In this paper, we study the controllability of the Bloch
equation, for an ensemble of non interacting half-spins, in a static
magnetic field, with dispersion in the Larmor frequency. The control
are two (time-varying) radio frequency fields which are orthogonal to
the static magnetic field. This system may be seen as a prototype for
infinite dimensional bilinear systems with continuous spectrum, whose
controllability are not well understood. We prove that this
system is not locally exactly controllable in finite time T: in fact
the states that one can be reached from a given state with small
control is of infinite codimension. Jr-Shin Li and Navin Khaneja
(2006, IEEE-CDC) proved that this control system is globally
approximately controllable. However if, for example, one starts with a
state with the spins all close to the North Pole (of the Bloch sphere),
one need, with Li-Khaneja's method, to go a very large number of
times from the North Pole to the South Pole and come back in order to
go closer to the North Pole. We propose a method to do go closer to the
North Pole which requires to go only once to the South Pole. The idea
is the following one. At the North Pole the linearized control systems
is not controllable: roughly one can control with the linearization
``half of the state space''. Similarly, at the South Pole, one can
control with the linearization at the South Pole ``half of the state
space''. But it turns out that these two halves are ``complementary''.
Hence starting from the North Pole, going only once to the South
Pole and then returning back to the North Pole allows the linearization
to control the state (locally).