Time optimal problems on Lie groups and applications to quantum control

Abstract

In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ([1], [2]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $. In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in [2]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ([3], [4]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.

Keywords

• symplectic
• manifolds
• lie-poisson bracket
• lie algebras
• co-adjoint orbits
• extremal curves
• integrable systems