Superfluid-Mott insulator transition and Bose-Einstein Condensation of phonons in ion traps

We show that transverse phonons in a set of trapped ions under the action of lasers are described by a Bose‐Hubbard model whose parameters can be externally adjusted. The hopping of the phonons between different phonons is provided by the Coulomb interaction. On the other hand, the nonlinear terms in the interaction of the ions with a standing‐wave provide us with the phonon‐phonon interaction. We investigate the possibility of observing several quantum many‐body phenomena, including (quasi) Bose‐Einstein Condensation as well as a superfluid‐Mott insulator quantum phase transition.