近日，意大利卡梅里诺大学的Paul M. Alsing及其研究团队取得一项新进展。他们对时变磁场中量子比特的量子演化曲率进行了研究。相关研究成果已于2025年1月6日在国际知名学术期刊《物理评论A》上发表。

据悉，在量子力学过程的几何描述中，量子演化的时变曲率系数由状态向量的，切向量的协变导数模的平方来定义。特别地，曲率系数衡量了在非定常哈密顿量（该哈密顿量规定了薛定谔演化方程）作用下，以幺正方式演化的平行传输纯量子态所描绘的量子曲线的弯曲程度。

该研究团队给出了一个置于时变磁场中的，二能级量子系统量子演化曲率的精确解析表达式。具体而言，研究人员探究了由具有单位速度效率的，两参数非定常厄米哈密顿量生成的动力学。这两个参数指定了用于表示演化纯态的布洛赫球中，极角和方位角的恒定时间变化率。

为了更好地理解曲率系数的物理意义，研究人员证实由于量子演化的测地线效率严格小于1，因此量子曲线是非测地线的，并通过调整两个哈密顿量参数，将曲率系数的时间行为，与系统在希尔伯特投影空间中的演化速度和加速度的时间行为，进行了比较。

此外，研究人员还将曲率系数的时间变化曲线，与平行和横向磁场强度之比平方的时间变化曲线进行了比较。最后，研究人员讨论了将该研究的几何方法扩展到，在任意时变厄米哈密顿量作用下幺正演化的高维量子系统时，寻找精确解析解所面临的挑战。

附：英文原文

Title: Curvature of quantum evolutions for qubits in time-dependent magnetic fields

Author: Carlo Cafaro1,2, Leonardo Rossetti1,3, and Paul M. Alsing1

Issue&Volume: 2025-01-06

Abstract: In the geometry of quantum-mechanical processes, the time-varying curvature coefficient of a quantum evolution is specified by the magnitude squared of the covariant derivative of the tangent vector to the state vector. In particular, the curvature coefficient measures the bending of the quantum curve traced out by a parallel-transported pure quantum state that evolves in a unitary fashion under a nonstationary Hamiltonian that specifies the Schrdinger evolution equation. In this paper, we present an exact analytical expression of the curvature of a quantum evolution for a two-level quantum system immersed in a time-dependent magnetic field. Specifically, we study the dynamics generated by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. The two parameters specify the constant temporal rates of change of the polar and azimuthal angles used in the Bloch sphere representation of the evolving pure state. To better grasp the physical significance of the curvature coefficient, showing that the quantum curve is nongeodesic since the geodesic efficiency of the quantum evolution is strictly less than 1 and tuning the two Hamiltonian parameters, we compare the temporal behavior of the curvature coefficient with that of the speed and the acceleration of the evolution of the system in projective Hilbert space. Furthermore, we compare the temporal profile of the curvature coefficient with that of the square of the ratio between the parallel and transverse magnetic field intensities. Finally, we discuss the challenges in finding exact analytical solutions when extending our geometric approach to higher-dimensional quantum systems that evolve unitarily under an arbitrary time-dependent Hermitian Hamiltonian.

DOI: 10.1103/PhysRevA.111.012408

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.111.012408