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。最先进的量子技术可以用来捕捉和实验单个原子(量子光学)，成像大脑和心脏(磁共振成像)，产生生物大分子的结构和动力学信息(核磁共振光谱)，以及处理信息和计算(量子信息和计算)。所有这些应用都可以通过施加单个电磁脉冲或多个脉冲序列来实现。一个典型的问题是设计时变脉冲，在允许的时间量内，同时将大量(例如，数十亿)相同的量子系统从初始态引导到期望的目标态，或者尽可能地接近。这样的脉冲设计任务是具有挑战性的，因为系综中的各个量子系统具有略有不同的动力学，例如不同的频率，但它们需要使用共同的控制输入来控制。这个项目将开发一个统一的控制理论框架，用于系统和有效地设计脉冲，以最佳地控制量子系统的时间演化。这一高度跨学科的研究不仅将对系统论和控制工程做出创新和重大贡献，而且将对生物化学和医学物理做出有希望的贡献。例如，这项研究将提高医学成像的分辨率，以实现精确的医疗诊断。该项目还将支持新的倡议，通过为密苏里州圣路易斯市当地高中的学生创造暑期研究机会，促进学生，特别是那些来自传统上服务不足的人群的跨学科教育。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。具体地说，我们将建立非线性量子自旋系统与线性简谐振子在最佳作用力下的动态映射，从而能够推导出未被发现的解析宽带脉冲，并将在Frenet标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联系。此外，还将设计一种基于奇异值分解的迭代算法和一种综合了多维伪谱离散化和约束划分技术的通用优化方法，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光谱和核磁共振的实用脉冲，以及用于便携式医学成像设备的低复杂性脉冲。这些脉冲将在实验中实现和验证。

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.