Optimal Control on a N-Level Quantum System

Abstract

We apply techniques of subriemannian geometry on Lie groups to laser-induced population transfer in a n-level quantum system. The aim is to induce transitions by n-1 laser pulses, of arbitrary shape and frequency, minimizing the pulse energy. For n = 3, we prove that the Hamiltonian system given by the Pontryagin Maximum Principle is completely integrable, since this problem can be stated as a "k ⊕ 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". For n ≥ 4 the integrability of the Hamiltonian system given by the Pontryagin Maximum Principle is an open question, but is possible to prove that optimal trajectories are still in resonance even without finding the explicit expressions for trajectories and controls. The amplitude of optimal controls is computed numerically for n = 4