Programmable Interactions and Emergent Geometry in a 1D Array of Atomic Ensembles

Tunable interactions are essential for building flexible platforms for quantum computation and simulation. We couple a 1D array of atomic ensembles to an optically driven cavity, generating an XY Hamiltonian. Precise control of magnetic field gradients and drive field modulation allow us to engineer spin-spin couplings with arbitrary dependence of the amplitude and phase on distance. This enables us to implement the XY Hamiltonian on a variety of different geometries, including 2D surfaces and a Moebius ladder. State-sensitive in-situ imaging of the individual ensembles allows us to directly reconstruct the effective Hamiltonian's dispersion relation, as well as the programmed geometry. These highly programmable interactions anticipate further study of frustrated systems, fast scrambling, and designer Hamiltonians for quantum-enhanced precision measurement.