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

Some properties of the nonlinear 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 ${D}_{h}(\ensuremath{\alpha})$ to the vacuum state. The unity resolutions in terms of the projectors $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha}{,h}^{\ensuremath{-}1}|,|\ensuremath{\alpha}{,h}^{\ensuremath{-}1}〉〈\ensuremath{\alpha},h|$ are presented together with a measure allowing a resolution in terms of $|\ensuremath{\alpha},h〉〈\ensuremath{\alpha},h|.$ ${D}_{h}(\ensuremath{\alpha})$ is also used for introducing the probability distribution funtion ${\ensuremath{\rho}}_{A,h}(z)$ while the existence of a measure is exploited for extending the P representation to these states. The weight of the $n\mathrm{th}\mathrm{}$ Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter $\ensuremath{\eta},$ oscillates so wildly as n grows up to infinity that the normalized NCS fill the open circle ${\ensuremath{\eta}}^{\ensuremath{-}1}$ in the complex $\ensuremath{\alpha}$ plane. In addition, this prevents the existence of a measure including normalizable states only. This difficulty is overcome by introducing a family of deformations that 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.