开放路径的几何相位与能级简并处的测地线选择规则
Geometric phase of open paths and a geodesic-selection rule at a level degeneracy
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中文总结 AI 辅助
该研究解决了开放路径几何相位在能级简并处的测地线选择模糊性，通过闭式推导确定了仅密切平面内的大圆能使几何相位与包围立体角等价，将启发式规则转化为可计算方案。
中文摘要 AI 辅助
当量子比特、偏振态或自旋-1/2系统的控制场扫过能级简并点时，其方向会在布洛赫球上描绘出一条端点为对跖点的开放曲线，此时开放路径几何相位的测地线规则变得模糊：无数条测地线可闭合该路径，不同的闭合方式包围不同的立体角。我们以闭式形式解决了这一模糊性。与坐标无关的单极子联络本质上定义了开放路径的立体角Ω[C]，将简并点沿εû方向位移可闭合路径，所包围的立体角为Ω(εû)=Ω[C]+2α+O(ε)，其中α是û的横向部分相对于控制曲线在交叉点处主法线的方位角。因此，几何相位与包围立体角的等价性仅对一条闭合测地线成立——即密切平面内的大圆（α=0），该大圆由简并点处的曲率提供。由此可推导出Berry在位移反转下的π不变量及反射对称下的±π/2值，有限温度Uhlmann相位的纯态极限会自动选择密切平面闭合方式，将开放路径文献中的启发式闭合规则转化为可计算的方案。
英文摘要
When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $Ω[C]$ intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where $α$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane ($α=0$)---supplied by the curvature at the degeneracy. Berry's $π$ invariant under reversal of the displacement and the values $\pmπ/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
