双光子量子拉比模型——谱简并与对称性
Two-photon quantum Rabi models - Spectral degeneracy and symmetries
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中文总结 AI 辅助
该研究探究双光子非对称量子拉比模型的谱简并与对称性，证明简并本征值和相关对称性算子存在，刻画谱简并，提出简并性与超椭圆曲线、数论相关的猜想及隐藏对称性与简并性的关系。
中文摘要 AI 辅助
量子拉比模型在量子光与物质的研究中具有重要地位，描述了最基本的相互作用。阐明其谱性质及其推广模型的谱性质，对量子光学和量子信息技术等领域的应用至关重要。本文研究双光子非对称量子拉比模型的能量简并的精确描述以及该系统对称性的本质，结果表明这些性质与某些代数曲线（超椭圆曲线）和数论的结构存在显著关系。具体而言，我们首先利用该系统本征值问题对应的富克斯型常微分方程的单值性数据，证明了简并本征值的存在性，以及相关对称性算子的存在性，这些算子此前在物理学中已被发现并进行了启发性研究。随后，我们对谱简并进行了完整刻画，并提出一系列猜想，表明简并性可确定整个能量谱结构。更确切地说，对于足够强的相互作用，我们通过与简并本征态相关的二元多项式的零点轨迹，得到了能量曲线的极佳近似。此外，我们提出了哈密顿量层面缺失的对称性（隐藏对称性）与双光子量子拉比模型简并性之间关系的明确描述。
英文摘要
The quantum Rabi model occupies a distinguished role in the study of quantum light and matter, describing the most fundamental interactions. Elucidation of its spectral properties, along with those of its generalizations, is necessary for quantum optics and applications in areas such as quantum information technologies. In this paper, we explore a precise description of energy degeneracy of the two-photon asymmetric quantum Rabi model and the nature of the system's symmetries; as a result, we demonstrate that these have significant relationships with the structure of certain algebraic curves (hyperelliptic curves) and number theory. Concretely, we first prove the existence of degenerate eigenvalues using the monodromy data of the Fuchsian ODE, which draws the eigenvalue problem of the system, and the existence of associated symmetry operators, previously discovered and studied heuristically in physics. Subsequently, we give a complete characterization of spectral degeneracies and present a series of conjectures showing that the degeneracy may determine the entire energy spectral structure. More precisely, for sufficiently strong interaction, we obtain an excellent approximation of the energy curves by the zero locus of a two-variable polynomial associated to degenerate eigenstates. In addition, we postulate an explicit description of the relation between the missing symmetry at the Hamiltonian level (hidden symmetry) and the degeneracy of the two-photon quantum Rabi model.
