AI 中文总结
研究 $k$ 光子光 - 物质哈密顿量自伴性，通过块雅可比分解等方法，证明 $\Sigma$ 正规时 $k\leq2$ 时 $H$ 自伴，$k\geq3$ 时计算亏指数等，还表明非正规耦合的 $k$ 光子杰恩斯 - 卡明斯模型对各 $k$ 都自伴，并在相关模型上说明结果。
AI 中文摘要
多光子光 - 物质相互作用是量子光学中真正非线性的来源，且在实验上越来越容易实现。本文研究了作用于 $\mathcal{H}\otimes L^2(\mathbb{R})$ 上的一类算符 $H = H_{\rm mat}\otimes I + I\otimes\omega a^\ast a + \Sigma\otimes(a^\ast)^k + \Sigma^\ast\otimes a^k$，它通过有界算符 $\Sigma$ 将单个玻色子模式与任意物质系统耦合。当 $\Sigma$ 正规且非零时，证明了 $H$ 自伴当且仅当 $k\leq2$；对于 $k\geq3$，计算了亏指数，参数化了所有自伴扩张，并表明当物质系统是有限维时每个扩张都有纯离散谱。分析基于块雅可比分解和依赖于 $\Sigma$ 极分解的合适酉变换。$\Sigma$ 的正规性是最优的，非正规耦合的 $k$ 光子杰恩斯 - 卡明斯模型对每个 $k$ 都保持自伴。还在 $k$ 光子拉比和迪克模型上说明了结果。
英文摘要
Multiphoton light-matter interactions, in which a bosonic mode exchanges $k$ excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators $H = H_{\rm mat}\otimes I + I\otimesωa^\ast a + Σ\otimes(a^\ast)^k + Σ^\ast\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $Σ$. When $Σ$ is normal and nonzero, we prove that $H$ is self-adjoint if and only if $k\leq2$; for $k\geq3$ we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of $Σ$. The normality of $Σ$ is optimal: a $k$-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every $k$. We illustrate our results on the $k$-photon Rabi and Dicke models.
Comments17 pages
