"Singly-charged ions in traps can form strings at low temperatures, where ordering is due to the interplay of the confining potential and the Coulomb repulsion. At zero temperature the ion string exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or the interparticle distance . The transition is driven by transverse, short wavelength vibrational modes . This is a quantum phase transition, which can be experimentally realized and probed. Indeed, by means of a mapping to the Ising model in a transverse field, the quantum critical point can be determined in terms of the system parameters, finding a finite, measurable deviation from the critical point predicted by the classical theory. A measurement procedure is suggested which can probe the effects of quantum fluctuations at criticality . We then consider the stability and dynamics of an ion chain confined inside a high-finesse optical resonator. When the dipolar transition of the ions strongly couples to one cavity mode, the mechanical effects of light modify the chain properties close to a structural transition. We focus on the linear chain close to the zigzag instability and show that linear and zigzag arrays are bistable for certain strengths of the laser pumping the cavity. For these regimes the chain is cooled into one of the configurations by cavity-enhanced photon scattering. The excitations of these structures mix photonic and vibrational fluctuations, which can be entangled at steady state. These features are signalled by Fano-like resonances in the spectrum of light at the cavity output ."  "Multiple-shell structures of laser-cooled 24Mg+ ions in a quadrupole storage ring" G. Birkl, S. Kassner, and H. Walther, Nature (London) 357, 310 (1992).  "Structural phase transitions in low-dimensional ion crystals", Sh. Fishman, G. De Chiara, T. Calarco, and G. Morigi, Phys. Rev. B 77, 064111 (2008).  "Quantum zigzag transition in ion chains", E. Shimshoni, G. Morigi, S. Fishman, Phys. Rev. Lett 106, 010401 (2011).  "Structural Transitions of Ion Strings in Quantum Potentials", Cecilia Cormick and Giovanna Morigi, Phys. Rev. Lett. 109, 053003 (2012).