  Tightly confined modes of light, as in optical nanofibers or photonic crystal
waveguides, can lead to large optical coupling in atomic systems, which
mediates long-range interactions between atoms. These one-dimensional systems
can naturally possess couplings that are asymmetric between modes propagating
in different directions. Strong long-range interaction among atoms via these
modes can drive them to a self-organized periodic distribution. In this paper,
we examine the self-organizing behavior of atoms in one dimension coupled to a
chiral reservoir. We determine the solution to the equations of motion in
different parameter regimes, relative to both the detuning of the pump laser
that initializes the atomic dipole-dipole interactions and the degree of
reservoir chirality. In addition, we calculate possible experimental signatures
such as reflectivity from self-organized atoms and motional sidebands.
