2000/03/28 by V. I. Manʹko, V. Man'ko, G. Marmo +3 · 46 citations
Computer Science · Physics and Astronomy · #Atomic physics #Coherent states #Cold Atom Physics and Bose-Einstein Condensates #Formalism (music) #Ion #Mathematical physics #Measure (data warehouse) #Physics #Quantum #Quantum Information and Cryptography #Quantum mechanics #Quantum optics and atomic interactions #quant-ph
paper · pdf · doi:10.1103/physreva.62.053407
published in Physical Review A 62(5) (American Physical Society) · submitted to Phys. Rev. A
arxiv created 2000/03/28 · openalex publication_date 2000/10/13 · arxiv updated 2012/11/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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) to the vacuum state. The unity resolutions in terms of the projectors |\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|. Dh(\ensuremathα) is also used for introducing the probability distribution funtion \ensuremathρA,h(z) while the existence of a measure is exploited for extending the P representation to these states. The weight of the nth Fock state of the NCS relative to a trapped ion with Lamb-Dicke parameter \ensuremathη, oscillates so wildly as n grows up to infinity that the normalized NCS fill the open circle \ensuremathη^\ensuremath-1 in the complex \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.