Spin Dynamics: A Paradigm for Time Optimal Control on Compact Lie Groups

The development of efficient time optimal control strategies for coupled spin systems plays a fundamental role in nuclear magnetic resonance (NMR) spectroscopy. In particular, one of the major challenges lies in steering a given spin system to a maximum of its so-called transfer function. In this paper we study in detail these questions for a system of two weakly coupled spin-½ particles. First, we determine the set of maxima of the transfer function on the special unitary group SU(4). It is shown that the set of maxima decomposes into two connected components and an explicit description of both components is derived. Related characterizations for the restricted optimization task on the special orthogonal group SO(4) are obtained as well. In the second part, some general results on time optimal control on compact Lie groups are re-inspected. As an application of these results it is shown that each maximum of the transfer function can be reached in the same optimal time. Moreover, a global optimization algorithm is presented to explicitly construct time optimal controls for bilinear systems evolving on compact Lie groups. The algorithm is based on Lie-theoretic time optimal control results, established in [15], as well as on a recently proposed hybrid optimization method. Numerical simulations show that the algorithm performs well in the case a two spin-½ system.

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