diff --git "a/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt" "b/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/ENE1T4oBgHgl3EQfqQWR/content/tmp_files/load_file.txt" @@ -0,0 +1,784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf,len=783 +page_content='Enantio-specific state transfer of chiral molecules through enantio-selective shortcut-to-adiabaticity paths Jian-Jian Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 Chong Ye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='3 and Yong Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ∗ 1Center for Theoretical Physics and School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hainan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Haikou 570228,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 2Beijing Computational Science Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beijing 100193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 3Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beijing Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 100081 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China 4Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Changsha 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2023) An interesting method of fast enantio-specific state transfer is proposed for cyclic three-level systems of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We show that the fast population transfer via shortcut to adiabaticity can be accomplished for the cyclic three-level system of a general (chiral) molecule with invariant- based inverse engineering of the coupling strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By choosing appropriate parameters, the two enantiomers, which are initially prepared in their ground states in the three-level systems, will evolve respectively along their enantio-selective shortcut-to-adiabaticity paths to different-energy final states simultaneously, namely achieving the fast enantio-specific state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' INTRODUCTION Since Pasteur first discovered chiral molecules in 1848, the theoretical and experimental studies of chiral molecules have proliferated in chemistry [1], biotechnolo- gies [2], and pharmaceutics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Chiral molecules contain two species, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' left- and right-handed ones [4], which are often called enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The two enantiomers are mir- ror images of each other but can be superposed on each other via translations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The enantiodiscrim- ination (as well as enantioseparation and enantioconver- sion) [5–8] of chiral molecules remains an enormous chal- lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The traditional method of enantiodiscrimination is to break the mirror symmetry of the enantiomers by using circularly polarized light [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Some commonly used chiroptical methods of enantiodiscrimination are circular dichroism [10], vibrating circular dichroism [11], optical rotation [9], and Raman optical activity [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' However, these methods rely on the interference between electric- dipole and weak magnetic-dipole (or electric-quadrupole) transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Alternatively, enantiodiscrimination methods that only use electric-dipole interactions [13, 14], have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The left- and right-handed chiral molecules can be modeled as cyclic three-level systems, where three electromagnetic (optical or microwave) fields couple respectively to three transitions via electric-dipole interactions [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Due to the intrinsic property of chi- ral molecules, the product of the corresponding three cou- pling strengths (Rabi frequencies) in the cyclic three-level systems can differ in signs for the two enantiomers [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' So the corresponding overall phases in the cyclic three-level systems differ by π with the enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Based on such cyclic three-level systems, one can use different schemes, such as enantio-selective three-wave ∗ yongli@hainanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='cn mixing [17–21], enantio-selective absorption [22], enantio- selective AC stark effect [23] and enantio-selective two- dimensional spectra [24], to discriminate the left- and right-handed molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Moreover, some more ingenious sources of modern optics physics, such as frequency en- tangled photons [25], quantized photons [26, 27], and cor- related photons in cavities [28], have been introduced to enhance the performance of enantiodiscrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Beyond the enantiodiscrimination, the cyclic three- level systems of chiral molecules have also been used in some more ambitious issues, such as the enantio-specific state transfer (ESST) [15, 29–38], enantioseparation [39– 42], and enantioconversion [16, 43–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The perfect ESST of chiral molecules can be realized by transferring the left- and right-handed chiral molecules from the same-energy initial states to different-energy final states by choosing suitable electromagnetic fields [15, 29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Recently, the feasibility of ESST based on the cyclic three-level sys- tems has been demonstrated experimentally in gaseous samples by using microwave fields [47–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' After the achievement of the ESST, one can further realize the enantiodiscrimination and spatial enantioseparation for the chiral molecules [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the original ESST method based on cyclic three- level systems of chiral molecules [15], the ESST was re- alized by using the adiabatic (and also diabatic) passage technique, which makes the ESST process slow and com- plicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' To overcome these defects, several theoretical methods of fast ESST were proposed and developed [29– 38] based on cyclic three-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Among them, an ingenious method [31] was proposed to achieve the fast ESST of chiral molecules by using the “shortcut to adi- abaticity” (STA) concept via adding a counterdiabatic field to accelerate the stimulated Raman adiabatic pas- sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Motivated by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' [31], here we propose to achieve the ESST by a different STA with invariant-based inverse engineering [51–53], instead of the STA with adding the counterdiabatic field [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The invariant-based inverse arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='03341v1 [quant-ph] 9 Jan 2023 2 engineering starts by introducing a Lewis-Riesenfeld in- variant in a time-dependent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The invariant can be used to derive a law that governs the evolution state for the designed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By means of the invariant- based inverse engineering of the time-dependent Hamil- tonians with designing appropriate control parameters, the left- and right-handed chiral molecules prepared ini- tially in their corresponding ground states would evolve (approximately) along their enantio-selective shortcut- to-adiabaticity paths to different-energy final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' CYCLIC THREE-LEVEL SYSTEMS A general chiral molecule can be modeled as the cyclic three-level system by choosing appropriate three electro- magnetic fields to couple with three electric-dipole tran- sitions [15, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, we only consider the case that all the three electromagnetic fields couple resonantly with the electric-dipole transitions respectively, as shown sim- ilar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the basis of {|1⟩, |2⟩, |3⟩}, the Hamil- tonian of the cyclic three-level system can be described in the interaction picture as (ℏ = 1) [31] ˆH(t) = � � 0 Ωx(t) Ωz(t)e−iφ Ωx(t) 0 Ωy(t) Ωz(t)eiφ Ωy(t) 0 � � (1) with |1⟩ = (1, 0, 0)T , |2⟩ = (0, 1, 0)T , |3⟩ = (0, 0, 1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here Ωj(t) (j = x, y, z) are the Rabi frequencies, which can be controlled by varying the amplitudes of the ap- plied electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' φ is the overall phase of the three Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we set φ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Without loss of generality, we have assumed Ωj are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Then the Hamiltonian can be expressed as ˆH(t) = Ωx(t) ˆKx + Ωy(t) ˆKy + Ωz(t) ˆKz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (2) Here, ˆKx, ˆKy, and ˆKz are the SU(2) angular-momentum operators [55] ˆKx = � � 0 1 0 1 0 0 0 0 0 � � , ˆKy = � � 0 0 0 0 0 1 0 1 0 � � , ˆKz = � � 0 0 −i 0 0 0 i 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (3) They satisfy the commutation relations [ ˆKx, ˆKy] = i ˆKz, [ ˆKy, ˆKz] = i ˆKx, [ ˆKz, ˆKx] = i ˆKy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (4) The fact that Hamiltonian (2) is written as the sum of three SU(2) operators, means it addresses the SU(2) al- gebraic structure [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For the two enantiomers of chiral molecules, the overall phases in the cyclic three-level systems under consider- ation differ by π [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For convenience, we specify that the signs before Ωx and Ωz are equal for the two enan- tiomers, while the sign before Ωy is opposite, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' |3〉L |3〉R |2〉L |2〉R |1〉L |1〉R Ωzeiϕ Ωy Ωx Ωzeiϕ Ωy Ωx (a) Left-handed (b) Right-handed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (a) Left- and (b) right-handed chiral molecules of cyclic three-level systems, where three electromagnetic fields couple resonantly to the three electric-dipole transitions, re- spectively, with Ωx, ±Ωy, and Ωzeiφ the corresponding Rabi frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, the Hamiltonians of the cyclic three-level systems for the two enantiomers in the basis {|m⟩L} and {|m⟩R} (m = 1, 2, 3) can be described as ˆHL,R(t) = Ωx(t) ˆKL,R x ± Ωy(t) ˆKL,R y + Ωz(t) ˆKL,R z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) Here, the indices L and R [which correspond, respec- tively, to the signs + and − in the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5)], denote the left- and right-handed chiral molecules, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ˆKQ j (j = x, y, z, Q = L, R) is just ˆKj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (3) for the two enantiomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this work, when refer- ring to left- or right-handed chiral molecules, we will add the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' When there is no index, we refer to general molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' INVARIANT DYNAMICS Shortcut to adiabaticity (STA) is a fast route to ac- celerate a slow adiabatic process by controlling the pa- rameters of a system [56], while keeping the same initial and final states as that in the adiabatic passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' A mo- tivation to apply the STA technique is to manipulate the quantum system on timescales shorter than decoherence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' There are two main STA techniques that have been proposed theoretically and implemented experimen- tally to inversely engineer the time-dependent Hamilto- nian of a quantum system for accelerating slow adiabatic process [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' One is the counterdiabatic driving method with adding an auxiliary field in a reference Hamilto- nian to cancel the nonadiabatic coupling, where the dy- namics follows exactly the adiabatic passage defined by the reference Hamiltonian [52, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The other one is the invariant-based inverse engineering method, which is based on the Lewis-Riesenfeld invariant that carries the eigenstates of a system from the initial state to the de- sired final state [52], with keeping the same initial and final states as those in the adiabatic passage, but without following the adiabatic passage at the intermediate time instants [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In what follows, we focus on how to use 3 the latter STA technique to achieve the ESST of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Commonly a Lewis-Riesenfeld invariant for a Hamilto- nian ˆH(t) is a Hermitian operator ˆI(t) that satisfies [59] dˆI(t) dt ≡ ∂ ˆI(t) ∂t − i[ˆI(t), ˆH(t)] = 0, (6) so that its eigenvalues remain constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Accord- ing to the Lewis-Riesenfeld theory [51, 52, 59], if {|φn(t)⟩} is a set of orthogonal eigenstates of the invariant ˆI(t), the solution to the time-dependent Sch¨ordinger equation can be constructed as |Ψ(t)⟩ = � n cneiαn(t)|φn(t)⟩, with cn being a time-independent coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here αn(t) = � t 0⟨φn(t′)|[i∂t′ − ˆH(t′)]|φn(t′)⟩dt′ is the Lewis-Riesenfeld phase [51, 52, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In general, ˆH(t) does not commute with the invari- ant ˆI(t) at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We only require the invariant and the Hamiltonian to commute at the initial and final time instants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=', [ ˆH(0), ˆI(0)] = 0 and [ ˆH(τ), ˆI(τ)] = 0 [51– 53, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The eigenstates of the Hamiltonian and the in- variant coincide at the initial and final time instants but may be different at the intermediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This leaves large freedom to choose how the state evolves in the in- termediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (6) to find the Hamilto- nian (2) that drives such a designed evolution of a given state in the cyclic three-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Moreover, we con- sider, respectively, the evolutions of the left- and right- handed chiral molecules with cyclic three-level structures by invariant-based inverse engineering of the Rabi fre- quencies (equivalently the amplitude of the electromag- netic fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By choosing appropriate Rabi frequencies, the fast ESST can be achieved by transferring the two enantiomers from their ground states to different-energy final states through their corresponding eigenstates of in- variants, following their enantio-selective STA paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Invariant dynamics for the left-handed chiral molecules We first consider the state transfer of the left-handed chiral molecules with the cyclic three-level structures by the invariant-based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Since ˆHL(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) possesses the SU(2) algebraic structure, the cor- responding invariant ˆIL(t) can be given as [53] ˆIL =Ω0 2 (cos γ sin β · ˆKL x + cos γ cos β · ˆKL y + sin γ · ˆKL z ) =Ω0 2 � � 0 cos γ sin β −i sin γ cos γ sin β 0 cos γ cos β i sin γ cos γ cos β 0 � � L (7) in the basis {|1⟩L, |2⟩L, |3⟩L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, Ω0 is an arbitrary constant with unit of frequency, and the time-dependent auxiliary parameters γ and β satisfy the equations ˙γ = Ωx cos β − Ωy sin β, ˙β = (Ωx sin β + Ωy cos β) tan γ − Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (8) The eigenstates of the invariant ˆIL(t), which satisfy ˆIL(t)|φn(t)⟩L = λL n|φn(t)⟩L (n = 0, ±), are |φ0⟩L = � � cos γ cos β −i sin γ − cos γ sin β � � L , (9) |φ±⟩L = 1 √ 2 � � sin γ cos β ± i sin β i cos γ − sin γ sin β ± i cos β � � L (10) with the corresponding (time-independent) eigenval- ues λL 0 = 0 and λL ± = ±Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this case, the Lewis-Riesenfeld phases are αL 0 (t) = 0, and αL ±(t) = ∓ � t 0[ ˙β(t′) sin β(t′) + Ωx(t′) sin β(t′) cos γ(t′) + Ωy(t′) cos β(t′) cos γ(t′) + Ωz(t′) sin γ(t′)]dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here, we take Ωx(t) = Ωz(t) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (8), we have Ωx = Ωz = ˙β sin β + ˙γ cos β tan γ tan γ − sin β , Ωy = ˙β cos β + ˙γ(1 − tan γ sin β) tan γ − sin β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (11) Once the appropriate boundary conditions for γ and β are fixed, one can insert a polynomial function to deter- mine Ωx, Ωy, and Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Our task is to design the Hamilto- nian ˆHL(t) to drive the initial state |1⟩L to the final state |3⟩L (up to a phase factor) along the invariant eigenstate |φ0(t)⟩L in a given time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, based on the invari- ant eigenstate |φ0(0)⟩L = (1, 0, 0)T L = |1⟩L at the initial instant time and |φ0(τ)⟩L = (0, 0, −1)T L = −|3⟩L at the final instant time τ, the boundary conditions for γ and β can be given as γ(0) = 0, β(0) = 0, γ(τ) = 0, β(τ) = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (12) On one hand, one needs to impose the boundary con- ditions to make ˆHL(t) and ˆIL(t) commute at t = 0 and t = τ so that they have common eigenstates at these time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' On the other hand, one requires the Rabi fre- quencies to vanish at the initial and final time instants to make the electromagnetic fields turn on and off smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' These requirements further imply the additional bound- ary conditions ˙γ(0) = 0, ˙β(0) = 0, ˙γ(τ) = 0, ˙β(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (13) There are many interpolating functions consistent with the boundary conditions at the initial and final time in- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' With these boundary conditions, we can simply choose γ(t) = 0, β(t) = 3π 2τ 2 t2 − π τ 3 t3 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (14) Here the small value η is set to avoid the infinite values of the Rabi frequencies at the initial time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus 4 the designed Rabi frequencies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (11) reduce to Ωx = Ωz = 3πt τ 2 � t τ − 1 � , Ωy = 3πt τ 2 � t τ − 1 � cot � 3π 2τ 2 t2 − π τ 3 t3 + η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) (a) Ωx (Ωz) Ωy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ Rabi frequencies (2π /τ) (b) P1 L P3 L P2 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ population FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The designed Rabi frequen- cies for the left-handed chiral molecules with Ωx = Ωz (red solid line) and Ωy (blue dashed line) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) Time evolution of corresponding populations in |1⟩L (red solid line), |2⟩L (black dotted line), and |3⟩L (blue dashed line) for the left-handed chiral molecules with the initial state |1⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 shows the designed Rabi frequencies for the left- handed chiral molecules and corresponding evolution of the populations in the states |m⟩L (m = 1, 2, 3) for the initial state |Ψ(0)⟩L = |1⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the ideal condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' the case of η = 0), the left-handed chiral molecules will evolve from the initial state |1⟩L (= |φ0(0)⟩L) to the final target state −|3⟩L (up to a phase factor), along the invariant eigenstate |φ0(t)⟩L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' For the case of small value η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2(b), the initial state |1⟩L ≈ |φ0(0)⟩L, thus the populations in the initial state |1⟩L with P L 1 (0) = 1 are finally transferred approxi- mately to that in the target state |3⟩L with probabil- ity P L 3 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='9991 for the left-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Correspondingly, P L 2 (0) = 0 = P L 3 (0), P L 1 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0005, and P L 2 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Invariant dynamics for the right-handed chiral molecules Then we consider the state transfer of the right-handed chiral molecules with the cyclic three-level structures by the invariant-based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Since the Hamiltonian ˆHR(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (5) of the right-handed chi- ral molecules has the same SU(2) algebraic structure as ˆHL(t) of the left-handed ones, similarly the invariant ˆIR(t) can be given in the basis {|1⟩R, |2⟩R, |3⟩R} as the form ˆIR= Ω0 2 (cos ξ sin χ · ˆKR x + cos ξ cos χ · ˆKR y + sin ξ · ˆKR z ) = Ω0 2 � � 0 cos ξ sin χ −i sin ξ cos ξ sin χ 0 cos ξ cos χ i sin ξ cos ξ cos χ 0 � � R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (16) Here the time-dependent auxiliary parameters ξ(t) and χ(t) satisfy the equations ˙ξ = Ωx cos χ + Ωy sin χ, ˙χ = (Ωx sin χ − Ωy cos χ) tan ξ − Ωz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (17) The eigenstates of the invariant ˆIR(t), which satisfy ˆIR(t)|φn(t)⟩R = λR n |φn(t)⟩R (n = 0, ±), are |φ0⟩R = � � cos ξ cos χ −i sin ξ − cos ξ sin χ � � R , (18) |φ±⟩R = 1 √ 2 � � sin ξ cos χ ± i sin χ i cos ξ − sin ξ sin χ ± i cos χ � � R (19) with the corresponding eigenvalues λR 0 = 0 and λR ± = ±Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here the Lewis-Riesenfeld phase is αR 0 (t) = 0, and αR ±(t) = ∓ � t 0[ ˙χ(t′) sin χ(t′) + Ωx(t′) sin χ(t′) cos ξ(t′) − Ωy(t′) cos χ(t′) cos ξ(t′) + Ωz(t′) sin ξ(t′)]dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we still take Ωx = Ωz for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (17), we have Ωx = Ωz = ˙χ sin χ + ˙ξ cos χ tan ξ tan ξ − sin χ , Ωy = ˙χ cos χ + ˙ξ(1 − tan ξ sin χ) sin χ − tan ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (20) Similar to the case of the left-handed chiral molecules in the above subsection, once the functions χ and ξ are fixed, we can construct Ωx, Ωy, and Ωz and thus the Hamiltonian HR(t) can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we aim to design the Hamiltonian ˆHR(t) to make the system evolve from the initial state |1⟩R to the finial state |2⟩R (up to a phase factor) along the invariant eigenstate |φ0(t)⟩R in a given time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, based on the invariant eigenstate |φ0(0)⟩R = (1, 0, 0)T R = |1⟩R at the initial time instant and |φ0(τ)⟩R = (0, −i, 0)T R = −i|2⟩R at the final time instant τ, the boundary conditions for ξ and χ can be given as ξ(0) = 0, χ(0) = 0, ξ(τ) = −π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (21) 5 (a) Ωx (Ωz) Ωy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ Rabi frequencies (2π /τ) (b) P1 R P2 R P3 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0 t /τ population FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The designed Rabi frequencies for the right-handed chiral molecules with Ωx = Ωz (red solid line) and Ωy (blue dashed line) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) Time evolution of corresponding populations in |1⟩R (red solid line), |2⟩R (black dotted line), and |3⟩R (blue dashed line) for the right-handed chiral molecules with the initial state |1⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here η′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Similarly, we set ˆHR(t) and ˆIR(t) commute at the ini- tial and final time instants (so that they have the same eigenstates at these time instants) and make the electro- magnetic fields (equivalently the Rabi frequencies) turn on and off smoothly for the right-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the additional boundary conditions for ξ(t) and χ(t) can be given as ˙ξ(0) = 0, ˙χ(0) = 0, ˙ξ(τ) = 0, ˙χ(τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (22) Consistent with these boundary conditions, we can choose χ(t) = 0, ξ(t) = − 3π 2τ 2 t2 + π τ 3 t3 + η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (23) Here the small value η′ is set to avoid the infinite values of the Rabi frequencies at the initial time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus the designed Rabi frequencies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (20) reduce to Ωx = Ωz = 3πt τ 2 � t τ − 1 � , Ωy = 3πt τ 2 � t τ − 1 � cot � 3π 2τ 2 t2 − π τ 3 t3 − η′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3 shows the designed Rabi frequencies of the right- handed chiral molecules and corresponding evolution of the populations in the states |m⟩R (m = 1, 2, 3) for the initial state |Ψ(0)⟩R = |1⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the ideal condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' the case of η′ = 0), the right-handed chiral molecules will evolve from the initial state |1⟩R (= |φ0(0)⟩R) to the final target state −i|2⟩L (up to a phase factor), along the invariant eigenstate |φ0(t)⟩R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' When we set the small value η′ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3(b), the initial state |1⟩R ≈ |φ0(0)⟩R, thus the populations in the initial state |1⟩R with P R 1 (0) = 1 are finally transferred approximately to that in the tar- get state |2⟩R with P R 2 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='9991 for the right-handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Correspondingly, P R 2 (0) = 0 = P R 3 (0), P R 1 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0005, and P R 3 (τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Achieving the fast enantio-specific state transfer So far we have designed the desired evolution for the left- and right-handed chiral molecules of the cyclic three- level systems via the STA technique with invariant-based inverse engineering in the above two subsections, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24), it can be found that the two groups of designed Rabi frequen- cies for the two enantiomers are exactly the same when η = −η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This means that the two enantiomers are driven by the same three electromagnetic fields indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In this case, the left-handed chiral molecule begins with |1⟩L and terminates approximately at −|3⟩L, almost along the in- variant eigenstate |φ0(t)⟩L, while the right-handed chi- ral molecule begins with |1⟩R and terminates approxi- mately at −i|2⟩R, almost along the invariant eigenstate |φ0(t)⟩R simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3, the left- and right-handed chiral molecules pre- pared in the same-energy initial states evolves (approx- imately) to the different-energy final states via the dif- ferent enantio-selective STA processes of invariant-based inverse engineering, driven by the same electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the fast ESST via enantio-selective STA is achieved (approximately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In the above ESST method via the enantio-selective STA with invariant-based inverse engineering, the enan- tiomeric excess of the ESST can be defined as [23, 38] ϵ ≡ ���P L 3 (τ) − P R 3 (τ) P L 3 (τ) + P R 3 (τ) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (25) Although the small values η and η′ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' η = −η′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02) have been introduced to avoid the infinite Ωy at the initial time instant, we still obtain a highly efficient ESST with enantiomeric excess ϵ = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='92% at the final time instant (with most of left-chiral molecule staying in |3⟩L and very few of the right-chiral molecule staying in the same-energy state |3⟩R, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In general, the final populations are effected by the small value η (or η′) and are independent of the param- eter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4(a), the population of the tar- 6 get state |3⟩L can be further decreased by increasing the small value η, while the population of the other target state |3⟩R would be commonly increased by increasing the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, it is possible to achieve a better enantiomeric excess with relatively small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (15) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (24), decreasing the small amount η (or η′) implies the tradeoff of requiring larger Rabi frequencies and laser intensities [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Here we define Ωmax=Max{|Ωx(t)|, |Ωy(t)|, |Ωz(t)|} as the maximum ab- solute value of the Rabi frequencies during the whole evo- lution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4(b), the maximum ab- solute value of the Rabi frequencies increase dramatically when decreasing the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (a) P3 R P3 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='04 η population (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='25 5 10 50 100 500 1000 η Ωmax (2π⨯MHz) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (Color online) (a) The corresponding populations in |3⟩L (red solid line) and |3⟩R (blue dashed line) at the final time versus the small value η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' The initial states are |1⟩L,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' (b) The maximum absolute value of the Rabi frequencies Ωmax versus the small value η with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' In experiments, the typical Rabi frequencies for the transitions of chiral molecules are about 2π×10 MHz [18, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' That means the evolution time can be shortened to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content='5 µs for the experimentally available Rabi fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, the decoherence effects (typically being about 5 ∼ 6 µs) [17, 47] will become negligable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' This is the advantage of our ESST method since it allows to manipulate the quantum system on the timescales much shorter than the typical decoherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Note that in the previous ESST method via STA [31], an auxiliary counterdiabatic field has been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' It works as a shortcut to adiabaticity for canceling the nonadiabatic coupling and induces perfect population transfer between the states |1⟩L and |3⟩L for the left- handed chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Simultaneously, it also acts oppositely for strengthening the nonadiabatic coupling for the right-handed chiral molecules and the population transfer between the states |1⟩R and |3⟩R is canceled com- pletely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Therefore, under such an ESST process, the left- handed chiral molecule begins with |1⟩L and terminates at −|3⟩L, following a STA path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' But the right-handed chiral molecule is subject to a free evolution, instead of following the STA path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' By contrast, in our ESST method via STA, the eigenstates of invariants for the two enantiomers define their corresponding enantio-selective STA paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Thus, our ESST can be achieved with trans- ferring the two enantiomers from their ground states to different-energy final states along their enantio-selective STA paths simultaneously, by choosing appropriate in- tensities of the three electromagnetic fields (that is, the Rabi frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' CONCLUSION In conclusion, we have proposed the fast ESST method of chiral molecules via the STA technique with invariant- based inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Based on the cyclic three-level systems, the ESST of chiral molecules can be achieved through enantio-selective STA paths: for the left- and right-handed chiral molecules prepared initially in their ground states, they will evolve (approximately) finally to the different-energy states almost along the eigenstates of the invariants within a short operation time simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' Hence, our fast ESST method via STA with invariant-based inverse engineering has promising appli- cations in discriminating molecular chirality and control- ling the dynamics of chiral molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Natural Science Foun- dation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12074030, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12274107, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' U1930402), National Science Foundation for Young Scientists of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQfqQWR/content/2301.03341v1.pdf'} +page_content=' 12105011), and Bei- jing Institute of Technology Research Fund Program for Young Scholars.' metadata={'source': 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