We show the propagation
of a photon from the excited emitter to the TB waveguide only when the SSH
waveguide is in the topological phase. Thus, the heterojunction acts as a
quantum switch or conditional propagation channel.
