两能级混合中的几何相位：从中性介子到全息量子比特与拓扑马约拉纳模式

中文摘要 AI 辅助

我们研究了一个由中性介子振荡启发的厄米两能级混合哈密顿量的几何结构，及其与量子比特、费米子和拓扑描述的联系。将该哈密顿量映射为有效的量子比特表示，我们分析了布洛赫球上循环参数演化所关联的几何相位。在不将复混合相位识别为物理的CP破坏可观测量前提下，我们将其解释为一个类CP几何参数，控制哈密顿量矢量的方位角取向及其在相位共轭下的反转。由本征态构造的三阶Bargmann不变量产生一个关联的离散相位，我们考察了该相位对混合参数的依赖，并与连续的几何构造进行了比较。循环演化也被表示为单量子比特的$R_z$相位操作，并通过马约拉纳双线性形式表达。将该构造推广到Kitaev链的动量依赖Bogoliubov-de Gennes哈密顿量，我们将动量空间绕数关联到拓扑区域以及实参数模型中的量子化Berry（Zak）相位，而相应的Bargmann构造在连续极限下趋近于全局相位。综合来看，这些结果提供了一个几何视角，将相关两能级哈密顿量中的相位结构、量子比特操作、费米子表示和动量空间拓扑联系起来。

英文摘要

We investigate the geometric structure of a Hermitian two-level mixing Hamiltonian motivated by neutral meson oscillations and its connections with qubit, fermionic, and topological descriptions. Mapping the Hamiltonian to an effective qubit representation, we analyze the geometric phase associated with cyclic parameter evolution on the Bloch sphere. Without identifying the complex mixing phase with a physical CP-violation observable, we interpret it as a CP-like geometric parameter controlling the azimuthal orientation of the Hamiltonian vector and its reversal under phase conjugation. Third-order Bargmann invariants constructed from the eigenstates yield an associated discrete phase whose dependence on the mixing parameters is examined alongside the continuous geometric construction. The cyclic evolution is also represented as a single-qubit $R_z$ phase operation and expressed through Majorana bilinears. Extending the construction to the momentum-dependent Bogoliubov-de Gennes Hamiltonian of the Kitaev chain, we relate momentum-space winding to the topological regimes and the quantized Berry (Zak) phase in the real-parameter model, while the corresponding Bargmann construction approaches the global phase in the continuum limit. Together, these results provide a geometric perspective connecting phase structure, qubit operations, fermionic representations, and momentum-space topology within related two-level Hamiltonians.

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21 pages, 4 figures