  Nonlinear optical phenomena are typically local. Here we predict the
possibility of highly nonlocal optical nonlinearities for light propagating in
atomic media trapped near a nano-waveguide, where long-range interactions
between the atoms can be tailored. When the atoms are in an
electromagnetically-induced transparency configuration, the atomic interactions
are translated to long-range interactions between photons and thus to highly
nonlocal optical nonlinearities. We derive and analyze the governing nonlinear
propagation equation, finding a roton-like excitation spectrum for light and
the emergence of order in its output intensity. These predictions open the door
to studies of unexplored wave dynamics and many-body physics with
highly-nonlocal interactions of optical fields in one dimension.
