The Lie Algebra Structure and Nonlinear Controllability of Spin Systems

In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 12 particles with different gyromagnetic ratios in an electromagnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also provide a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We analyze low dimensional problems (number of particles less then or equal to three) with possibly equal gyromagnetic ratios. This provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.