  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, 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 2-D manifolds. Let $(-E,E)$ be the two
energy levels, and $|\Omega(t)|\leq M$ 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. Fixed $E$ we also prove that for $M\to\infty$ 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 byproduct we prove
that the qualitative shape of the time optimal synthesis presents different
patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a
conjecture formulated in a previous paper.
