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

Abstract

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.

submitted to Phys. Rev. A

References in corpus (1)

• Time dependent nonclassical properties of even and odd nonlinear coherent states

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