Jaynes-Cummings模型中的经典Bethe参数和规范形
Classical Bethe Ansatz and Normal Forms in the Jaynes-Cummings Model
摘要：
Jaynes-Cummings-Gaudin模型描述了耦合到谐振子的$n$自旋的集合。已知它是可积的，因此可以定义一个矩映射，该矩映射将$n+1$守恒哈密顿量的值列表与相空间中的每个点相关联。我们确定了这个映射的所有临界点，并使用模型的Lax矩阵表示来计算相应的二次范式。正规坐标是由一个过程，这似乎是一个经典版本的贝特Anglands用于解决量子模型。我们发现，只有椭圆形或焦点焦点奇点存在于这个模型中，这提供了一个有趣的例子，辛复曲面行动的奇点。为了探索这些，我们详细研究了$n=1$和$n=2$的情况下的光谱曲线的退化。这给出了动量图（IMM）和相关分叉图的完整图像。对于$n=2$，我们特别发现了一些秩为1的线，它们一方面位于IMM的边界上，在那里它们表现得像分隔两个面的边，另一方面则在IMM内部。
英文摘要：
The Jaynes-Cummings-Gaudin model describes a collection of $n$ spins coupled to an harmonic oscillator. It is known to be integrable, so one can define a moment map which associates to each point in phase-space the list of values of the $n+1$ conserved Hamiltonians. We identify all the critical points of this map and we compute the corresponding quadratic normal forms, using the Lax matrix representation of the model. The normal coordinates are constructed by a procedure which appears as a classical version of the Bethe Ansatz used to solve the quantum model. We show that only elliptic or focus-focus singularities are present in this model, which provides an interesting example of a symplectic toric action with singularities. To explore these, we study in detail the degeneracies of the spectral curves for the $n=1$ and $n=2$ cases. This gives a complete picture for the image of the momentum map (IMM) and the associated bifurcation diagram. For $n=2$ we found in particular some lines of rank 1 which lie, for one part, on the boundary of the IMM, where they behave like an edge separating two faces, and which go, for another part, inside the IMM.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
