Time Minimal Trajectories for two-level Quantum Systems with Drift

On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|\Omega(t)|\leq M$ the bound on the field amplitude. For each values of $E$ and $M$, we provide the explicit expression of the time optimal trajectory steering the state one to the state two in terms of a parameter that should be computed numerically. For $M<<E$, every time optimal trajectory is periodic (and in particular bang-bang) with frequency of the order of the resonance frequency $\omega_R=2E$. On the other side, for $M>E$ the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained in the Rotating Wave Approximation.