Non-integrability of the optimal control problem for n-level quantum systems

We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory.