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.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.

Physica ScriptaDec 1, 20250 citations

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

Vladimir I. Man’ko, Giuseppe Marmo, A. Porzio +2 more

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.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.

arXiv:quant-ph/0003125Physical Review AOct 13, 200046 citations

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.