Fast enantioconversion of chiral mixtures based on a four-level double-$\Delta$ model

Based on the four-level double-$\Delta$ model composed of two degenerated (left- and right-handed) chiral ground states and two achiral excited states, we propose a purely optical method for enantio-conversion of chiral mixture. By choosing appropriate parameters, the original four-level model will be simplified to two effective two-level sub-systems with each of them involving one chiral ground state. Then, with the help of well-designed optical operations, the initial unwanted and wanted chiral ground states are converted, respectively, to the wanted chiral ground state and an auxiliary chiral excited state with the wanted chirality, i.e., achieving the enantioconversion of the chiral mixture. Comparing with the original work of enantioconversion based on the four-level double-$\Delta$ model with the requirement of the time-consuming relaxation step and repeated operations, our method can be three orders of magnitude faster since we use only purely optical operations. Thus, it offers a promising candidate for fast enantioconversion when the total operation time is limited due to the racemization and/or experimental conditions.

Over years, laser-assisted enantioconversion [54][55][56][57][58] has also been investigated theoretically.It is devoted to converting chiral mixtures to enantiopure samples with the help of optical operations.That means the unwanted enantiomer, which may be inefficient [6] or even cause serious side effects [7,8], is converted to the wanted enantiomer.Such a promising feature makes the laser-assisted enantioconversion to be an ambitious issue related to chiral molecules.Previously, the four-level double-∆ model of chiral molecules [55][56][57][58] has been introduced to investigate the laser-assisted enantioconversion.In this fourlevel model, two achiral excited states and each of two degenerated chiral ground states are coupled by three optical fields in a form of ∆-type transitions.The sign of the product of coupling strengths in each ∆-type transition is dependent on the chirality of its corresponding chiral ground state.This fact offers the possibility of breaking the left-right symmetry in the dynamics of molecules [56], i.e., the molecules initially in the two degenerated chiral ground states can be de-populated differently.
In these original works of laser-assisted enantioconversion [55][56][57][58], the chirality-dependent de-population is obtained in an excitation step by applying three optical fields to construct the four-level double-∆ model.After that, the three optical fields are turned off, the system experiences a relaxation step, where the molecules in the achiral excited states equally relax back to the two degenerated chiral ground states [55][56][57][58].Then, some population is converted from one chiral ground state to the other after applying a single pair of excitation and relaxation steps.By repeating the pair of two steps until equilibrium is reached, the laser-assisted enantioconversion can be realized in the scenario called "laser distillation of chiral enantiomers" [55][56][57][58].However, in the laser-distillation method [55][56][57][58], the relaxation step is usually time-consuming and the requirement of repeated operations makes it need more total operation time.This greatly prevents the realization of the enantioconversion in the reality when the total operation time is limited due to the racemization and/or experimental conditions.Thus, the discussions about laser-assisted enantio-conversion are almost in silence now.
In this Letter, we propose a theoretical fast method using three purely optical operations to convert a chiral mixture to an enantiopure sample of the wanted chirality.Comparing with the laser-distillation method [55][56][57][58], our method, which does not require the time-consuming relaxation step and repeated operations, can work fast to realize enantioconversion of a chiral mixture and thus greatly promote the realization of laser-assisted enantioconversion based on the four-level double-∆ model in the reality.In our method, by well designing the three optical fields, the original four-level double-∆ model is simplified to two effective two-level sub-systems with each of them including one chiral ground state and one of two mutual orthogonal superposition states of the two achiral excited states.Then, the dynamics of the two chiral ground states are constrained in their own two-dimensional subspaces and are subjected to the same effective coupling strengths but different effective detunings.These properties make the left-right symmetry breaking in the dynamics of chiral molecules stand out and play center roles in our purely optical method for enantioconversion.
Model.-The considered four-level double-∆ model of chiral molecules [55][56][57][58]  Here, we are interested in reducing the original four-level double-∆ model to the simplified four-level model composed of two separated two-level sub-systems as shown in Figs.1(b) and 1(c) by selecting appropriate parameters of the optical fields.The frequencies of the three optical fields should be chosen following the three-photon resonance condition and the one-photon resonance condition of the transition |S ↔ |A as ( When the overall phase is tuned to be φ = 0, the dynamics of the two chiral ground states are restricted in their corresponding two-dimensional subspaces and described by the two-level sub-systems with Hamiltonian ĤL =( Optical-operation method.-Followingthese properties of our simplified models of two-level sub-systems, we now present our optical method for enantioconversion of a racemic mixture with each molecule initially depicted by the density matrix ρ0 = (|L L| + |R R|)/2.Our optical method includes three purely optical operations.For simplicity, we assume that the coupling strengths and the detunings are time-independent in each operation.In the first optical operation, we apply three optical fields to realize the simplified four-level model composed of two separated two-level sub-systems as shown in Fig. 2 For this, we assume that the coupling strengths and the detunings satisfy the following conditions [42] with integers n L > n R ≥ 0.Then, the operation ends when the chiral ground state |L evolves back to itself by experiencing integer n L periods of its corresponding Rabi oscillation.In the meanwhile, the chiral ground state |R experiences half-integer (n R + 1/2) periods of its corresponding Rabi oscillation and evolves to the state |D − , since its corresponding two-level model is on-resonance.Thus, by applying such an operation to the initial racemic mixture, we can achieve the finial state of ρ1 .
In the second optical operation, we turn off the two optical fields coupling with the transitions |Q ↔ |S and |Q ↔ |A , keep the optical field coupling with transition |A ↔ |S , and apply an auxiliary optical field to resonantly couple with the transition between each chiral ground state and a corresponding auxiliary chiral excited state with the same chirality as shown in Fig. 2 In the third optical operation, we turn off the auxiliary optical field and construct the simplified four-level model composed of two separated two-level sub-systems as shown in Fig. 2(c) with the overall phase φ = π [similar to Fig. 1(c)] by applying three optical fields.By taking the resonant condition in the two-level sub-system including the states |L and |D − Ω SA = ∆, (7) the occupied dressed state |D − can evolve to the wanted chiral ground state |L by experiencing half-integer periods of its corresponding Rabi oscillation.Thus, each molecule in the initial chiral mixture is converted to the one with the same chirality through In Fig. 2(d), we give an example of our opticaloperation method for converting a racemic sample to an enantiopure sample of the wanted left-handed chirality by choosing Ω aux = √ 2Ω and ∆ = √ 6Ω.We note that in each operation the (effective) coupling strengths can be chosen independently.Alternatively, when the righthanded chirality is wanted, we can realize the enantioconversion with the similar process as shown in Fig. 2 by replacing the overall phases φ in the first and third optical operations with π and 0, respectively, and replacing the auxiliary state |L ′ by the auxiliary one |R ′ .Then, each molecule will be finally depicted with ρ′ Advantage of our method.-Now,we have obtained the enantioconversion based on our optical-operation method within the simplified four-level model composed of two separated two-level sub-systems.The time cost of our purely optical method is of order of Ω −1 .For the laserdistillation method [55][56][57][58], the main time cost is attributed to the relaxation step.Generally, the relaxation time should be much larger than the time cost of optical operations since the typical coupling strengths are much larger than the relaxation rate.Then, our method can be much faster than the laser-distillation method [55][56][57][58] with the requirement of the relaxation step and repeated operations Specifically, the typical coupling strength Ω can be of order of 10 MHz [44][45][46][47][48][49][50][51][52] or even stronger [37,40], which means the time cost of our optical method is about (or less than) 0.1 µs.For the case of the buffer-gas experiments [44,45,51], the characteristic relaxation time is of order of 10 µs.The requirement of repeated operations (e.g. 10 times) in the laser-distillation method [55][56][57][58] will make its more time-consuming (e.g. 100 µs).Therefore, our purely optical method, which can be finished within about 0.1 µs, will be about three order of magnitude faster than the laser-distillation method [55][56][57][58].This is a promising advantage in the reality, since some chiral molecules rapidly racemize, and even for the chiral molecules with long enough racemization time, the overall operation time may be limited by experimental conditions.For an example, in the buffer-gas experiments [44,45,51], the overall operation time should be less than the diffusion time, out of which the molecules will lost.
Conclusions.-Inconclusion, we have proposed theoretically the optical-operation method for enantioconversion by simplifying the original four-level double-∆ model to two separated effective two-level sub-systems.With three well-designed purely optical operations, the initial chiral mixture can be converted to an enantiopure sample of the wanted chirality.Specifically, the initial unwanted chiral ground state is converted to the wanted chiral ground state and in the meanwhile the initial wanted chiral ground state is transferred to the auxiliary chiral excited state of the wanted chirality.Our purely optical method can work much faster than the laser-distillation method [55][56][57][58], where the relaxation step and repeated operations are required.This advantage is essential for the realization of enantioconversion when the operation time is limited due to the racemization and/or experimental conditions.

Figure 1 .
Figure 1.(a) The schematic of four-level double-∆ model of chiral molecules.The chiral ground state |Q (Q = L, R), the asymmetric achiral excited state |A , and the symmetric achiral excited state |S are coupled in ∆-type transitions |Q ↔ |A ↔ |S ↔ |Q by three optical fields with frequencies ω2, ω0, and ω1, respectively.Here Ω jk (j, k = L, R, A, S) are the corresponding (real) coupling strengths and satisfy ΩLS = ΩRS > 0, ΩLA = −ΩRA > 0 and ΩSA > 0; φ (φ + π) is the overall phase of the coupling strengths in the ∆-type transitions related to the left-handed chiral ground state |L (the right-handed one |R ).By assuming the three-photon condition of the ∆-type transitions, the one-photon resonance condition of the transition |A ↔ |S (that means ∆A = ∆S ≡ ∆ with ∆A and ∆S being the detunings corresponding to the transitions |Q ↔ |A and |Q ↔ |S ), and ΩLS = ΩLA ≡ Ω, the original model in (a) can be simplified to two effective two-level sub-systems when the overall phase of the coupling strengths in the ∆-type transitions is well adjusted as (b) φ = 0 or (c) φ = π.
) Then, the detunings ∆ S ≡ ω S − ω 1 and ∆ A ≡ ω A − ω 2 , which correspond to the transitions |Q ↔ |S and |Q ↔ |A respectively, satisfy ∆ A = ∆ S ≡ ∆.Under the rotating wave approximation, the system can be described by the following Hamiltonian in the interaction picture with respect to Ĥ0 = ω 2 |A A| + ω 1 |S S| as ( = 1) Ĥ =∆(|A A| + |S S|) + (Ω SA |S A| + H.c.) + Q=L,R (Ω QA e iφ |Q A| + Ω QS |Q S| + H.c.).(2) The symmetry breaking in the chirality is reflected in the coupling strengths [55-58] with Ω LS = Ω RS and Ω LA = −Ω RA .Without loss of generality, we assume Ω LA , Ω LS , and Ω SA are positive.Here φ is the overall phase of the ∆-type transitions |L ↔ |A ↔ |S ↔ |L .It can be determined by modifying the phases of the three optical fields.Moreover, the intensities of the three optical fields should be adjusted so that the coupling strengths of the transitions satisfy Ω LS = Ω LA ≡ Ω > 0.
where the dressed states and the corresponding detunings are |D ± = (|S ± |A )/ √ 2 and ∆ ± = ∆ ± Ω SA .Then, the original four-level double-∆ model is simplified to two separated effective two-level sub-systems in the basis {|L , |D + } and {|R , |D − } as shown in Fig. 1(b).Similarly, when φ = π, the original four-level double-∆ model can also be simplified to the two separated effective two-level sub-systems with the basis {|L , |D − } and {|R , |D + }, respectively, as shown in Fig. 1(c).The two-level sub-systems for the two chiral ground states are, respectively, described by Ĥ′ L = ( √ 2Ω|L D − | + H.c.) + ∆ − |D − D − | and Ĥ′ R = ( √ 2Ω|R D + | + H.c.) + ∆ + |D + D + |.Within our simplified models of two-level sub-systems as shown in Figs.1(b) and 1(c), the dynamics of the two chiral ground states are constrained in their own subspaces and are subject to the same coupling strengths but different detunings.These clearly demonstrate the left-right symmetry breaking in the dynamics of chiral molecules.Moveover, by changing the overall phase φ from 0 to π, the subspace for the two sub-systems change from {|L , |D + } and {|R , |D − } to {|L , |D − } and {|R , |D + }, which offers the possibility of converting one chiral ground state to another.

Figure 2 .
Figure 2. Optical-operation method of enantioconversion by simplifying the original four-level double-∆ model to two separated two-level sub-systems and introducing an auxiliary state |L ′ .(a,b,c) denote, respectively, the first, second, and third optical operations, where the ellipses stand for the occupied states at the end of each operation in the initial racemic sample with each molecule described by ρ0 = (|L L| + |R R|)/2.The corresponding evolution of the population in each state is plotted in panel (d).The dashed lines divide the evolution into three regions labeled with I, II, and III, which correspond to the operations in panels (a), (b), and (c), respectively.Here the optical fields applied in panels (a) and (c) are respectively similar to those in Fig. 1(b) [under the condition (9)] and Fig. 1(c) with ∆ = ΩSA [under the condition (12)].The other parameters are chosen to be Ωaux = √ 2Ω and ∆ = √ 6Ω.
(a) with the overall phase φ = 0 [similar to Fig. 1(b)].We aim to achieve that the wanted chiral ground state |L evolves back to the state |L and in the meanwhile the unwanted chiral ground state |R is transferred to the dressed state |D − during the first opera-That is, the initial density matrix of each molecule, ρ0 , becomes ρ1 = (|L L| + |D − D − |)/2 accordingly.
(b).The working Hamiltonian in the interaction picture under the rotating-wave approximation Ĥaux = (Ω aux |L ′ L| + H.c.) + 2Ω SA |D + D + |. (6) In this case, the wanted chiral ground state |L can evolve to the axillary chiral excited state of the wanted chirality |L ′ , e.g.experiencing half-integer periods of the Rabi oscillation governed by the following effective Hamiltonian (Ω aux |L ′ L| + H.c.).Meanwhile, the dressed state |D − remains unchanged as indicated by Eq. (6).Thus, at the end of the second operation, the density matrix of each molecule becomes ρ2 = (|L ′ L ′ | + |D − D − |)/2.