General methods to control right-invariant systems on compact Lie groups and multilevel quantum systems

Abstract

For a right-invariant system on a compact Lie group G, I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate assumptions, achieves exact control to the target but requires estimation of the ‘size’ of a neighborhood of the identity in G and solution of a nonlinear algebraic equation. The second method does not involve any mathematical difficulty and obtains control to a desired target with arbitrary accuracy. A third method is then given combining the main ideas of the previous methods. This is also very simple in its formulation and turns out to be generically more efficient as illustrated by one of the examples I consider. The methods described in the paper provide arbitrary constructive control for any right-invariant system on a compact Lie group. In particular, the results can be applied to the coherent control of general multilevel quantum systems to an arbitrary target.