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

Nonlinear Coherent States and Some of Their Properties

J. Récamier, M. Gorayeb, Wolf Luis Mochán +1 more

International Journal of Theoretical PhysicsAug 8, 200733 citations

On theq-deformed coherent states of a generalizedf-oscillator

Marcelo A. Marchiolli

Adopting the framework of the generalized q -deformed Heisenberg–Weyl algebra , we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q -deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f -oscillator. In particular, we establish the properties of a small group of q -deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson–Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.Adopting the framework of the generalized q -deformed Heisenberg–Weyl algebra , we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q -deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f -oscillator. In particular, we establish the properties of a small group of q -deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson–Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.

Physica ScriptaDec 22, 200510 citations

Generalized Coherent States

Thalanayar S. Santhanam

Springer eBooksJan 1, 19802 citations

Generalized coherent states

A. D. Jannussis, G. Karayannis, Παναγιώτης Παναγόπουλος +1 more

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoOct 1, 19834 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.