Optimal Pulse Design in Quantum Control

精确控制量子系统动力学的能力是量子科学技术进步的重要一步。利用最先进的量子技术，可以对单个原子进行捕获和实验（量子光学），对大脑和心脏进行成像（磁共振成像），生成生物大分子的结构和动态信息（核磁共振波谱），处理信息和计算（量子信息和计算）。所有这些应用都是通过应用单个电磁脉冲或多个脉冲序列来实现的。一个典型的问题是设计时变脉冲，同时引导一个大的集合（例如，数十亿）相同的量子系统从初始状态到期望的目标状态，或者在允许的时间内尽可能接近目标状态。这样的脉冲设计任务是具有挑战性的，因为集合中的单个量子系统具有略微不同的动力学，例如不同的频率，但它们需要通过使用共同的控制输入来控制。该项目将开发一个统一的控制理论框架，用于系统有效地设计脉冲，以最佳地控制量子系统的时间演化。这项高度跨学科的研究不仅会对系统理论和控制工程产生创新和重大贡献，而且还会对生物化学和医学物理学产生有希望的贡献。例如，这项研究将提高医学成像的分辨率，以实现精确的医学诊断。该项目还将通过为密苏里州圣路易斯市当地高中的学生创造夏季研究机会，支持促进学生跨学科教育的新举措，特别是那些传统上服务不足的人群。通过将集合系统理论与几何和计算最优控制联系起来，将制定通用和通用的量子控制框架。具体而言，将构建非线性量子自旋系统与线性谐振子在最优强迫下的动态映射，从而推导出未发现的解析宽带脉冲，并在Frenet框架上建立具有螺旋的量子自旋系综演化的直观而优雅的几何连接。此外，还设计了基于奇异值分解的迭代算法和多维伪谱离散化与约束分割技术相结合的通用优化过程，用于设计无约束和有约束的最优脉冲。与美国和海外不同机构的实验量子控制小组合作，开发的方法将用于设计用于蛋白质核磁共振波谱和核磁共振成像的实用脉冲，以及用于便携式医疗成像设备的低复杂度脉冲。这些脉冲将被实验实现和验证。

The capability to precisely control the dynamics of quantum systems constitutes a significant step in the advancement of quantum science and technology. State-of-the-art quantum technology can be used to trap and experiment with individual atoms (quantum optics), image brains and hearts (magnetic resonance imaging), generate structural and dynamical information of biological macromolecules (nuclear magnetic resonance spectroscopy), and process information and computation (quantum information and computation). All of these applications are enabled by the application of a single electromagnetic pulse or multiple pulse sequences. A typical problem is to engineer time-varying pulses that simultaneously steer a large ensemble (e.g., billions) of identical quantum systems from an initial state to a desired target state, or as close as possible, in a permissible amount of time. Such pulse design tasks are challenging because individual quantum systems in an ensemble have slightly different dynamics, such as different frequencies, but they need to be controlled by the use of a common control input. This project will develop a unified control-theoretic framework for systematically and effectively design pulses that optimally control the time evolution of quantum systems. This highly transdisciplinary research will not only result in innovative and significant contributions to systems theory and control engineering, but also generate promising contributions to biochemistry and medical physics. For example, this research will enhance resolution in medical imaging for precise medical diagnosis. The project will also support new initiatives to promote interdisciplinary education for students, in particular for those from traditionally underserved populations through the creation of summer research opportunities for students from local high schools in the city of St. Louis, MO.By bridging ensemble systems theory with geometry and computational optimal control, general and versatile frameworks for quantum control will be formulated. Specifically, a dynamic mapping between nonlinear quantum spin systems and linear harmonic oscillators under optimal forcing will be constructed, which enables the derivation of undiscovered analytical broadband pulses, and an intuitive and elegant geometric connection of the evolution of a quantum spin ensemble with a helix on the Frenet frame will be established. In addition, an iterative algorithm based on the singular value decomposition and a universal optimization procedure integrating multidimensional pseudospectral discretizations and constraint partitioning techniques for designing unconstrained and constrained optimal pulses will be devised. In collaboration with experimental quantum control groups at different institutions in the United States and overseas, the developed methods will be employed to design practical pulses for protein NMR spectroscopy and MRI, as well as low-complexity pulses for portable medical imaging devices. These pulses will be experimentally realized and verified.