Trapped ions are considered one of the best candidates to perform quantum information processing. By interacting them with laser beams they are, somehow, easy to manipulate, which makes them an excellent choice for the production of nonclassical states of their vibrational motion, the reconstruction of quasiprobability distribution functions, the production of quantum gates, etc. However, most of these effects have been produced in the so-called low intensity regime, this is, when the Rabi frequency (laser intensity) is much smaller than the trap frequency. Because of the possibility to produce faster quantum gates in other regimes it is of importance to study this system in a more complete manner, which is the motivation for this contribution. We start by studying the way ions are trapped in Paul traps in and review the basic mechanisms of trapping. Then we show how the problem may be completely solved for trapping states; i.e., we find (exact) eigenstates of the full Hamiltonian. We show how in the low intensity regime Jaynes-Cummings and anti-Jaynes-Cummings interactions may be obtained, without using the rotating wave approximation and analyze the medium and high intensity regimes were dispersive Hamiltonians are produced. The traditional approach (low intensity regime) is also studied and used for the generation of non-classical states of the vibrational of the vibrational wavefunction. In particular, we show how to add and subtract vibrational quanta to an initial state, how to produce specific superpositions of number states and how to generate NOON states for the two-dimensional vibration of the ion. It is also shown how squeezing may be measured. The time dependent problem is studied by using Lewis-Ermakov methods, we give a solution to the problem when the time dependence of the trap is considered and also analyze an specific (artificial) time dependence that produces squeezing of the initial vibrational wave function. A way to mimic the ion-laser interaction via classical optics is also introduced.

We show that in the trapped ion-laser interaction all the regimes may be considered analytically. We may solve not only for different laser intensities, but also away from resonance and from the Lamb-Dicke regime. It is found a dispersive Hamiltonian for the high intensity regime, that, being diagonal, its evolution operator may be easily calculated

We present a phase formalism that passes the Barnett-Pegg acid test, i.e. phase fluctuations for a number state are the expected value π 2 /3 which are the fluctuations for a classical random phase distribution. The formalism is shown to have consistency subjected to different approaches.

We show set of transformations which allow to obtain analytic solutions in several quantum-optical problems. We start with the ion-laser (time dependent) interaction, continue with the problem of a slow atom interacting with a quantized field to end with a master equation that describes losses. In all cases it is shown that one may find useful transformations that simplify the problems.