The Hamiltonian describing a single ion placed in a potential trap in interaction with a laser beam is studied by means of a suitable perturbative approach. It is shown, in particular, that the rotating wave approximation does not provide the correct expression, at the first perturbative order, of the evolution operator of the system.
Some properties of the non--linear coherent states (NCS), recognized by Vogel and de Matos Filho as dark states of a trapped ion, are extended to NCS on a circle, for which the Wigner functions are presented. These states are obtained by applying a suitable displacement operator $D_{h}(α) $ to the vacuum state. The unity resolutions in terms of the projectors $| α, h> < α, h^{-1}| ,| α, h^{-1}> < α, h| $ are presented together with a measure allowing a resolution in terms of $| α, h> < α, h| $. $D_{h}(α) $ is also used for introducing the probability distribution funtion $ρ_{A,h}(z) $ while the existence of a measure is exploited for extending the P-representation to these states. The weight of the n-th Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter $η,$ oscillates so wildly as $n$ grows up to infinity that the normalized NCS fill the open circle $η^{-1}$ in the complex $α$-plane. In addition this prevents the existence of a measure including normalizable states only. This difficulty is overcome by introducing a family of deformations which are rational functions of n, each of them admitting a measure. By increasing the degree of these rational approximations the deformation of a trapped ion can be approximated with any degree of accuracy and the formalism of the P-representation can be applied.