Optimal Pulse Design in Quantum Control

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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.