Optimal control theory (OCT) is applied to the problem of cooling molecular
rotations. The optimal field gives rise to a striking behavior, in which t
here is no noticeable increase in the lowest rotational state population un
til the last percent or so of the control interval,:at which point the popu
lation jumps to 1. Further analysis of the intermediate time interval revea
ls that cooling is taking place all along, in the sense that the purity of
the system, as measured by Tr(rho(2)), is increasing monotonically in time.
Once the system becomes almost completely pure, the external control field
can transfer the amplitude to the lowest rotational state by a completely
Hamiltonian manipulation. This mechanism is interesting because it suggests
a possible way of accelerating cooling, by exploiting the cooling induced
by spontaneous emission to all the ground electronic state levels, not just
the lowest rotational level. However, it also raises a major paradox: it m
ay be shown that external control fields, no matter how complicated, cannot
change the value of Tr(rho(2)); changing this quantity requires spontaneou
s emission which is inherently uncontrollable. What place is there then for
control, let alone optimal control, using external fields? We discuss the
resolution to this paradox with a detailed analysis of cooling in a two-lev
el system.