  Single-photon super- and subradiance are important for the quantum memory and
quantum information. We investigate one-dimensional atomic arrays under the
spatially periodic magnetic field with a tunable phase, which provides a
distinctive physics aspect of revealing exotic two-dimensional topological
phenomena with a synthetic dimension. A butterfly-like nontrivial bandstructure
associated with the non-Hermitian physics involving strong long-range
interactions has been discovered. It leads to pairs of topologically-protected
edge states, which exhibit the robust super- or subradiance behavior, localized
at the boundaries of the atomic arrays. This work opens an avenue of exploring
an interacting quantum optical platform with synthetic dimensions pointing to
potential implications for quantum sensing as well as the super-resolution
imaging.
