Sub-Riemannian Geodesics in $SU(n)/S(U(n-1) \times U(1))$ and Optimal Control of Three Level Quantum Systems

We study the time optimal control problem for the evolution operator of an $n$ -level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian $K\hbox{--}P$ problem, (as introduced in articles by U. Boscain and by V. Jurdjevic), whose underlying symmetric space is $SU(n)/S(U(n-1) \times U(1))$ . Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of $S(U(n-1) \times U(1))$ on $SU(n)$ as a conjugation $X \rightarrow KXK^{-1}$ . This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels ( $\Lambda$ -systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data.