Numerous applications in control of quantum systems involve controlling a continuum of dynamical systems with different dynamics by using the same control field. This forms a new class of control problems and we call them Ensemble Control. We propose a fundamental investigation of such problems and plan to develop new methods and applicable theorems, employing Lie algebras and the noncommutativity of vector fields, to understand the controllability conditions and optimal control of such systems. The motivation for looking into these problems comes from simultaneous manipulation of dynamics of quantum ensembles, where, in many cases, the elements of the ensemble show variations in the parameters that characterize the system dynamics. The resulting methodology obtained from our analytical work can be directly applied to the design of optimal pulse sequences in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI), with applications to structural biology, medical diagnosis, and NMR quantum computing.
The PI has introduced notion of ensemble control and proposes to develop new methods for investigating fundamental properties of ensemble control systems including controllability and optimal control. The derived methodology provides systematic and revolutionary approaches for pulse sequence design in various magnetic resonance applications. The work promises enhanced sensitivity and improved experiments in protein NMR spectroscopy and medical imaging. The new mathematical structures appearing in ensemble control are excellent motivation for new developments in control and systems theory.
Broader Impacts: The PI's research plan will advance state-of-the-art methods of mathematical control theory. The applications of these methods extend beyond the scope of pulse design in magnetic resonance. They are applicable to all areas involving coherent control of quantum dynamics, including various fields of coherent spectroscopy (electron spin resonance (ESR), laser spectroscopy, etc.) as well as quantum information processing. The coupled research/education proposal will expose undergraduate and graduate students to interdisciplinary knowledge in both theoretical and practical viewpoints.