  The fields of quantum simulation with cold atoms [1] and quantum optics [2]
are currently being merged. In a set of recent pathbreaking experiments with
atoms in optical cavities [3,4] lattice quantum many-body systems with both, a
short-range interaction and a strong interaction potential of infinite range
-mediated by a quantized optical light field- were realized. A theoretical
modelling of these systems faces considerable complexity at the interface of:
(i) spontaneous symmetry-breaking and emergent phases of interacting many-body
systems with a large number of atoms $N\rightarrow\infty$, (ii) quantum optics
and the dynamics of fluctuating light fields, and (iii) non-equilibrium physics
of driven, open quantum systems. Here we propose what is possibly the simplest,
quantum-optical magnet with competing short- and long-range interactions, in
which all three elements can be analyzed comprehensively: a Rydberg-dressed
spin lattice [5] coherently coupled to a single photon mode. Solving a set of
coupled even-odd sublattice Master equations for atomic spin and photon
mean-field amplitudes, we find three key results. (R1): Superradiance and a
coherent photon field can coexist with spontaneously broken magnetic
translation symmetry. The latter is induced by the short-range nearest-neighbor
interaction from weakly admixed Rydberg levels. (R2): This broken even-odd
sublattice symmetry leaves its imprint in the light via a novel peak in the
cavity spectrum beyond the conventional polariton modes. (R3): The combined
effect of atomic spontaneous emission, drive, and interactions can lead to
phases with anomalous photon number oscillations. Extensions of our work
include nano-photonic crystals coupled to interacting atoms and multi-mode
photon dynamics in Rydberg systems.
