Time minimal trajectories for two-level quantum systems with two bounded controls

In this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude. After projection on the so-called Bloch sphere, we tackle the problem with well-developed techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for a complete optimal synthesis, are illustrated by numerical simulations.