  Transferring the state of a quantum system to a given distribution of
populations is an important problem with applications to Quantum Chemistry and
Atomic Physics. In this work we consider exact population transfers that
minimize the L^2 norm of the control which is typically the amplitude of an
electromagnetic field. This problem is analytically and numerically
challenging. Except for few exactly solvable cases, there is no general
understanding of the nature of optimal controls and trajectories. We find that
by examining the limit of large transfer times, we can uncover such general
properties. In particular, for transfer times large with respect to the time
scale of the free dynamics of the quantum system, the optimal control is a sum
of components, each being a Bohr frequency sinusoid modulated by a slow
amplitude, i.e. a profile that changes considerably only on the scale of the
transfer time. Moreover, we show that the optimal trajectory follows a "mean''
evolution modulated by the fast free dynamics of the system. The calculation of
the "mean'' optimal trajectory and the slow control profiles is done via an
"averaged'' two-point boundary value problem which we derive and which is much
easier to solve than the one expressing the necessary conditions for optimality
of the original optimal transfer problem.
