Evading thermal population influence in enantiomeric-specific state transfer based on a cyclic three-level system via ro-vibrational transitions
EEvading thermal population influence on enantiomeric-specific statetransfer based on a cyclic three-level system via ro-vibrational transitions
Quansheng Zhang, Yu-Yuan Chen, Chong Ye,
2, 1, a) and Yong Li
1, 3, b) Beijing Computational Science Research Center, Beijing 100193, China Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of Technology,100081 Beijing, China Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081,China (Dated: November 16, 2020)
Optical methods of enantiomeric-specific state transfer had been proposed theoretically based on a cyclic three-levelsystem of chiral molecule. According to these theoretical methods, recently the breakthrough progress has been re-ported in experiments [S. Eibenberger et al. , Phys. Rev. Lett. , 123002 (2017); C. P´erez et al. , Angew. Chem. Int.Ed. , 12512 (2017)] for cold gaseous chiral molecules but with achieving low state-specific enantiomeric enrichment.One of the limiting factors is the influence of the thermal population in the selected cyclic three-level system based onpurely rotational transitions in experiment. Here, we theoretically explore the improvement of the enantiomeric-specificstate transfer at finite temperature by introducing ro-vibrational transitions for the cyclic three-level system of chiralmolecules. Then, at the typical temperature in experiments, approximately only the lowest state in the desired cyclicthree-level system is thermally occupied and the optical method of enantiomeric-specific state transfer works well.Comparing with the case of purely rotational transitions where all the three states are thermally occupied, this modi-fication will remarkably increase the obtained state-specific enantiomeric enrichment with enantiomeric excess beingapproximately 100%. I. INTRODUCTION
Molecular chirality has played a crucial role in the enantio-selective biological and chemical processes , homo-chiralityof life , and even fundamental physics . It has attracted con-siderable interests to realize enantioseparation and enantiodis-crimination not only in chemistry but also in atomic, molec-ular, and optical physics . Among these, solelyoptical (or microwave) methods with the framework of cyclicthree-level ( ∆ -type) system (CTLS) based on electronic-dipole transitions, have caught the attention to realize enan-tioseparation as well as enantiodiscrimination andenantioconversion .The CTLS of chiral molecules is special since the electric-dipole transition moments for the three transitions are propor-tional to the projections of the electric-dipole transition mo-ment onto the three inertial axes of gaseous chiral molecules,respectively. It is well known that their product changes signfor the enantiomers . This inherent symmetry of the sys-tem can be mapped on the coupling strengths in the CTLSof chiral molecules , i.e., the product of the three couplingstrengths changes sign for the enantiomers. Using this feature,one can achieve the enantiomeric-specific state transfer (also named as inner-state enantioseparation ), i.e., for aninitial racemic chiral mixture, finally the populations in cer-tain energy levels (e.g. the ground states) will be differentfor the enantiomers. Based on the CTLS via optical meth-ods such as adiabatical processes , a dynamic operationof ultrashort pulses , and shortcuts-to-adiabaticity opera-tions , in principle the perfect enantiomeric-specific state a) Electronic mail: [email protected] b) Electronic mail: [email protected] transfer with 100% enantiomeric excess can be achieved withthe two enantiomers occupying different-energy levels. Fur-thermore, the left- and right-handed molecules in different-energy levels can be spatially separated by a variety of energy-dependent processes . In addition, based on the similarCTLS, one can also achieve directly the spatial enantiosep-aration by mean of the chiral generalized Stern-Gerlach ef-fect , that is, the two enantiomers move along differentspatial trajectories.Recently, breakthrough experiments have reportedthe enantiomeric-specific state transfer for gaseous chiralmolecules with the similar ideas proposed in Refs. . How-ever, the obtained enantiomeric enrichment is only about6% . One of the important factors limiting the obtainedenantiomeric enrichment is that each of the three selected ro-tational states in the CTLS may have multiple degeneratedmagnetic sub-levels. That means the considered CTLS willbe of multi loops. Thus, the ability of enantiomeric-specificstate transfer will be suppressed . Later on, it was pointedout in the theoretical work that the real single-loop CTLSof gaseous chiral molecules of asymmetric top can be con-structed by appropriately choosing three optical (microwave)fields. Another important factor limiting the obtained enan-tiomeric enrichment is the thermal population on the threelevels of CTLS. In those experiments , the CTLS con-sists of only rotational transitions, whose transition frequen-cies are usually at microwave wavelengths. Thus, the popula-tions in the three levels will approximately have the same or-der of magnitude at the typical effective rotational temperature ∼
10 K according to the Boltzmann distribution. This willbring an adverse influence on the efficiency of enantiomeric-specific state transfer based on the CTLS.In this paper, we aim to evade the adverse influence of thethermal population on the enantiomeric-specific state trans- a r X i v : . [ qu a n t - ph ] N ov fer. For this purpose, we re-construct the desired CTLS of(gaseous) chiral molecules by choosing the ground state andother two excited states to have different vibrational sub-levels. Then the transition frequencies between the groundstate of the desired CTLS and the two excited states are muchlarger than that of the previous ones as used in the experi-ments , where all the three states have the same vibra-tional sub-level. Thus, in the desired CTLS, approximatelyonly the ground state is occupied and the adverse influenceof the thermal population on the enantiomeric-specific statetransfer can be evaded. We use the 1,2-propanediol as an ex-ample to demonstrate our idea. Specifically, the vibrationalsub-level of the ground state (the two excited states) in thedesired CTLS is chosen to be the ground state (first excitedstate) of the vibrational degree of freedom corresponding toOH-stretch. The transition frequency between the two chosenvibrational sub-levels is about π × . THz . It is in-deed that almost only the ground state in the desired CTLSis thermally occupied initially in a wide range of temperature( − K). Thus, the influence of the thermal population onthe enantiomeric-specific state transfer is evaded.The structure of this paper is organized as following. InSec. II, we describe the optical method of the enantiomeric-specific state transfer based on a dynamic operation of ultra-short pulses. In Sec. III, we investigate the influence of finitetemperature on the enantiomeric-specific state transfer, andcompare the obtained state-specific enantiomeric enrichmentvia ro-vibrational transitions with that in the same vibrationalcase via purely rotational transitions. Finally, we summarizethe conclusions in Sec. IV.
II. DYNAMIC OPERATION OF ULTRASHORT PULSES
We now consider the CTLS of a chiral molecule coupledwith three classical electromagnetic (optical or microwave)fields as shown in Fig. 1(a). Then the Hamiltonian forthe system in the interaction picture with respect to H Q = (cid:80) n (cid:126) ω n | n (cid:105) QQ (cid:104) n | reads H Q ( t ) = (cid:88) m>n =1 (cid:126) Ω Qnm ( t ) e i ∆ mn t | n (cid:105) QQ (cid:104) m | + H . c ., (1)where (cid:126) ω n is the eigen-energy of state (level) | n (cid:105) Q with Q = L ( Q = R ) denoting the left-handed (right-handed) chi-ral molecule. The detuning for the transition | m (cid:105) Q → | n (cid:105) Q is defined as ∆ mn = ν mn − ω m + ω n , where ν mn is thefrequency of the classical electromagnetic field coupling tothe corresponding transition with the coupling strength (alsocalled Rabi frequency) Ω Qmn ( t ) . We can specify the chirality-dependence of our model via choosing the coupling strengthsof the left- and right-handed molecules as Ω Lmn ( t ) =Ω mn ( t ) , Ω R ( t ) = − Ω ( t ) , Ω R ( t ) = Ω ( t ) , and Ω R ( t ) =Ω ( t ) . That is, the overall phase in the CTLS differs with π for the two enantiomers: φ R = φ L + π , as seen in Fig. 1(a).In the following, we will briefly give the protocol of dy-namic operation of ultrashort pulses for the enantiomeric-specific state transfer, which is very similar to that of current (cid:28595) Figure 1. (Color online) (a) Model of CTLSs for left- and right-handed molecules. The three-level system is resonantly coupled tothe three classical fields with coupling strengths ± Ω , Ω , and Ω , respectively. The overall phase is depicted along the circlearrow with φ L and φ R for left- and right-handed molecules, re-spectively. They satisfy φ R = φ L + π . (b) Schematic represen-tation of ultrashort optical pulses to achieve perfect enantiomeric-specific state transfer with the molecule initially prepared in theground state, e.g. | (cid:105) Q . The related coupling strengths satisfy (cid:82) t A t Ω ( t ) dt = π/ (cid:82) t B t A Ω ( t ) dt/ − (cid:82) t C t B Ω ( t ) dt with Ω ( t ) = | Ω ( t ) | = − i Ω ( t ) = Ω ( t ) / √ . experiments . As depicted in Fig. 1(b), this protocol con-sists of three steps. In each step, the transitions of interest | m (cid:105) Q → | n (cid:105) Q are driven resonantly, i.e. ∆ mn = 0 .In step A, only the pump pulse Ω ( t ) is turned on. The cor-responding Hamiltonian reads H QA ( t ) = (cid:126) Ω Q ( t )( | (cid:105) QQ (cid:104) | +H . c . ) by assuming coupling strength Ω Q . By controlling thepulse to make it satisfy (cid:82) t A t Ω ( t ) dt = π/ , a general state(density matrix) ρ Q at time t will evolve to ρ QA = U QA ρ Q U Q † A at time t A according to the unitary evolution operator U QA =exp( − i (cid:82) t A t H QA dt/ (cid:126) ) . In the basis {| (cid:105) Q , | (cid:105) Q , | (cid:105) Q } , theunitary evolution operators U QA are U LA = √ − i √ − i √ √ , U RA = √ i √ i √ √ . (2)In step B, we turn off the pump pulse Ω [i.e. Ω ( t ) = 0 ]and introduce two ultrashort pulses Ω ( t ) and Ω ( t ) with Ω ( t ) = | Ω ( t ) | = − i Ω ( t ) ≡ Ω ( t ) / √ . The Hamilto-nian for this step reads H QB ( t ) = (cid:126) Ω ( t )( | D (cid:105) QQ (cid:104) | + H . c . ) with | D (cid:105) Q = ( i | (cid:105) Q + | (cid:105) Q ) / √ . Under the condition (cid:82) t B t A Ω ( t ) dt = π/ , the state will go to ρ QB = U QB ρ QA U Q † B with the unitary evolution operators U LB = U RB =
12 1 √ − i − √ − i √ i − i √ (3)in the basis {| (cid:105) Q , | (cid:105) Q , | (cid:105) Q } .The step C is realized by taking Ω ( t ) = Ω ( t ) = 0 andre-turning on the pump pulse Ω ( t ) . The pump pulse Ω ( t ) fulfills (cid:82) t C t B Ω ( t ) dt = − π/ [or equivalently ( k + 3 / π with integer k ]. Then, the final state reads ρ QC = U QC ρ QB U Q † C with the unitary evolution operators U LC = √ i √ i √ √ , U RC = √ − i √ − i √ √ . (4)With the above three operational steps, the initial state ρ Q will go to the final one ρ QC = U Q ρ Q U † Q with the total unitaryevolution operator U Q = U QC U QB U QA given as U L = − i − i , U R = − . (5)This indicates that the two enantiomers will suffer differentevolutions even when their initial states have the same form.Specially, U L will exchange the populations of left-handedmolecules in states | (cid:105) L and | (cid:105) L , while U R will exchange thepopulations of right-handed molecules in states | (cid:105) R and | (cid:105) R .When the above protocol is applied to the chiral mixture com-posed of chiral molecules in the two ground states | (cid:105) Q , theperfect enantiomeric-specific state transfer is achieved accord-ing to Eq. (5), since the initial state | (cid:105) L will evolve back toitself and the initial state | (cid:105) R in the meanwhile will be trans-ferred to | (cid:105) R . After that, the left- and right-handed moleculesin different-energy states can be further spatially separated bya variety of energy-dependent processes . III. TRANSITION BETWEEN DIFFERENTRO-VIBRATIONAL STATES
As discussed in the above section, the perfect enantiomeric-specific state transfer can be achieved if the left- and right-handed molecules has been initially prepared in the groundstate | (cid:105) Q of the CLTS. In experiments, the effective rotationaltemperature of chiral mixture is usually cooled to be ∼
10 Kwith recent technologies, such as buffer gas cooling andsupersonic expansions . The initial thermal population ineach state of the CTLS based on purely rotational transitionswill have the same order of magnitude. This will bring anadverse influence on achieving perfect enantiomeric-specificstate transfer .Now, we investigate the thermal population influence onthe enantiomeric-specific state transfer based on the desiredCTLS of chiral molecules at finite temperature. For thegaseous molecules, the rotational and vibrational degrees offreedom should be taken into account by adopting the ro-vibrational state | ψ (cid:105) = | ψ vib (cid:105) | ψ rot (cid:105) , (6)with the product form of the vibrational state | ψ vib (cid:105) and therotational state | ψ rot (cid:105) . Here, we have taken the rigid-rotor approximation , which means the coupling between vibra-tional and rotational states under field-free conditions can beneglected. This assumption is available for some kinds of chi-ral molecules, but is not available for some others. In thispaper, our discussions focus only on the case where the rigid-rotor approximation is available. A. Enantiomeric-specific state transfer at presence ofthermal population
In realistic experiments using buffer gas cooling , thevibrational relaxation cross sections of molecule-buff gas areusually smaller than rotational ones. This indicates the effec-tive vibrational temperature T vib of chiral molecules is typi-cally higher than the effective rotational temperature T rot .Accordingly, we assume that the effective rotational temper-ature to be T rot ∼
10 K and the effective vibrational tempera-ture T vib = 300 K.For the three-level system, the initial thermal populationin the ro-vibrational state | n (cid:105) Q with the eigen-energy (cid:126) ω n = (cid:126) ( ω n, vib + ω n, rot ) has the Boltzmann-distribution form p n = 1 Z P n, vib P n, rot . (7)Here, the factor P n, vib ( P n, rot ) is defined as P n, vib =exp( − (cid:126) ω n, vib /k B T vib ) [ P n, rot = exp( − (cid:126) ω n, rot /k B T rot ) ]with the vibrational (rotational) eigen-energy (cid:126) ω n, vib ( (cid:126) ω n, rot ), and Z = (cid:80) n =1 P n, vib P n, rot is the partitionfunction. The initial state of the CTLS for enantiomers reads ρ Q = p | (cid:105) QQ (cid:104) | + p | (cid:105) QQ (cid:104) | + p | (cid:105) QQ (cid:104) | , (8)with p + p + p = 1 .By means of the dynamic operation of ultrashort pulses asdiscussed in the above section, the states ρ L,R for enantiomerswill evolve differently under the different evolution operators U L,R . Then the final state for the left-handed molecules isgiven as ρ LC = p | (cid:105) LL (cid:104) | + p | (cid:105) LL (cid:104) | + p | (cid:105) LL (cid:104) | , (9)and the final state for the right-handed molecules will go to ρ RC = p | (cid:105) RR (cid:104) | + p | (cid:105) RR (cid:104) | + p | (cid:105) RR (cid:104) | . (10)By comparing Eq. (9) with Eq. (10), there are different pop-ulations in energy-degenerated states | n (cid:105) L and | n (cid:105) R ( n =1 , , . Then, the (imperfect) enantiomeric-specific statetransfer has been achieved. Specially, we now focus on thepopulations in the state | (cid:105) by introducing the enantiomericexcess (cid:15) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ LC, − ρ RC, ρ LC, + ρ RC, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) p − p p + p (cid:12)(cid:12)(cid:12)(cid:12) , = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −
21 + P , vib P , vib × P , rot P , rot (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11)with ρ QC, = Q (cid:104) | ρ QC | (cid:105) Q . It gives the excess of oneenantiomer over the other in the state | (cid:105) . The perfectenantiomeric-specific state transfer is achieved, when the state | (cid:105) is occupied by enantio-pure molecules with (cid:15) = 100% .For the selected three states of the CTLS with same vibra-tional state ( ω , vib = ω , vib = ω , vib ) , (cid:15) will be only depen-dent on the rotational transition frequency ( ω , rot − ω , rot ) .If we choose different vibrational states with ω , vib (cid:54) = ω , vib ,the ratio ( P , vib / P , vib ) , which is usually much larger than 1,will bring an enormously contribution to get a much higher (cid:15) .This offers the possibility of evading the thermal populationinfluence on enantiomeric-specific state transfer by introduc-ing ro-vibrational transitions in the CTLS. B. Examples of 1,2-propanediol
Particularly, we will demonstrate the above idea by usingchiral molecules of 1,2-propanediol as an example. Suchmolecules have been considered in the experimental worksof enantiomeric-specific state transfer and enantiodiscrim-ination . The rotational constants for 1,2-propanediolmolecules are A/ π = 8 . GHz, B/ π = 3 . GHz,and C/ π = 2 . GHz . The corresponding three ro-vibrational states are selected as | (cid:105) = | g (cid:105)| J τM = 0 (cid:105) , | (cid:105) = | e (cid:105) | (cid:105) , and | (cid:105) = | e (cid:105) | (cid:105) , where | g (cid:105) and | e (cid:105) are, respectively, the vibrational lowest and first-excitedstates for the motion of OH-stretch with the transition fre-quency ω vib / π = 100 . THz . Here, we have adoptedthe | J τM (cid:105) nomenclature to designate the rotational states ofasymmetric-top molecules. J is the total angular momentum, τ runs from − J to J in unit step in order of ascending energy,and M is the magnetic quantum number corresponding tothe degenerated magnetic sub-levels.Figure 2(a) displays the CTLS of 1,2-propanediol for thecase of ro-vibrational transitions. The corresponding initialthermal population with different rotational temperature T rot for each selected state of the CTLS is numerically shown inFig. 2(c) at the effective vibrational temperature T vib = 300 Kaccording to Eq. (7). As the vibrational transition frequencyof OH-stretch is about π × . THz, there will be littleinitial thermal occupation for the excited states even at highrotational temperature (e.g. T rot = 300 K). In other words,among the selected three states | (cid:105) , | (cid:105) , and | (cid:105) in the CTLS,we can assume that initially only the ground state | (cid:105) is ther-mally populated.As a comparison, we consider the case of purely rota-tional transitions in the CTLS with the same vibrational state | ψ vib (cid:105) = | g (cid:105) . The rotational transitions among | (cid:105) = | g (cid:105)| (cid:105) , | (cid:105) = | g (cid:105)| (cid:105) , and | (cid:105) = | g (cid:105)| (cid:105) of the CTLS are dis-played by Fig. 2(b), and the corresponding thermal popula-tions are shown in Fig. 2(d). As expected, the excited states | (cid:105) = | g (cid:105)| (cid:105) and | (cid:105) = | g (cid:105)| (cid:105) have a considerable pop-ulation at the typical rotational temperature T rot = 10 K inexperiments. (cid:28595)
Figure 2. (Color online) The CTLS of 1,2-propanediol for (a) thecase of ro-vibrational transitions among | (cid:105) = | g (cid:105)| J τM = 0 (cid:105) , | (cid:105) = | e (cid:105)| (cid:105) , and | (cid:105) = | e (cid:105)| (cid:105) ; (b) the case of purely ro-tational transitions among | (cid:105) = | g (cid:105)| (cid:105) , | (cid:105) = | g (cid:105)| (cid:105) , and | (cid:105) = | g (cid:105)| (cid:105) . Here, | g (cid:105) and | e (cid:105) are, respectively, the vibra-tional lowest and first-excited states for the vibrational motion ofOH-stretch with the corresponding vibrational transition frequency ω vib / π = 100 . THz. By choosing three special polarizedelectromagnetic fields, the real single loop CTLS come true . Pan-els (c) and (d) show the thermal populations of the three levels in theCTLS versus the effective rotational temperature with the effectivevibrational temperature T vib = 300 K according to the cases (a) and(b), respectively.
Ro-vibrational Transition Purely Rotational Transition
Figure 3. (Color online) The enantiomeric excess for the case ofro-vibrational transitions (red dashed line) or purely rotational tran-sitions (blue solid line) as a function of rotational temperature T rot with vibrational temperature T vib = 300 K. The other parametersare the same as those in Fig. 2.
For the racemic mixture of chiral molecules, the initialthermal populations in the states | n (cid:105) L and | n (cid:105) R will be thesame. According to the method of the dynamic operation ofultrashort pulses as discussed in Sec. III A, the enantiomeric-specific state transfer can be realized. Specifically, we givethe enantiomeric excess (cid:15) to depict the obtained state-specificenantiomeric enrichment. The enantiomeric excess (cid:15) as afunction of rotational temperature T rot with vibrational tem-perature T vib = 300 K is shown in Fig. 3. It shows the enan-tiomeric excess (cid:15) is nearly (see the blue solid line inFig. 3) for the case of ro-vibrational transitions with the effec-tive rotational temperature T rot in a wide temperature range,even at K. However, for the case of purely rotational tran-sitions, (cid:15) will rapidly decrease when T rot (cid:38) . K (see the reddashed line in Fig. 3). For instance, at T rot = 10 K, whichis near to the cooled rotational temperature achieved in theexperiment for the chiral molecules of 1,2-propanediol , (cid:15) isonly about . C. Ratio of pure enantiomers to the chiral mixture
Based on the chiral-molecule CTLS composed of ro-vibrational transitions instead of purely rotational ones, wehave shown above that the thermal population influence onenantiomeric-specific state transfer is negligible. In the realis-tic case, besides the selected three states (i.e., | (cid:105) = | g (cid:105)| (cid:105) , | (cid:105) = | e (cid:105) | (cid:105) , and | (cid:105) = | e (cid:105) | (cid:105) ) in the desired CTLS, theother states out of the CTLS will also be thermally occupied.For the left- or right-handed molecules, we can introduce P n ( n = 1 , , ) to describe the proportion of state | n (cid:105) to the re-lated total population as P n = 1 Z tot P n, vib P n, rot , (12)where Z tot = (cid:80) j P j, vib P j, rot is the total partition functionwith j summing for all the states | v (cid:105) | J τ M (cid:105) . Here | v (cid:105) repre-sents the vibrational ground state | g (cid:105) , vibrational first excitedstate | e (cid:105) , and vibrational higher-energy excited ones.The proportions P n of the selected ro-vibrational states | (cid:105) , | (cid:105) , and | (cid:105) are shown in Fig. 4(a). When T rot ≤ . K, P approximates to 1 and P , approximates to 0, which meansmost molecules are in the ground state of the CTLS. Then, af-ter a single operation of enantiomeric-specific state transfer asgiven in Sec. III A, most right-handed molecules will occupy | (cid:105) R , while the left-handed molecules will be approximatelynot populated in | (cid:105) L . Moreover, via the energy-dependentspatially separated processes , the pure enantiomers withnearly all the right-handed molecules can be obtained fromthe chiral mixture. When T rot > . K, the total proportionsin the CTLS ( P + P + P (cid:39) P ) decrease rapidly with theincrease of the temperature. For an example, at T rot = 10 K, P (cid:39) . ( P + P (cid:39) ). Then, only 0.1% pure enantiomerswith right-handed molecules can be obtained. Figure 4. (Color online) (a) The proportion for the selected threestates | (cid:105) (black dashed line), | (cid:105) (red solid line), and | (cid:105) (blue dottedline) when all of the other vibrational and rotational states out ofthe CTLS are considered; (b) The ratio of pure enantiomers to theracemic mixture at different rotational temperature T rot . The otherparameters are the same as those in Fig. 2. It is clearly that the ratio of the obtained pure enantiomersto the chiral mixture, η , is determined by the initial populationin | (cid:105) R . Specifically, we have η = 12 P , (13)where the factor / results from the fact that the racemicmixture consists of equal amounts of left- and right-handedmolecules. Accordingly, we show in Fig. 4(b) the ratio η at different rotational temperature T rot . When the rotationaltemperature T rot > η will decrease rapidly. Inorder to get more pure enantiomers from the chiral mixture,the rotational temperature should be further lowered and/orthe processes of the enantiomeric-specific state transfer aswell as the subsequent spatial separation should be repeated. IV. CONCLUSION
In conclusion, we have theoretically explored the thermalpopulation influence on the enantiomeric-specific state trans-fer based on the CTLS of chiral molecules by considering thero-vibrational transitions rather than purely rotational tran-sitions. Correspondingly, two infrared pulses and one mi-crowave pulse are used to respectively couple with the ro-vibrational and rotational transitions. This scheme was firstproposed for enantiomeric-specific state transfer in Ref. .Besides the promising features of this scheme emphasized inRef. , we have thoroughly discussed its advantage in evad-ing thermal population influence on the enantiomeric-specificstate transfer over the three-microwave-pulses scheme by cou-pling purely rotational transitions with the example of 1,2-propanediol. The adverse influence of thermal populationin other methods of enantioseparation and enantiodis-crimination based on the CTLS can also be evadedby adopting our similar idea.Usually, the ro-vibrational transition momenta correspond-ing to infrared pulses are typically smaller than the rotationaltransition momenta corresponding to microwave pulses. Sincethe intensity of experimental available infrared pulse will bemuch stronger than that of microwave pulse, the magnitude ofthe coupling strengths of these three transitions in our schemecan be comparable . It is worth noting that the operationtime of enantiomeric-specific state transfer based on the CTLScan be about ns in experiments . That means such amethod is implementable comparing with the relaxation time µ s . In addition, the precise control of relative phases of thethree fields plays an important role for enantiomeric-specificstate transfer, which has been realized experimentally in thethree-microwave-pulses scheme . As for the scheme byconsidering the ro-vibrational transitions, it had been theoret-ically proposed in Ref. that the precise control of relativephases of the three fields may be realized by phase-lockingtwo infrared pulses to a common reference standard (e.g., fre-quency comb) controlled by a microwave reference pulse.Moreover, the polarizations of the three electromagneticpulses can not be in the same plane when the molecularrotations are considered . This will bring the phase-matching problem in realistic cases and eventually reducethe achieved enantiomeric enrichment greatly . In experi-ments , phase-matching condition is approximately satis-fied, when the length scale of the practical excitation volumesis much smaller than the largest wavelength of three appliedfields driving the chiral molecules . Considering this, ourscheme with the largest wavelength in the micreowave re-gion has the advantage over the three-infrared-pulses schemewhich can be also used to evade thermal population influencein enantiomeric-specific state transfer. ACKNOWLEDGMENTS
This work was supported by the National Key R&D Pro-gram of China grant (2016YFA0301200), the Science Chal-lenge Project (under Grant No. TZ2018003) and the NaturalScience Foundation of China (under Grants No. 11774024,No. 11534002, No. U1930402, and No. 11947206).
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