On the $K+P$ Problem for a Three-level Quantum System: Optimality Implies Resonance

We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a three-level quantum system. The aim is to induce transitions by two laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. We prove that the Hamiltonian system given by the Pontryagin Maximum Principle is completely integrable, since this problem can be stated as a $\k\oplus\p$ problem'' on a simple Lie group. Optimal trajectories and controls are exhausted. The main result is that optimal controls correspond to lasers that are in resonance’'.