Fast chiral resolution with optimal control

In this work, we formulate the problem of achieving in minimum time perfect chiral resolution with bounded control fields as an optimal control problem on two noninteracting spins $1/2$. We assume the same control bound for the two Raman fields (pump and Stokes) and a different bound for the field directly connecting the two lower-energy states. Using control theory, we show that the optimal fields can only take the boundary values or be zero, with the latter corresponding to singular control. Subsequently, using numerical optimal control and intuitive arguments, we identify some three-stage symmetric optimal pulse sequences for relatively large values of the ratio between the two control bounds and analytically calculate the corresponding pulse timings as a function of this ratio. For smaller values of the bound ratio, numerical optimal control indicates that the optimal pulse sequence loses its symmetry and the number of stages increases in general. In all cases, the analytical or numerical optimal protocol achieves a faster perfect chiral resolution than other pulsed protocols, mainly because of the simultaneous action of the control fields. The present work is expected to be useful in the wide spectrum of applications across the natural sciences in which enantiomer separation is a crucial task.