双周期光子几何中的对称固定完整环绕量与谱隔离：量子比特的方形母流形与量子三态的六角形母流形
Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold
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中文总结 AI 辅助
该研究针对双周期光子几何，分析了对称固定完整环绕量的谱特性，提出8×8微环晶格的有限器件实现方案，给出方形和六角晶格的谱间隙等定量结果，是一项定量光谱学研究。
中文摘要 AI 辅助
具有两个周期性方向的系统携带两个可交换的完整环绕量a=(u,v)∈ℝ²/ℤ²。我们在分离晶格、算术和可观测效应的同时确定其特征值。最大化最低扭转本征值会将a置于动量晶格的深孔处。对于每个矩形环面，最大化器为反周期的，因此复乘法足以实现扭转最优值，但并非必要条件。设G_τ⁻¹为对偶度量，D_τ(a)为扭转拉普拉斯算子的归一化zeta行列式。在方形和六角晶格的旋转固定深孔处，对称性给出精确的行列式响应：-Hess_a log D_τ=2π(Im τ)G_τ⁻¹。在谱波数κ=2π时，最低流形为四重和三重，间隙分别为2κ²和4κ²/3；相位误差会线性分裂它们，而其质心保持静止。随后我们给出有限器件实现：一个由两个相位控制缝闭合的8×8微环晶格。在报告的16 GHz耦合尺度下，其精确方形晶格谱具有17.32 GHz的壳层间隙，以及0.1完整环绕量误差对应的1.92 GHz双重态分裂；三角连接构型产生三重量子三态流形，间隙为18.11 GHz。这是一个定量光谱学方案，而非拓扑保护或完整器件的声明。
英文摘要
A system with two periodic directions carries two commuting holonomies \(a=(u,v)\in\R^2/\Z^2\). We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places \(a\) at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let \(G_τ^{-1}\) be the dual metric and \(D_τ(a)\) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response \(-\operatorname{Hess}_a\log D_τ=2π(\Imτ)G_τ^{-1}\). With spectral wavenumber \(κ=2π\), the lowest manifolds are fourfold and threefold, with gaps \(2κ^2\) and \(4κ^2/3\); phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an \(8\times8\) microring lattice closed by two phase-controlled seams. At a reported coupling scale of \(16\) GHz, its exact square-lattice spectrum has a \(17.32\) GHz shell gap and a \(1.92\) GHz doublet separation for a \(0.1\) holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an \(18.11\) GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.
