  Electromagnetically-induced transparency (EIT) with Rydberg atoms enables
strong interactions between atoms to be controlled and measured by pulses of
light. Consequently, Rydberg dark polaritons are a promising platform for
quantum simulations. Of particular interest are simulations of one-dimensional
systems, known to exhibit unique properties compared to their
higher-dimensional counterparts. One limitation of standard EIT is that
reducing the polariton group velocity to bring the dark polaritons to rest also
implies activating strong Rydberg interactions. To achieve independent control
of polariton speed and interaction strength, we propose a stationary
light-scheme to bring Rydberg polaritons to a stand-still. This allows them to
remain primarily photonic and, therefore, non-interacting, before Rydberg
interactions are turned on. Once activated, in the dilute regime and in the
presence of strong transverse confinement, the strong polarizabilities of
Rydberg atoms give rise to an effective contact interaction between polaritons.
By tuning the various parameters of the stationary-light scheme, the effective
one-dimensional scattering length may be adjusted to yield various kinds of
physics, including the regime of a Tonks-Girardeau gas, when polaritons exhibit
fermionic behavior. We outline a protocol to observe the resulting spatial
correlations between neighboring polaritons.
