Time optimal control in coupled spin systems: a second order analysis

In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.