In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector $|T〉$ to the state vector $|\mathrm{\Psi }〉$ and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector $|T〉$ to the state vector $|\mathrm{\Psi }〉$, orthogonal to $|T〉$ and $|\mathrm{\Psi }〉$ and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.

在本文中，我们提出了如何量化非平稳哈密顿量下演化的状态向量所追踪的量子曲线的弯曲和扭曲的几何视角。具体来说，依靠现有的平稳哈密顿量的几何观点，我们讨论了将我们的理论构造推广到与时间相关的量子力学场景，其中时变曲率和扭转系数都起着关键作用。具体来说，我们提出了 Frenet-Serret 装置的量子版本，用于投影希尔伯特空间中的量子轨迹，该轨迹由平行传输的纯量子态在指定薛定谔演化方程的依赖于时间的哈密顿量下统一演化而绘制。时变曲率系数由状态向量 $|\mathrm{\Psi }〉$ 的切向量 $|T〉$ 的协变导数的大小平方指定，并测量量子曲线的弯曲度。相反，时变扭转系数由切向量 $|T〉$ 的协变导数到状态向量 $|\mathrm{\Psi }〉$ 的投影的幅度平方给出，与 $|T〉$ 和 $|\mathrm{\Psi }〉$ ，此外还测量量子曲线的扭曲。我们发现，从统计的角度来看，时变环境表现出更丰富的结构。例如，与时间无关的配置不同，我们发现广义方差的概念非常重要地出现在由非平稳哈密顿量下演化的量子态所描绘的曲线的挠率的定义中。为了从物理上说明我们的构造的重要性，我们将其应用于由正弦振荡时间相关势指定的精确可解的时间相关二态拉比问题。 在这种情况下，我们证明曲率和扭转系数的解析表达式仅由两个实数三维向量完全描述，即指定量子系统的布洛赫向量和外部施加的时变磁场。尽管我们表明，对于任意时间相关的单量子位哈密顿演化，挠率都为零，但我们研究了不同动态场景下曲率系数的时间行为，包括非共振和共振状态，此外，和弱驱动配置。虽然我们的形式主义适用于任意维度的纯量子态，但随着维度的增加，相关曲率和轨道模拟的分析推导可能会变得非常复杂。因此，最后我们简要评论了将我们的几何形式主义应用于在一般非平稳哈密顿量下统一演化的高维量子系统的可能性。