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arXiv:quant-ph/0003125 · Physical Review A

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

Vladimir I. Man’ko, Giuseppe Marmo, A. Porzio, S. Solimeno, F. Zaccaria

Published Oct 13, 2000• 46 citations
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Abstract

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.

DOI
10.1103/physreva.62.053407

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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 Dh(\ensuremathα){D}_{h}(\ensuremath{\alpha})Dh​(\ensuremathα) to the vacuum state. The unity resolutions in terms of the projectors ∣\ensuremathα,h〉〈\ensuremathα,h\ensuremath−1∣,∣\ensuremathα,h\ensuremath−1〉〈\ensuremathα,h∣|\ensuremath{\alpha},h〉〈\ensuremath{\alpha}{,h}^{\ensuremath{-}1}|,|\ensuremath{\alpha}{,h}^{\ensuremath{-}1}〉〈\ensuremath{\alpha},h|∣\ensuremathα,h〉〈\ensuremathα,h\ensuremath−1∣,∣\ensuremathα,h\ensuremath−1〉〈\ensuremathα,h∣ are presented together with a measure allowing a resolution in terms of ∣\ensuremathα,h〉〈\ensuremathα,h∣.|\ensuremath{\alpha},h〉〈\ensuremath{\alpha},h|.∣\ensuremathα,h〉〈\ensuremathα,h∣. Dh(\ensuremathα){D}_{h}(\ensuremath{\alpha})Dh​(\ensuremathα) is also used for introducing the probability distribution funtion \ensuremathρA,h(z){\ensuremath{\rho}}_{A,h}(z)\ensuremathρA,h​(z) while the existence of a measure is exploited for extending the P representation to these states. The weight of the nthn\mathrm{th}\mathrm{}nth Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter \ensuremathη,\ensuremath{\eta},\ensuremathη, oscillates so wildly as n grows up to infinity that the normalized NCS fill the open circle \ensuremathη\ensuremath−1{\ensuremath{\eta}}^{\ensuremath{-}1}\ensuremathη\ensuremath−1 in the complex \ensuremathα\ensuremath{\alpha}\ensuremathα 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.
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Abstract In this article, we introduce generalized Gilmore-Perelomov-type coherent states of a quantum harmonic oscillator on a circle and demonstrate that their underlying algebra is a deformed su(1,1). The effects of the circle’s spatial curvature are inherently encoded in these states, making them suitable for investigating curvature-induced phenomena in quantum optical processes at laboratory scales. Furthermore, we propose an experimental scheme to realize these states based on the intensity-dependent version of the Jaynes–Cummings model appearing in a laser-driven trapped ion. Employing the nonlinear coherent states of the atomic center of mass motion of a trapped ion, it is possible to control the laser-ion interaction in such a way that Gilmore-Perelomov-type circle coherent states are obtained as motional dark states of the system. Finally, by examining quadrature squeezing, we demonstrate that unlike standard coherent states which exhibit no squeezing, these curvature-dependent states display curvature-tunable squeezing effects. This property makes them particularly valuable for applications in high-precision metrology.
Using the nonlinear coherent states approach, a general formalism for the construction of various classes of nonlinear trio coherent states in the context of multi-mode quantum states has been proposed. In particular, taking into account the most popular nonlinearity function associated with f -deformed coherent states, i.e. the nonlinearity function of the centre-of-mass motion of a trapped ion, it is illustrated that the corresponding trio coherent state possesses some interesting nonclassical properties. To establish this observation, sub-Poissonian statistics, three-mode squeezing, the behaviour of Vogel's characteristic function and the existence of entanglement are investigated. Due to the intrinsic non-classicality nature of the considered three-mode states, their physical production may be of high interest in the quantum optics field. Thus, we have finally demonstrated how nonlinear motional trio coherent states can be generated in three-dimensional anisotropic traps, appropriately.