Discrimination of Chiral Molecules through Holonomic Quantum Coherent Control Teng Liu1Fa Zhao1Pengfei Lu1Qifeng Lao1Min Ding1Ji Bian1Feng Zhu1 2and Le Luo1 2 3 4 1School of Physics Astronomy Sun Yat-sen University Zhuhai Guangdong 519082 China

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Discrimination of Chiral Molecules through Holonomic Quantum Coherent Control

Teng Liu,1, ∗Fa Zhao,1, ∗Pengfei Lu,1Qifeng Lao,1Min Ding,1Ji Bian,1Feng Zhu,1, 2 and Le Luo1, 2, 3, 4, †

1School of Physics & Astronomy, Sun Yat-sen University, Zhuhai, Guangdong, 519082, China

2Shenzhen Research Institute of Sun Yat-Sen University, Nanshan Shenzhen 518087, China

3State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou 510275, China

4International Quantum Academy, Shenzhen, 518048, China

A novel optical method for distinguishing chiral molecules is proposed and validated within a quantum sim-

ulator employing a trapped-ion qudit. This approach correlates the sign disparity of the dipole moment of chiral

molecules with distinct cyclic evolution trajectories, yielding the unity population contrast induced by the dif-

ferent non-Abelian holonomies corresponding to the chirality. Harnessing the principles of holonomic quantum

computation (HQC), our method achieves highly efﬁcient, non-adiabatic, and robust detection and separation of

chiral molecules. Demonstrated in a trapped ion quantum simulator, this scheme achieves nearly 100% contrast

between the two enantiomers in the population of a speciﬁc state, showcasing its resilience to the noise inherent

in the driving ﬁeld.

Introduction.—Since Pasteur’s discovery of chirality in his

graceful tartaric acid experiment [1,2], omnipresent chiral

molecules have been realized and have profoundly inﬂuenced

many ﬁelds, including chemistry, biochemistry, pharmacol-

ogy, and materials science. A chiral molecule, as known as an

enantiomer, refers to the one that can overlap with its coun-

terpart with the transformation of mirror symmetry (cyclo-

hexylmethanol molecules shown in Fig. 1(a)). Two enan-

tiomers usually share numerous physical properties like den-

sity and viscosity[3,4], but the signiﬁcant differences in chi-

rality could emerge. An enantiomer drug (like R-thalidomide

) may be a fairly efﬁcient medicament, while its counterpart

(like S-thalidomide) may cut no ice or even result in detrimen-

tal reactions for living organisms [5,6]. Therefore, it is im-

perative to differentiate enantiomers quickly and accurately.

The early-stage methods[7] for chiral molecule detection

including crystallization, derivatization, kinetic resolution are

typically complicated, expensive and laborious. Alterna-

tively, optical methods, such as optical rotary dispersion

[8,9], circular dichroism [10–16], and Raman optical ac-

tivity [17], offer advantages in terms of simplicity and con-

venience, and are widely applied. The differential optical

signal for the chiral molecules mainly originates from weak

magnetic dipole or electric quadrupole interactions. Hence

a variety of strategies for enhancing the optical signals have

been developed, such as enhanced strong anti-Stokes Raman-

scattering ﬁeld[18], circularly polarized X-ray light[19], plas-

monic metamaterials[20], and various microwave-driven co-

herent population transfer techniques [21–25].

One notable approach to enhancing the signals is based

on quantum coherent control (QCC) techniques, where chi-

ral molecules are differentiated by precisely controlling the

phases of external optical ﬁelds, allowing the quantum states

of the molecules to evolve into completely different states.

The most typical scheme is enantio-selective cyclic popula-

tion transfer (CPT), proposed by Shapiro et al. in Ref. [26].

In this scheme, three optical ﬁelds are applied to couple three

∗These authors contributed equally

†luole5@mail.sysu.edu.cn

levels, respectively. The state evolution paths are separated

due to the contrary signs of transition dipoles in the mirror-

symmetric conﬁgurations, as shown in Figs. 1(a) and 1(b).

Subsequently, based on CPT, a two-step asymmetric synthe-

sis scheme is proposed [27,28], demonstrating the signiﬁcant

potential of QCC approaches not only for chiral discrimina-

tion but also for chiral puriﬁcation and asymmetric synthe-

sis. According to this design, one type of chiral molecule

is excited through the CPT process, and subsequently, these

“marked” molecules undergo a coherent process to achieve

conversion. This process is summarized in Fig. 1(c). De-

spite the requirement of molecular-scale quantum coherence

time to be as long as possible in their adiabatic processes,

these two schemes shed light on consecutive QCC methods

[27,29–38]. The central issue of QCC approaches lies in how

to rapidly and robustly induce molecules of different chirality

to distinct energy levels under the same external ﬁeld. Most

existing QCC methods mentioned above are time-consuming

and strongly depend on precise experimental control, lacking

optimization for robustness against experimental noise, thus

limiting the feasibility of their experimental implementation.

In this work, we present a fast, robust and fewer pulse

modulated high selectivity scheme using the method of ge-

ometric coherent control techniques, referred as Geometric

QCC (GQCC), and validate it with a qudit of trapped 171Y b+

ion. Firstly, we correlate the different signs of the dipole mo-

ment of chiral molecules with different geometric cyclic tra-

jectories, constructing chiral-dependent quantum holonomies.

Molecules with different chirality can thus be induced to

highly distinguishable orthogonal ﬁnal states under the same

external ﬁelds. The geometric cyclic evolution and the process

of geometric phase accumulation signiﬁcantly enhance the ro-

bustness of our scheme to local control errors[39,40]. Second,

we map the energy level structure of chiral molecules onto a

qudit in a trapped 171Y b+ion and demonstrate the robustness

and selectivity of our scheme via a experiment of quantum

simulation. Our work, for the ﬁrst time, introduces the geo-

metric coherent control techniques into chiral molecule dis-

crimination. It not only explores the feasibility of QCC ap-

proaches in the realms of chiral discrimination and asymmet-

ric synthesis but also provides a valuable paradigm for ex-

ploring quantum simulation and quantum control techniques

arXiv:2210.11740v4 [quant-ph] 8 Mar 2024

(a)

(c)

(b)

Enantio-

converter

Enantio-

discriminator

L R L R RR

50：50 100：0

The same external field

FIG. 1. (a) Schematic diagram of chiral cyclohexylmethanol

molecule’s partial structure. The difference of the two enantiomers

can be reduced to the discrepancy of electric dipole moments µiin

one of the three directions, here is µ1. (b) The electric dipole inter-

action model of chiral molecules with three different energy levels.

L- and R-handedness share same coupling with Ω2,Ω3and opposite

sign of coupling Ω1. (c) Schematic of two-step asymmetric synthesis

based on quantum coherent control.

in ion traps applied to speciﬁc chemical processes.

Holonomic coherent control for chiral molecules.—In the

general CPT based scheme, the closed-loop population trans-

fer exhibits phase sensitivity, as their phase has a πdiffer-

ence for different enantiomers due to the opposite transition

dipoles. In this process, three overlapped and modulated res-

onate pulses are required. We propose that the opposite tran-

sition dipoles can also be mapped onto different pure geo-

metrical quantum holonomy targeting |2⟩and |1L⟩(|1R⟩) sub-

spaces. Population can be naturally induced onto the opposite

levels by different paths. Our scheme relaxes the constraints

of the three-pulse joint modulation, obtaining more robust and

higher contrast chiral discrimination

Our path design is grounded in quantum holonomy

theory[41]. For an initial state |ψ(0)⟩following the

Schr¨

odinger equation, we can represent the ﬁnal states as

|ψ(t)⟩=Te−iRt

0H(τ0)dτ0|ψ(0)⟩, where Tis the time order-

ing operator and ℏ= 1 for simplicity. At each moment, we

can deﬁne a complete set of basis {|ζm(t)⟩}, therefore the

states can be represented as |ψ(t)⟩=Pmαm(t)|ζm(t)⟩.

The time-dependent bases |ζm(t)⟩characterizes the local ge-

ometric properties of the evolution trajectory at each moment.

When t=T, the state vector has completed a cyclic evolution

along closed trajectories with ζm(T) = ζm(0), and the ﬁnal

state is |ψ(T)⟩=Pmαm(T)|ζm(0)⟩.Compare to the initial

state |ψ(0)⟩=Pmαm(0) |ζm(0)⟩,αm(T)contains the geo-

metric information of the cyclic evolution path. The dynamics

of αm(t)follow:

dtαm(t) = iX

[Amn(t)−Hmn(t)] αn(t)(1)

where Hmn(t) = ⟨ζm(t)|H(t)|ζn(t)⟩is the element of dy-

namical part matrix H,Amn(t) = i⟨ζm(t)|(d/dt)|ζn(t)⟩is

element of geometric part Awhich depends on the change of

bases owing to the local curvature. Through appropriate pa-

rameter conﬁgurations, the dynamical part can be set to zero,

thereby achieving purely geometric quantum evolution.

In the following, we construct two holonomies for differ-

ent enantiomers, which transfer the sign of transition dipoles

of different chiral molecules onto two different bases. For

the three levels in Fig. 1(b), we drive the time-dependent

couplings Ω1(t) = Ω(t) sin θ

2eiΦ(t)between |0⟩and |1⟩and

Ω2(t) = Ω(t) cos θ

2ei(Φ(t)+ϕ), then incorporate the different

transition dipole moment signs into the Hamiltonian,

HL/R(t) = 



0 0 ±Ω1(t)/2

0 0 Ω2(t)/2

±Ω1(t)∗/2 Ω2(t)∗/2 ∆(t)



(2)

where ∆(t),Ω(t)and Φ(t)are modulated detuning, Rabi fre-

quency and phase respectively (In supplementary material S1

for more details). The ”+” represents L-handedness and ”-”

represents R-handedness here. According to Eq. (2), we can

deﬁne the basis in dark and bright state space with |DL⟩=

−cos(θ/2)|1⟩+ sin(θ/2)eiϕ|2⟩,|DR⟩= cos(θ/2)|1⟩+

sin(θ/2)eiϕ|2⟩, and |BL⟩= sin(θ/2) |1⟩+ cos(θ/2)eiϕ |2⟩,

|BR⟩=−sin(θ/2) |1⟩+ cos(θ/2)eiϕ |2⟩. Thus Eq. (2) can

be reduce to

HL/R(t) = ∆(t)|0⟩⟨0|+Ω(t)

2eiΦ(t)

BL/R⟨0|+H.c. 

(3)

Within this framework, the bright state |BL/R⟩couples

with |0⟩while the dark state |DL/R⟩is decoupled. We can

deﬁne the invariant bases |ζ0L⟩=|DL⟩,|ζ0R⟩=|DR⟩, and

the time-dependent bases along the trajectories of cyclic evo-

lution for different handedness,

|ζ1L/R(t)⟩= sin k(t)

2|BL/R⟩+ cos k(t)

2e−iβ(t)|0⟩(4)

where k(t)and β(t)indicate the characteristics of the cyclic

path (In supplementary material S1 for details). The bases in

Eq. (4) satisfy the cyclic evolution condition |ζ1L/R(0)⟩=

|ζ1L/R(T)⟩with k(T) = k(0) = π, where Tis the cyclic

evolution time, shown in Fig. 2. Substitute Eq. (4) into Eq.

(1), if condition for diagonalization of matrix A−His satis-

ﬁed, we can obtain αm(t) = eiRt

0[Amm (τ)−Hmm (τ)]dτ . At the

end of cyclic evolution, if the dynamic phase Rt

0Hmn = 0,

the time dependent bases |ζ1L/R(0)⟩will acquire a pure ge-

ometric phase γ=Rt

0iζ1L/R(τ)|(d/dτ)|ζ1L/R(τ)dτ, and

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DiscriminationofChiralMoleculesthroughHolonomicQuantumCoherentControlTengLiu,1,∗FaZhao,1,∗PengfeiLu,1QifengLao,1MinDing,1JiBian,1FengZhu,1,2andLeLuo1,2,3,4,†1SchoolofPhysics&Astronomy,SunYat-senUniversity,Zhuhai,Guangdong,519082,China2ShenzhenResearchInstituteofSunYat-SenUniversity,NanshanShenzhen51...

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Discrimination of Chiral Molecules through Holonomic Quantum Coherent Control Teng Liu1Fa Zhao1Pengfei Lu1Qifeng Lao1Min Ding1Ji Bian1Feng Zhu1 2and Le Luo1 2 3 4 1School of Physics Astronomy Sun Yat-sen University Zhuhai Guangdong 519082 China

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