Fast chiral resolution with optimal control

TL;DR

The paper tackles minimum-time perfect chiral resolution in a two-spin-1/2 model under bounded control fields, formulating it as a time-optimal control problem on a closed-loop three-level molecular system. By applying Pontryagin's Maximum Principle, it proves the optimal controls are bang or singular, then derives explicit three-stage symmetric pulse sequences that are optimal for Ω1/Ω0 ≥ 1 and analytically characterizes their timing as a function of the bound ratio. For Ω1/Ω0 < 1, it shows a similar three-stage structure with the Q-field active throughout, with symmetry breaking and additional stages appearing below a critical ratio; in all cases the derived protocols yield faster chiral resolution than standard pulsed schemes. The findings highlight the advantage of simultaneous control-field action and provide time formulas that facilitate experimental design for rapid enantiomer separation, with potential applications across chemistry, physics, and related areas.

Abstract

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 non-interacting spins-$1/2$. We assume the same control bound for the two Raman fields (pump and Stokes) and a different bound for the field connecting directly the two lower-energy states. Using control theory, we show that the optimal fields can only take the boundary values or be zero, 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 larger values of the ratio between the two control bounds, and analytically calculate the corresponding pulse timings as functions of this ratio. For smaller values of the bounds 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 where enantiomer separation is a crucial task.

Figures (10)

• Figure 1: Closed loop three-level system for the $L$- and $R$-molecules. The pump and Stokes Raman fields are common for both chiralities, while the $Q$-field has opposite sign.
• Figure 2: Case $\Omega_1/\Omega_0=1$, (a) optimal controls, (b) optimal trajectories on the Bloch sphere (upper panel) and time evolution of level $\ket{3}$ populations (lower panel) for the $L$- (red) and $R$- (blue) molecules.
• Figure 3: Case $\Omega_1/\Omega_0\rightarrow\infty$, (a) optimal controls, (b) optimal trajectories on the Bloch sphere (upper panel) and time evolution of level $\ket{3}$ populations (lower panel) for the $L$- (red) and $R$- (blue) molecules.
• Figure 4: Case $\Omega_1/\Omega_0=2$, (a) optimal controls, (b) optimal trajectories on the Bloch sphere (upper panel) and time evolution of level $\ket{3}$ populations (lower panel) for the $L$- (red) and $R$- (blue) molecules.
• Figure 5: Case $\Omega_1/\Omega_0=0.9$, (a) optimal controls, (b) optimal trajectories on the Bloch sphere (upper panel) and time evolution of level $\ket{3}$ populations (lower panel) for the $L$- (red) and $R$- (blue) molecules.