  Emitters strongly coupled to a photonic crystal provide a powerful platform
for realizing novel quantum light-matter interactions. Here we study the
optical properties of a three-level artificial atomic chain coupled to a
one-dimensional superconducting microwave photonic crystal. A sharp
minimum-energy dip appears in the transmission spectrum of a weak input field,
which reveals rich behavior of the long-range interactions arising from
localized bound states. We find that the dip frequency scales linearly with
both the number of the artificial atoms and the characteristic strength of the
long-range interactions when the localization length of the bound state is
sufficiently large. Motivated by this observation, we present a simple model to
calculate the dip frequency with system parameters, which agrees well with the
results from exact numerics for large localization lengths. We observe
oscillation between bunching and antibunching in photon-photon correlation
function of the output field. Furthermore, we find that the model remains valid
even though the coupling strengths between the photonic crystal and artificial
atoms are not exactly equal and the phases of external driving fields for the
artificial atoms are different. Thus, we may infer valuable system parameters
from the dip location in the transmission spectrum, which provides an important
measuring tool for the superconducting microwave photonic crystal systems in
experiment. With remarkable advances to couple artificial atoms with microwave
photonic crystals, our proposal may be experimentally realized in currently
available superconducting circuits.
