arXiv quant-ph/04100032004-10quant-phhep-thmath-phmath.MP

Tavis-Cummings模型的量子对角化方法

Quantum Diagonalization Method in the Tavis-Cummings Model

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato, Tatsuo Suzuki, Yukako Wada

备注：: Latex files, 21 pages; minor changes. To appear in International Journal of Geometric Methods in Modern Physics
发表信息：: Int.J.Geom.Meth.Mod.Phys. 2 (2005) 425-440
DOI：: 10.1142/S021988780500065X

摘要：

为了得到Tavis-Cummings模型中演化算符的显式形式，必须显式计算${e}^{-itg(S_{+}\otimes a+S_{-}\otimes a^{\dagger})}$项，这是非常困难的。本文尝试将量子矩阵$A\equiv S_{+}\otimes a+S_{-}\otimes a^{\dagger}$对角化，计算出${e}^{-itgA}$，进而了解模型的深层结构。对于一个，两个和三个原子的情况下，我们给出了这样的对角化，这是第一个非平凡的例子，据我们所知，并重现计算${e}^{-itgA}$在quant-ph/0404034。我们也给出了一个应用于非交换微分几何的提示。然而，量子对角化不是唯一的，并且受到量子物理中算子的非交换性所产生的一些模糊性的影响。我们的方法可能在数学物理或量子物理中开辟一个新的观点。

英文摘要：

To obtain the explicit form of evolution operator in the Tavis-Cummings model we must calculate the term ${e}^{-itg(S_{+}\otimes a+S_{-}\otimes a^{\dagger})}$ explicitly which is very hard. In this paper we try to make the quantum matrix $A\equiv S_{+}\otimes a+S_{-}\otimes a^{\dagger}$ diagonal to calculate ${e}^{-itgA}$ and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is first nontrivial examples as far as we know, and reproduce the calculations of ${e}^{-itgA}$ given in quant-ph/0404034. We also give a hint to an application to a noncommutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the noncommutativity of operators in quantum physics. Our method may open a new point of view in Mathematical Physics or Quantum Physics.

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