Coherent control and geometrical phases for trapped ions

In this work, a method to implement the pushing gate and the fast gates based on laser pulses is shown. An alternative and more general model of fast gate based on continuous off-resonant lasers is proposed. As further applications, a way to induce collective and nearest neighbor lasing interactions, and use them to produce squeezing, and generate cluster and GHz states of up to 30 atoms of the order of a period of the trap is demonstrated.