  Quantum emitters interacting with photonic band-gap materials lead to the
appearance of qubit-photon bound states that mediate decoherence-free, tunable
emitter-emitter interactions. Recently, it has been shown that when these
band-gaps have a topological origin, like in the photonic SSH model, these
qubit-photon bound states feature chiral shapes and certain robustness to
disorder. In this work, we consider a more general situation where the emitters
interact with an extended SSH photonic model with longer range hoppings that
displays a richer phase diagram than its nearest-neighbour counterpart, e.g.,
phases with larger winding numbers. In particular, we first study the features
of the qubit-photon bound states when the emitters couple to the bulk modes in
the different phases, discern its connection with the topological invariant,
and show how to further tune their shape through the use of giant atoms, i.e.,
non-local couplings. Then, we consider the coupling of emitters to the edge
modes appearing in the different topological phases. Here, we show that giant
atoms dynamics can distinguish between all different topological phases, as
compared to the case with local couplings. Finally, we provide a possible
experimental implementation of the model based on periodic modulations of
circuit QED systems. Our work enriches the understanding of the interplay
between topological photonics and quantum optics.
