Time minimal trajectories for a spin 1/2 particle in a magnetic field

In this paper we consider the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on two-dimensional (2-D) manifolds. Let (-E,E) be the two energy levels, and parallel to Omega(t)parallel to <= M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level -E, and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M/E < 1, every time optimal trajectory is bang-bang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency omega(R)=2E. On the other side, for M/E>1, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. For fixed E, we also prove that for M ->infinity the time needed to reach the state two tends to zero. In the case M/E>1 there are time optimal trajectories containing a singular arc. Finally, we compare these results with some known results of Khaneja, Brockett, and Glaser and with those obtained by controlling the magnetic field both on the x and y directions (or with one external field, but in the rotating wave approximation). As a byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns that cyclically alternate as M/E -> 0, giving a partial proof of a conjecture formulated in a previous paper.