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

Some properties of the nonlinear coherent states (NCS), recognized by Vogel and de Mates 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 nth 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 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.