Generating Graph States in an Array of Atomic Ensembles

Graph states constitute a broad class of resource states with applications in quantum computation, simulation and metrology. Their generation demonstrates versatile control over the structure of entanglement in a quantum system. We generate graph states of atomic ensembles by combining two ingredients: local spin rotations and cavity-mediated all-to-all interactions. Using this general protocol, we alternatively localize entanglement within subsystems or enhance non-local entanglement to enable Einstein-Podolsky-Rosen steering between the two halves of our system. We further scale this protocol to generate a four mode square cluster state. We thus demonstrate programmable spatial entanglement and open the door to applications ranging from multimode quantum sensing to measurement based quantum computation.