On the interplay of control fields and spontaneous emission in laser cooling

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.