Constructive controllability of one and two spin ½ particles

We consider the problem of steering control for the systems of one spin ½ particle and two interacting spin ½ particles in an electromagnetic field. The describing models are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. For both systems, we describe their controllability properties and provide constructive control algorithms. The algorithms based on optimal control theory consider the time of transfer as cost functional. Explicit expressions are derived for the time optimal controls. The algorithms based on Lie group decompositions improve over previously presented algorithms in that they allow an explicit analytic calculation of the parameters involved which does not require any numerics