arXiv quant-ph/04102012004-10quant-phhep-thmath-phmath.MP

Jaynes-Cummings模型与非交换“几何”：几个值得注意的问题

Jaynes-Cummings Model and a Non-Commutative "Geometry" : A Few Problems Noted

Kazuyuki Fujii

备注：: Latex files, 16 pages. Talk at "Yamagata Conference on Mathematical Sciences" (4~6/November/2004). An appendix added

摘要：

本文指出，不带重谐振条件的Jaynes-Cummings模型给出了M处理的简单自旋模型（包括参数$x$，$y$和$z$）的一个非对易形式.维·贝里该模型不同于通常的非对易模型，因为x-y坐标是量子化的，而z坐标不是。在我们的非对易模型中，一个新的有趣的现象是Berry模型中对应于Dirac弦的弦只存在于包含基态（${\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}$）的态中，而对于其它激发态（${\cal F}\times {\cal F} \setminus {\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}$），它们不存在.一个非对易模型可能使相应经典模型中的奇异对象（奇点或奇异线等）变得温和或部分消除。

英文摘要：

In this paper we point out that the Jaynes-Cummings model without taking a renonance conditon gives a non-commutative version of the simple spin model (including the parameters $x$, $y$ and $z$) treated by M. V. Berry. This model is different from usual non-commutative ones because the x-y coordinates are quantized, while the z coordinate is not. One of new and interesting points in our non-commutative model is that the strings corresponding to Dirac ones in the Berry model exist only in states containing the ground state (${\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}$), while for other excited states (${\cal F}\times {\cal F} \setminus {\cal F}\times \{\ket{0}\} \cup \{\ket{0}\}\times {\cal F}$) they don't exist. It is probable that a non-commutative model makes singular objects (singular points or singular lines or etc) in the corresponding classical model mild or removes them partly.

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英文摘要：

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