  The coupled dynamics of the atom and photon fields in optical ring cavities
with two counter-propagating modes give rise to both spin-orbit interactions as
well as long-ranged interactions between atoms of a many-body system. At zero
temperature, the interplay between the two-body and cavity-mediated
interactions determines the ground state of a Bose-Einstein condensate. In this
work, we find that cavity quantum electrodynamics in the weak-coupling regime
favors a stripe-phase state over a plane-wave phase as the strength of
cavity-mediated interactions increases. Indeed, the stripe phase is
energetically stabilized even for condensates with attractive intra- and
inter-species interactions for sufficiently large cavity interactions. The
elementary excitation spectra in both phases correspond to linear dispersion
relation at long wavelengths, indicating that both phases exhibit
superfluidity, though the plane-wave phase also displays a characteristic
roton-type feature. The results suggest that even in the weak coupling regime
cavities can yield interesting new physics in ultracold quantum gases.
