Programmable interactions and emergent geometry in an array of atom clouds.

Interactions govern the flow of information and the formation of correlations between constituents of many-body quantum systems, dictating phases of matter found in nature and forms of entanglement generated in the laboratory. Typical interactions decay with distance and thus produce a network of connectivity governed by geometry-such as the crystalline structure of a material or the trapping sites of atoms in a quantum simulator1,2. However, many envisioned applications in quantum simulation and computation require more complex coupling graphs including non-local interactions, which feature in models of information scrambling in black holes3-6 and mappings of hard optimization problems onto frustrated classical magnets7-11. Here we describe the realization of programmable non-local interactions in an array of atomic ensembles within an optical cavity, in which photons carry information between atomic spins12-19. By programming the distance dependence of the interactions, we access effective geometries for which the dimensionality, topology and metric are entirely distinct from the physical geometry of the array. As examples, we engineer an antiferromagnetic triangular ladder, a Möbius strip with sign-changing interactions and a treelike geometry inspired by concepts of quantum gravity5,20-22. The tree graph constitutes a toy model of holographic duality21,22, in which the quantum system lies on the boundary of a higher-dimensional geometry that emerges from measured correlations23. Our work provides broader prospects for simulating frustrated magnets and topological phases24, investigating quantum optimization paradigms10,11,25,26 and engineering entangled resource states for sensing and computation27,28.