Recent experimental projects in quantum control using finite-dimensional systems as the control of spin systems in nuclear magnetic resonance and references therein) are motivating new theoretical studies in the case where the system interacts with its environment. The primary objective of the proposed research is to apply techniques of geometric optimal control theory to the control of the spin dynamics by magnetic fields in Nuclear Magnetic Resonance (NMR). Through interaction with a magnetic field, NMR involves the manipulation of nuclear spins. It has many potential applications extending from the determination of molecular structures (NMR spectroscopy) and quantum computing, where NMR remains one of the most promising road in the construction of a scalable quantum computer, to medical imagery (MRI). The control technology developed over the past 50 years allows the use of sophisticated control fields and permits the implementation of complex quantum algorithms such as the Deutsh-Jozsa and the Grover algorithms. NMR is therefore an ideal experimental testbed for quantum control. The proposed research will also impact the domain of quantum mechanics. First, solving the contrast imaging problem can potentially have a profound impact on how medical imaging is done in hospitals. Indeed, by designing magnetic fields to maximize the distance between the two spin we increase the image resolution and therefore improve its quality which improves patient care. Second, by using geometric techniques our approach will complement existing efficient numerical tools for pulse sequence optimization, such as the GRAPE (gradient ascent pulse engineering) by providing an understanding of the qualitative structures of the dynamics of the system. In particular, the physicists will gain insight about the control mechanism that lead to the optimal solutions.

Starting as a tool for characterization of organic molecules, the use of nuclear magnetic resonance (NMR) has spread to areas as diverse as pharmaceutics, medical diagnostics (medical resonance imaging) and structural biology. The principles of NMR have served as a paradigm for other physical methods that rely on interaction between radiation and matter. It is therefore not surprising that experiments in NMR also serve as good model problems in control of quantum systems. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. Until now, the analysis of nonlinear optimal control has been mostly concerned with the class of single-input systems. Via this application in quantum control, it is proposed to extend the analysis to the multidimensional case. The proposal research will also impact significantly the field of quantum mechanics. First, medical imagery can be drastically improved through the contrast problem which will directly impact patient care in hospitals. Second, by using geometric tools to develop optimal synthesis, we provide physicists with an additional understanding about the control mechanism of the optimal solutions which in turn will help developing improved numerical schemes. We also propose an initiative to address the underrepresentation of minorities, and particularly of Native Hawaiian women, in the fields of science, technology, engineering, and mathematics (STEM disciplines) at UH Manoa and nationwide. The goal is to address the dual barriers of gender and ethnicity that Native Hawaiian females face in these fields.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Michael H. Steuerwalt
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University of Hawaii
United States
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