Synthetic gauge fields for vibrational excitations of trapped ions

The vibrations of a collection of ions in a microtrap array can be described in terms of tunneling phonons. We show that the vibrational couplings may be tailored by using a gradient of the trap frequencies together with a periodic driving of the trapping potentials. These ingredients allow us to induce effective gauge fields on the vibrational excitations, such that phonons mimic the behavior of charged particles in a magnetic field. In particular, microtrap arrays are well suited to realize a quantum simulator of the famous Aharonov-Bohm effect and observe the paradigmatic edge states typical from quantum-Hall samples and topological insulators.