Geometric Analysis and Optimal Control of Quantum Systems in the KP Configuration; Generalizations to nonlinear Systems with Symmetries

精确和有效地控制量子力学系统的状态，如原子、原子核和电子，是这些系统的大多数应用的要求。这种控制通常是通过与外部适当形状的电磁场相互作用获得的。此外，人们通常不仅希望将状态驱动到理想值，还希望优化可用资源。时间的最小化尤为重要。在计算应用中，快速动态导致实现算法的速度加快。此外，一般来说，演化必须在数学模型可以被认为有效的时间范围内发生，在未建模的动力学的影响变得相关之前。在此背景下，本研究将实现三个相互关联的目标：1)它将为在重要应用中非常常见的大型量子力学系统提供明确的最优控制设计算法。2)它将对对称性在量子控制系统中的作用进行深入的数学分析，以及如何使用这些对称性来简化量子系统的数学模型，从而大大扩展现有的理论。3)与实验实验室合作，通过量子光学和核磁共振实验对数学结果进行验证。总体结果将是一个丰富的工具箱，用于量子力学系统的最佳操作，可用于安全通信，强大的量子计算，测量设备设计，医疗诊断以及一般情况下使用量子系统的每个设备。这些活动将涉及一个由工程师、物理学家和数学家组成的跨学科研究小组，其目标是发展一种共同的语言和科学。由此产生的知识将成为一个新的工程领域和量子工程课程的基础，随着量子力学在日常生活中的应用不断扩大，这个领域在未来将变得非常重要。本研究中使用和发展的主要数学工具来自微分几何领域，特别是黎曼和次黎曼几何。起点是所谓的KP数学模型，这种模型的状态在李群上变化，其动力学方程对应于相关李代数的Cartan分解。相应的最优控制问题一方面在应用中很常见，另一方面又显式可解。在这个项目中，它们作为一个试验台来研究量子控制系统的一般特性。这些涉及到对称性的作用，最优轨迹（测地线）的定性行为和可达集的几何。该数学方法的一个关键技术成分是使用对称约简作为工具来分析低维商空间上的控制问题。在这个空间中，可以提出一个更简单的控制问题，并经常得到明确的解决。这一过程将大大扩大现有的量子控制工具箱，这往往局限于小维系统。由此产生的控制设计的实验实现将用于量子光学和核磁共振系统，这是构建量子计算机最有前途的候选者之一。此外，数学分析将需要引入奇异和分层空间理论中的元素，这在除量子力学之外的其他控制应用领域中很重要。在此背景下，本项目也将有助于经典系统控制理论的发展。

The precise and efficient control of the state of quantum mechanical systems, such as atoms, nuclei and electrons, is a requirement in most applications of these systems. Such a control is typically obtained through the interaction with external, appropriately shaped, electromagnetic fields. Moreover, one often wants not only to drive the state to a desired value but also to optimize the available resources. The minimization of time is especially important. In applications to computation, fast dynamics result in the speed-up of the implemented algorithms. Furthermore, in general, the evolution has to occur within the time frame during which the mathematical model can be considered valid, before the effect of un-modeled dynamics becomes relevant. In this context, this research will accomplish three inter-related objectives: 1) It will provide explicit optimal control design algorithms for a large class of quantum mechanical systems very common in important applications. 2) It will provide an in depth mathematical analysis of the role of symmetries in quantum control systems and how these symmetries can be used to simplify mathematical models of quantum systems, thus considerably extending the existing theory. 3) It will validate the mathematical results through experiments in quantum optics and nuclear magnetic resonance via the collaboration with experimental laboratories. The overall result will be a rich toolbox for the optimal manipulation of quantum mechanical systems to be used in applications in secure communication, powerful quantum computing, design of measurement devices, medical diagnostics and, in general, every device which uses quantum systems. The activities will involve an interdisciplinary research team composed of engineers, physicists and mathematicians with the objective of developing a common language and science. The resulting knowledge will be the basis of a new area of engineering and a curriculum in Quantum Engineering, a field that will become very important in the future as the applications of quantum mechanics in everyday life continue to expand. The main mathematical tools used and developed in this research come from the field of differential geometry and in particular Riemannian and sub-Riemannian geometry. The starting point are the so-called KP mathematical models which are models whose state varies on a Lie group and whose dynamical equations correspond to a Cartan decomposition of the associated Lie algebra. The corresponding optimal control problems are, on one hand, very common in applications, and, on the other hand, explicitly solvable. In the project they serve as a test-bed to investigate properties of quantum control systems in general. These involve the role of symmetries, the qualitative behavior of optimal trajectories (geodesics) and the geometry of the reachable sets. A key technical ingredient of the mathematical approach is the use of symmetry reduction as a tool to analyze the control problem on a lower dimensional quotient space. On this space a simpler control problem can be posed and often explicitly solved. This procedure will substantially enlarge the existing toolbox in quantum control, which is frequently restricted to small dimensional systems. The experimental implementations of the resulting control design will be for systems in quantum optics and nuclear magnetic resonance, which are among the most promising candidates for the construction of quantum computers. Furthermore, the mathematical analysis will require the introduction of elements from the theory of singular and stratified spaces, which is important in other areas of applications of control besides quantum mechanics. In this context, this project will contribute to the development of control theory for classical systems as well.