  Topological quantum optical states in one-dimensional (1D) quasiperiodic cold
atomic chains are studied in this work. We propose that by introducing
incommensurate modulations on the interatomic distances of 1D periodic atomic
chains, the off-diagonal Aubry-Andr\'e-Harper (AAH) model can be mimicked,
although the crucial difference is the existence of long-range dipole-dipole
interactions. The discrete band structures with respect to the modulation
phase, which plays the role of a dimension extension parameter, are calculated
for finite chains beyond the nearest-neighbor approximation. It is found that
the present system indeed supports nontrivial topological states localized over
the boundaries. Despite the presence of long-range dipole-dipole interactions
that leads to an asymmetric band structure, it is demonstrated that this system
inherits the topological properties of two-dimensional integer quantum Hall
systems. The spectral position, for both real and imaginary frequencies, and
number of these topologically protected edge states are still governed by the
gap-labeling theorem and characterized by the topological invariant, namely,
the (first) Chern number, indicating the validity of bulk-boundary
correspondence. Due to the fractal spectrum arising from the quasiperiodicity
in a substantially wide range of system parameters, our system provides a large
number of topological gaps and optical states readily for practical use. It is
also revealed that a substantial proportion of the topological edge states are
highly subradiant with extremely low decay rates, which therefore offer an
appealing route for controlling the emission of external quantum emitters and
achieving high-fidelity quantum state storage.
