  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 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 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 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 time
dependence that produces squeezing of the initial vibrational wave function.
