Uniformly locally chiral light for global control of chiral light-matter interaction

Locally chiral light (LCL) is a promising tool for probing and controlling molecular chirality. The pioneering tricolor LCLs' degrees of chirality (DOCs) change periodically in space and vanish by integrating over the whole spatial period, making them cease to be globally effective (i.e., globally achiral). Locally and globally chiral light (LGCL) is a type of more efficient LCL whose DOCs survive by integrating but still change periodically in space. Here, we propose a scheme to generate a new type of LCL called uniformly locally chiral light (ULCL). ULCLs are globally chiral and have spatially uniform DOCs, which makes them superior to current types of LCLs in studies of chiral molecules. By applying the ULCLs in an optical molecular-chirality switch, the racemic mixtures can be converted to enantiopure samples without size restrictions, where the perfect control of molecular chirality with a global efficiency of $100\%$ is predicted, which is twice that of LGCLs. Our work potentially constitutes the starting point for developing more efficient chiroptical techniques for probing and controlling molecular chirality.

Nowadays, developing more efficient LCLs for probing and controlling molecular chirality is becoming a new frontier in the studies of chiral molecules [21,22,[70][71][72][73][74][75]. In this context, chirality polarized light (CPL) [74] and locally and globally chiral light (LGCL) [21] were proposed. CPL is still globally achiral but has the polarization of chirality, just like a neutral and polarized onedimensional medium of alternating negative and positive charges. Although the angle-integrated emitted signals are the same for two enantiomers, the emission directions are highly enantioselective. By applying angle-resolved detection methods [74], two enantiomers can be distinguished without sample size restrictions.
LGCLs are globally chiral and thus more efficient than globally achiral LCLs.
LGCLs can be used in probing molecular chirality without sample size restrictions or the requirement of challenging angle-resolved detection [7,21]. Due to their global chirality, they are proposed to control molecular chirality [22] and chiral nuclear dynamics in chemical reactions [75], also without sample size restrictions. The LGCLs' DOCs still change periodically in space, which gives rise to a considerable reduction in the global efficiency of controlling molecular chirality. In other words, in the final product of the whole spatial period, there is still a considerable proportion of unwanted enantiomers. This effect limits the application of LGCLs in practice because unwanted enantiomers are usually inefficient or even cause serious side effects in pharmacology.
In this letter, we propose a scheme to generate a novel type of LCL that is globally chiral and whose DOCs remain invariant in space. Thus, we name this LCL uniformly locally chiral light (ULCL). Because ULCLs' DOCs remain invariant in space, the enantioselective responses of chiral molecules are spatially independent. ULCLs can perform better than current types of LCLs in studies of chiral molecules, especially in pursuing perfect control of molecular chirality. By applying the UL-CLs in a scheme similar to the optical molecular-chirality switch introduced by Král et al. [29], the perfect control of molecular chirality with a global efficiency of 100% is predicted, which means the racemic mixture can be converted to an enantiopure sample of the selected chirality without sample size restrictions. By comparison, the predicted global efficiencies obtained by using LGCLs and tricolor LCLs in the switch are 50% and 0%, respectively.
Uniformly locally chiral light.-Our scheme consists of two non-collinear sub-beams. The first sub-beam is polychrome with 2n i=0 ω i = 0, where the absolute values of ω i are the angular frequencies of the carrier waves. The overall field of the first sub-beam is linearly polarized. The second sub-beam is monochrome and exhibits circular polarization in the plane perpendicular to the polarization of the first sub-beam. Nevertheless, the ULCLs of our scheme can be realized with just two frequency components, as shown in Fig. 1 (a). The first sub-beam [see the left half-panel of Fig. 1 (a), labeled with c = 1], consists of the ω and 2ω carrier waves (2ω − ω − ω = 0); they are z-polarized and propagate in the x-direction. The second sub-beam [see the right half-panel of Fig. 1 (a), labeled with c = 2], is a circularly polarized ω field and propagates in the z-direction. The initial phases and amplitudes of the carrier waves are φ i and E i , respectively, with i = 0, 1, 2.
The ULCLs in Fig. 1 (a) are non-collinear and exhibit polarization in three dimensions. They have both even and odd harmonics of the fundamental frequency in the carrier waves. The even and odd harmonics are not perpendicular. According to the dynamical symmetry description of LCLs [21], these properties ensure that the ULCLs in Fig. 1 (a) are locally chiral. The chirality of LCL is also reflected in the time evolution of its electric field vector at a fixed position in space, whose tip draws a chiral 3D Lissajous curve [see the orange solids Fig. 1 (b)]. Similar to chiral molecules, the chiral 3D Lissajous curve [see the left half-panel of Fig. 1 (b)] can not be superposed on its mirror image [see the right half-panel of Fig. 1

(b)] by spatial rotations and temporal translations.
This property can be more clearly represented in the colored 2D version of the Lissajous curves [see the middle of Fig. 1 (b)]. The chiral Lissajous curves in Fig. 1 (b) further confirm that the ULCLs in Fig. 1 (a) are locally chiral. The overall fields of the ULCLs in Fig. 1 (a) at different positions can be superposed on each other by spatial rotations and temporal translations [see the colored 2D Lissajous curves in Fig. 1 (a)]. This property can be demonstrated analytically because the overall fields can be rearranged by using the rotated direction vectors of electric fieldsẽ µ (θ) (µ = X, Y, Z) with θ = k(z − x) and the translated time t ′ = t − kx/ω as Here, the components of the electric field vector are . Specifically, the rotated direction vectors of the electric fields areẽ Z (θ) = e z ,ẽ X (θ) = e x cos θ − e y sin θ, andẽ Y (θ) = e x sin θ+e y cos θ with the unrotated direction vectors e x , e y , and e z . According to the symmetry-based overlap measure of the degree of (local) chirality [21][22][23][24], Eq. (1) indicates that the DOCs at different positions in space are identical. In other words, the ULCLs in Fig. 1 (a) are uniformly locally chiral. In contrast, the DOCs for the tricolor LCLs and the LGCLs change periodically in space; they are not uniformly locally chiral. The LCLs' DOCs can also be characterized by the chiral correlation functions (CCFs) defined as h (n) (ω 0 , ..., [22], where F (ω) is a Fourier transform of the electric field vector at the angular frequencȳ ω. According to Eq. (1), all orders of the CCFs for the ULCLs in Fig. 1 (a) are spatially identical. For more details on the proof of this conclusion based on the unrotated direction vectors of electric fields, see Sec. I of the Supplementary Materials. In this sense, we reconfirm that the DOCs for the ULCLs in Fig. 1 (a) are spatially uniform.
Enantioselective induced polarization.-Moreover, the CCFs also predict the strengths of enantioselectivity in molecular responses resulting from electric-dipole interactions between LCLs and chiral molecules [22]. Specifically, we focus on h (5) , which is the lowest order of CCFs for two-color LCLs [21]. For the ULCLs in Fig. 1 (a), we have with φ = φ 2 − 2φ 1 . As expected, they are nonzero and spatially uniform. We have assumed that the spatial envelope of each electromagnetic field is negligible in the interaction region, which is made available by using light beams with appropriate waist radii. The CCF in Eq. (2) governs the enantioselectivity in the induced polarization |P(2ω) [21]. The linear and fourth-order susceptibilities in randomly oriented samples for two enantiomers are χ 1,L = χ 1,R and χ 4,L = −χ 4,R , where subscripts Q(= L, R) denote the molecular chirality. Then, the induced polarizations are different for two enantiomers [see Fig. 2 (a)]. The molecular chirality can be decided by monitoring the electromagnetic fields attributed to the induced polarizations [34]. Figure 2 (a) shows the spatially uniform optimization of enantioselectivity in |P(2ω)| 2 . In other words, the full suppression in one enantiomer (blue dotted line) and maximal enhancement in the other with opposite chirality (red solid line) are simultaneously obtained at any point in space. For the ULCLs in Fig. 1 (a), such a case can be achieved at |χ 1,L |E 2 = |χ 4,L |E 2 0 E 2 1 and φ = φ χ1,L − φ χ4,L ∓ π/2 with χ i,L = |χ i,L | exp (iφ χi,L ) by tuning the relative phases and strengths between the carrier waves in the c = 1 sub-beam. In contrast, the tricolor LCLs and the LGCLs give rise to spatially oscillated induced polarizations |P(2ω)| 2 [see Fig. 2 (b) and Fig. 2 (c)]. The optimization of enantioselectivity in |P(2ω)| 2 is only achieved at some points in space. The tricolor LCLs' global responses obtained by integrating over the whole spatial period will even become identical Globally perfect molecular-chirality switch.-The UL-CLs can also be used to control the molecular chirality of racemic samples by using a two-step molecular-chirality switch similar to that introduced by Král et al. [29]. In the first step, we consider a transition of chiral molecules |g Q ↔ |e Q with ω eg = 2ω. The effective coupling strength attributed to the ULCLs in Fig. 1 (a) is proportional to the induced polarization as |Ω| ∝ |P(2ω)|. For more details about the derivation of this relation, see Sec. IV of the Supplementary Materials. By well choosing the relative phases and strengths of the carrier waves, the effective coupling strength of one enantiomer is zero and that of the other with opposite chirality is nonzero. The results in Fig. 2 (a) correspond to the case of |Ω R | = 0 and |Ω L | = 0. When the decoherence becomes negligible, after a π-pulse, the right-handed molecules are transferred to the chiral excited state, and the left-handed molecules remain in the ground state (i.e., the perfectly enantioselective excitation of chiral molecules).
In the second step, we apply another light to couple the chiral excited states to an achiral excited state in the large-detuning limit, yielding an effective coupling between two chiral excited states [51]. After a π-pulse, we transfer the racemic sample of ρ i = |g L g L | + |g R g R | to the enantiopure sample of ρ f = |g L g L | + |e L e L |. Equation (2) also indicates that the ULCLs change to their mirror images, and the responses of two enantiomers exchange with each other by changing the overall phase φ by π or changing the sign of σ. By tuning the applied ULCL to its mirror image, the final state of the system will become ρ f = |g R g R | + |e R e R |. Then, we can switch the racemic samples to enantiopure samples of the selected chirality. The efficiency of the molecularchirality switch is given by the enantiomeric excess defined as e.e. ≡ |N L − N R |/(N L + N R ), where N L and N R are the particle numbers of two enantiomers in the final product. As shown in Fig. 3 (a), the optimized global efficiency of the ULCLs in controlling molecular chirality is 100% at any position in space; in other words, the globally perfect control of molecular chirality can be obtained without sample size restrictions. For simplicity and without loss of generality, we have used a toy model to describe the enantioselective excitation. The analyses based on this model can be extended to other types of LCLs, which provides a simple way to compare the global efficiencies of different types of LCLs. For this purpose, we apply the tricolor LCLs and the LGCLs, which are typical types of globally achiral and chiral LCLs, in the first step of the molecularchirality switch. After the two steps, the corresponding optimized results for the two cases are shown in Fig. 3 (b) and Fig. 3 (c), respectively. The perfect control of molecular chirality is only achieved at some points in space. For the two cases, the corresponding global efficiencies obtained by integrating over the whole spatial period are 0% and 50%, respectively. Therefore, the ULCLs are superior to current types of LCLs in pursuing perfect control of molecular chirality. Perfectly enantioselective excitation without sample size restrictions can also be realized by using a four-level microscopic model of chiral molecules, which is driven by other ULCLs of our scheme. For more details on the perfectly enantioselective excitation by using the microscopic model, see Sec. V of the Supplementary Materials. Physical mechanism of uniformly locally chiral light.-To illustrate the physical mechanism of ULCLs, we rewrite the nonzero CCFs as the summation of terms corresponding different pathways h (n) (ω 0 , ...,ω n−1 ) = , where c i (= 1, 2) denotes the Fourier transform F ci coming from the c i sub-beam. In Fig. 4 (a), we show that the pathway of the lowest order of CCFs for the ULCLs can be divided into cyclic odd-and even-photon processes [see the dotted and solid cycles therein]. The two cyclic processes are evoked by the two sub-beams, respectively. The spacedependent phases k i · r of F 1 (−2ω), F 1 (ω), and F 1 (ω) in the dotted cycle cancel out each other, as do the spacedependent phases of F 2 (ω) and F 2 (−ω) in the solid cycle. Here, k i and r are the wave vector of each carrier wave and the position vector of the chiral molecule in space, respectively. Then, the corresponding terms of such pathways in the summation of CCFs are spatially identical. This property of the pathways for ULCLs is attributed to the well-designed polarization structure of the two sub-beams. For more details on the proof of this concept, see Sec. I of the Supplementary Materials. Therefore, in the general cases of our scheme, the CCFs are spatially identical (i.e., the generated electromagnetic fields are uniformly locally chiral).
For the tricolor LCLs and the LGCLs, there is another type of pathway [see Fig. 4 (b) and the right half-panel of Fig. 4 (c)]. The cyclic odd-photon process cannot be divided into two sub-cycles respectively attributed to the two sub-beams (see the dashed-dotted cycles therein). In such pathways, the space-dependent phases cannot be canceled out, which makes the corresponding terms in the summation of CCFs change periodically in space and vanish by integrating over the whole spatial period. In the tricolor LCLs [see Fig. 4 (b)], the CCFs' pathways are the second type, so the tricolor LCLs are locally chiral but globally achiral. The LGCLs' CFFs contain both types of pathways [see Fig. 4 (c)], so the LGCLs are globally and locally chiral but not uniformly chiral. In this sense, Figure 4 reveals the fundamental difference in the physical mechanism between the ULCLs, the globally achiral LCLs, and the LGCLs.
Conclusions.-We have proposed a scheme to generate ULCLs, whose DOCs are identical in space. By using a two-color example of ULCLs, we have demonstrated that the ULCLs can generate enantioselective and spatially uniform responses of chiral molecules. These properties make the ULCLs superior to other types of LCL in probing and controlling molecular chirality in terms of concepts including enantiodetection, enantio-specific state transfer, enantioseparation, and enantioconversion. Our work potentially constitutes the starting point for developing more efficient chiroptical techniques for studies of chiral molecules, which will eventually benefit the developing related areas in chemical reactions, biological activities, drug industries, and life's chemistry.