Trapped ions in laser fields: A benchmark for deformed quantum oscillators

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 $D_{h}(\alpha)$ to the vacuum state. The unity resolutions in terms of the projectors $| \alpha, h> < \alpha, h^{-1}| ,| \alpha, h^{-1}> < \alpha, h|$ are presented together with a measure allowing a resolution in terms of $| \alpha, h> < \alpha, h|$. $D_{h}(\alpha)$ is also used for introducing the probability distribution funtion $\rho_{A,h}(z)$ 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 $\eta ,$ oscillates so wildly as $n$ grows up to infinity that the normalized NCS fill the open circle $\eta ^{-1}$ in the complex $\alpha$-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.