Ising interaction with arbitrarily tunable distance in a trapped ion ring*

We propose an experimental platform for quantum simulation using a ring ion trap. This trap generates a ring-shaped potential capable of confining a circular array of equally-spaced ons and thus provides a unique geometry for quantum simulation. In this system, programmable Ising interactions at arbitrary distances can be achieved through frequency modulation of a vertically oriented global laser beam, with couplings mediated by collective vertical motional modes. We discuss how a class of spin-spin interactions preserving periodic symmetry can be realized. The fully periodic boundary conditions inherent to the ring geometry allow for interaction patterns that are inaccessible in linear ion chains, providing a high tunability for studying exotic many-body physics. Moreover, incorporating light shifts or ion-state shelving allows for controllable breaking of translational symmetry and thus enables simulation of more complex interaction topologies. This versatility makes ring traps a powerful tool for exploring a wide range of quantum phenomena, including the Haldane–Shastry model, topological phase transitions, and many-body quantum-enhanced sensing.