dorsal/arxiv
View SchemaTrapped ions in laser fields: A benchmark for deformed quantum oscillators
| Authors | V. Man'ko, G. Marmo, A. Porzio, S. Solimeno, F. Zaccaria |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0003125 |
| URL | https://arxiv.org/abs/quant-ph/0003125 |
| DOI | 10.1103/PhysRevA.62.053407 |
| Journal | Physical Review A, VOLUME 62, 053407 (2000) |
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.
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"abstract": "Some properties of the non--linear coherent states (NCS), recognized by Vogel\nand de Matos Filho as dark states of a trapped ion, are extended to NCS on a\ncircle, for which the Wigner functions are presented. These states are obtained\nby applying a suitable displacement operator $D_{h}(\\alpha) $ to the vacuum\nstate. The unity resolutions in terms of the projectors $| \\alpha, h\u003e \u003c \\alpha,\nh^{-1}| ,| \\alpha, h^{-1}\u003e \u003c \\alpha, h| $ are presented together with a measure\nallowing a resolution in terms of $| \\alpha, h\u003e \u003c \\alpha, h| $. $D_{h}(\\alpha)\n$ is also used for introducing the probability distribution funtion\n$\\rho_{A,h}(z) $ while the existence of a measure is exploited for extending\nthe P-representation to these states. The weight of the n-th Fock state of the\nNCS relative to a trapped ion with Lamb-Dicke parameter $\\eta ,$ oscillates so\nwildly as $n$ grows up to infinity that the normalized NCS fill the open circle\n$\\eta ^{-1}$ in the complex $\\alpha $-plane. In addition this prevents the\nexistence of a measure including normalizable states only. This difficulty is\novercome by introducing a family of deformations which are rational functions\nof n, each of them admitting a measure. By increasing the degree of these\nrational approximations the deformation of a trapped ion can be approximated\nwith any degree of accuracy and the formalism of the P-representation can be\napplied.",
"arxiv_id": "quant-ph/0003125",
"authors": [
"V. Man\u0027ko",
"G. Marmo",
"A. Porzio",
"S. Solimeno",
"F. Zaccaria"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.62.053407",
"journal_ref": "Physical Review A, VOLUME 62, 053407 (2000)",
"title": "Trapped ions in laser fields: A benchmark for deformed quantum oscillators",
"url": "https://arxiv.org/abs/quant-ph/0003125"
},
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