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 $∣Ω(t)∣≤M$ 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 $ωR=2E$⁠. 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 $M→∞$ 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.