Many-body systems of atomic ions confined by electro-magnetic traps crystallize at very low  densities, due to the long-range nature of their repulsive interaction. Such ion Coulomb crystals are experimentally feasible and constitute the natural playground for Wigner crystals in-depth analysis.  Strongly-correlated regimes and the appearance of long-range ordering are achievable by laser-cooling the ion system. Typical lattice spacings span around the order of magnitude of microns,  and yet they retain a sensitive amount of quantum character.  Such mesoscopic distances enhance the capability of addressing single ions, thus leading to improved probing and measurement, as well as helping control by external sources.

In fact, by coupling the ion crystal to optical cavities and laser pulses, it is possible to tune the microscopic motion of the ions, and even couple their spatial with internal degrees of freedom. Many quantum technological applications have been proposed and developed for ion crystals,  but their full physical characterization is still far from complete.

When the trapping confinement is sufficiently anisotropic, the ions organize in a purely one-dimensional structure, i.e. a linear configuration. As the transverse potential is released, the equilibrium state is instead a zigzag-shaped chain:

 

 

 

By  means  of  numerical  simulations  ( employing  density  matrix renormalization group technique ) we studied the  quantum picture for  this  second order  phase transition.  We were able to draw the phase diagram as well as characterize the universality class of the criticality beyond doubt. Moreover,  we  developed a numerical suite capable of performing real-time out-of-equilibrium coherent dynamics of the ion Coulomb chain. This computational tool  will be instrumental for the optimal preparation of an arbitrary quantum state,  and  will  have  a  wide application in experimental contexts.