AI 中文总结
本文针对开放量子系统，通过定态与动力学生成元的有序对构建几何，利用自动微分计算联络与测地线，实现定态响应的协变传输，为可观测量响应跨模型传递提供了内在结构。
AI 中文摘要
我们开发了一种几何，用于在开放量子系统的控制空间中传输定态响应。物理模型由其定态和动力学生成元的有序对表示，将这些对嵌入公共环境空间会在控制流形上诱导出度量、响应一次形式和闭二次形式。该环境空间具有可交换态和生成元方向的典范复结构，但刘维尔零态条件将物理模型限制在不一定保持该结构的子流形上。对于振幅阻尼光学布洛赫模型，在单点ω=0和g/Γ=1/√2处，度量与响应二次形式相容；在其他位置，物理流形是非凯勒的。诱导的度量定义了列维-奇维塔联络、测地线和平行传输，无需凯勒相容性。我们通过定态刘维尔求解的自动微分计算联络，并以机器精度恢复符号结果。两点计算表明，所得测地线与控制空间中的线性插值不同，且在态-生成元模型族中遵循更短路径。该几何提供了内在导数和传输结构，用于在可接受的定态模型之间传递包括多维光谱在内的可观测量响应。
英文摘要
We develop a geometry for transporting stationary-state response across the control space of an open quantum system. A physical model is represented by the ordered pair of its stationary state and dynamical generator. Embedding these pairs in a common ambient space induces a metric, a response one-form, and a closed two-form on the control manifold. The ambient space admits a canonical complex structure that exchanges state and generator directions, but the Liouvillian null-state condition restricts physical models to a submanifold that need not preserve this structure. For an amplitude-damped optical Bloch model, the metric and response two-form are compatible at the single point $ω=0$ and $g/Γ=1/\sqrt{2}$; the physical manifold is non-Kähler elsewhere. The induced metric defines the Levi--Civita connection, geodesics, and parallel transport without requiring Kähler compatibility. We compute the connection by automatic differentiation through the stationary Liouvillian solve and recover the symbolic result to machine precision. A two-point calculation then shows that the resulting geodesic differs from linear interpolation in control space and follows a shorter path through the family of state--generator models. This geometry supplies the intrinsic derivative and transport structure needed to carry observable response, including multidimensional spectra, between admissible stationary models.
