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

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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 Dₕ(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|. Dₕ(α) is also used for introducing the probability distribution funtion ρₐ,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.