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

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

Chong Ye, Quansheng Zhang, Yu-Yuan Chen, and Yong Li

Phys. Rev. Research 2, 033064 – Published 14 July 2020

Abstract

Based on the four-level double- $Δ$ model composed of two degenerated (left- and right-handed) chiral ground states and two achiral excited states, we propose a purely coherent-operation method for enantioconversion of chiral mixtures. By choosing appropriate parameters, the original four-level model will be simplified to two effective two-level subsystems with each of them involving one chiral ground state. Then, with the help of well-designed coherent 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 chiral mixtures. Comparing with the original works of enantioconversion based on the four-level double- $Δ$ 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 coherent operations. Thus our method offers a promising candidate for fast enantioconversion of chiral molecules with short enantiomeric lifetime or in the experimental conditions where the operation time is limited.

Received 21 November 2019
Revised 10 February 2020
Accepted 17 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033064

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Coherent control

Molecules

Atomic, Molecular & Optical

Authors & Affiliations

Chong Ye ¹, Quansheng Zhang ¹, Yu-Yuan Chen¹, and Yong Li ^1,2,*

¹Beijing Computational Science Research Center, Beijing 100193, China
²Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China

^*liyong@csrc.ac.cn

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Vol. 2, Iss. 3 — July - September 2020

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Atomic and Molecular Physics

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Figure 1
(a) The schematic of four-level double- $Δ$ 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 $Δ$ -type transitions $| Q 〉 \leftrightarrow | A 〉 \leftrightarrow | S 〉 \leftrightarrow | Q 〉$ by three-photon resonance condition fields with frequencies $ω_{2}$ , $ω_{0}$ , and $ω_{1}$ , respectively. Here, $Ω_{j k}$ ( $j, k = L, R, A, S$ ) are the corresponding (real) coupling strengths and satisfy $Ω_{L S} = Ω_{R S} > 0$ , $Ω_{L A} = - Ω_{R A} > 0$ and $Ω_{S A} > 0$ ; $ϕ$ is the overall phase of the coupling strengths in the $Δ$ -type transitions related to the left-handed chiral ground state $| L 〉$ . By assuming the three-photon resonance condition of the $Δ$ -type transitions, the one-photon resonance condition of the transition $| A 〉 \leftrightarrow | S 〉$ (that means $Δ_{A} = Δ_{S} \equiv Δ$ with $Δ_{A}$ and $Δ_{S}$ being the detunings corresponding to the transitions $| Q 〉 \leftrightarrow | A 〉$ and $| Q 〉 \leftrightarrow | S 〉$ ), and $Ω_{L S} = Ω_{L A} \equiv Ω$ , the original model in (a) can be simplified to two effective two-level subsystems when the overall phase of the coupling strengths in the $Δ$ -type transitions is well adjusted as (b) $ϕ = 0$ or (c) $ϕ = π$ .
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Figure 2
Coherent-operation method of enantioconversion by simplifying the original four-level double- $Δ$ model to two separated two-level subsystems and introducing an auxiliary state $| L^{'} 〉$ . (a,b,c) denote, respectively, the first, second, and third coherent 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 ${\hat{ρ}}_{0} = (| L 〉 〈 L | + | R 〉 〈 R |) / 2$ . 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 (a), (b), and (c), respectively. Here the electromagnetic fields applied in (a) and (c) are respectively similar to those in Fig. 1 [under the condition (9)] and Fig. 1 with $Δ = Ω_{S A}$ [under the condition (12)]. The other parameters are chosen to be $Ω_{aux} = \sqrt{2} Ω$ and $Δ = \sqrt{6} Ω$ .
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Figure 3
Four-level double- $Δ$ model of HSOH molecule (related transitions denoted by solid arrowed lines. The ro-vibrational states denoted by solid line are $| L 〉 = | 0^{L} 〉 \otimes | 0_{000} 〉$ , $| R 〉 = | 0^{R} 〉 \otimes | 0_{000} 〉$ , $| A 〉 = | 2 v_{HSOH}^{-} 〉 \otimes | 1_{010} 〉$ , and $| S 〉 = | 2 v_{HSOH}^{+} 〉 \otimes (| 1_{101} 〉 + | 1_{10 - 1} 〉) / \sqrt{2}$ . The energies of $| Q 〉$ , $| S 〉$ , and $| A 〉$ are 0, $2 π \times 25493.7135$ , and $2 π \times 25400.5897 GHz$ . The transitions denoted by arrowed lines of the same color are coupled by the same electromagnetic field. The polarizations of the electromagnetic fields are denoted by the colors of the arrowed lines (red, Z-polarized), (blue, Y-polarized), and (black, X-polarized). We also give the transition dipole in the molecule-frame for each transition. The transitions with the most possibility to challenge the four-level double- $Δ$ model denoted by dashed arrowed lines ( $| S^{'} 〉 = | 2 v_{HSOH}^{+} 〉 \otimes (| 1_{111} 〉 + | 1_{11 - 1} 〉) / \sqrt{2}$ ). The energy of $| S^{'} 〉$ is $2 π \times 25493.2718 GHz$ . They are negligible under the large-detuning condition.
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Fast enantioconversion of chiral mixtures based on a four-level double-Δ model

Chong Ye, Quansheng Zhang, Yu-Yuan Chen, and Yong Li

Phys. Rev. Research 2, 033064 – Published 14 July 2020

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