The fields of quantum simulation with cold atoms and quantum optics
are currently being merged. In a set of recent pathbreaking experiments
with atoms in optical cavities , 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 \inftyN→∞,
(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
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 appears in combination 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.