1. Introduction

It is known that a chiral molecule cannot be transformed into its mirror counterpart by only spatial translations and/or rotations [1]. The left- and right-handed chiral molecules, also known as enantiomers, are ubiquitous and play important roles in chemistry [2], biotechnologies [3], and pharmaceutics [4] due to their peculiar properties. Even though the two enantiomers may exhibit identical physical properties, they can however have completely distinct chemical properties. This holds critical significance in the field of pharmaceutics. Therefore, enantiodiscrimination [5] (as well as enantioseparation [6,7] and enantioconversion [8]) of chiral molecules is an essential and urgent task. Conventional methods of enantiodiscrimination are based on the interference between magnetic-dipole (or electrical-quadrupole) and electronic-dipole interactions [9–12] between light and molecules, which suffers from the fact that magnetic-dipole or electrical quadrupole interactions are typically very weak.

About twenty years ago, cyclic three-level systems of chiral molecules were proposed and investigated [13,14], where three electromagnetic fields are applied to couple to the three electronic-dipole transitions, respectively. Such cyclic three-level systems are similar for the left- and right-handed chiral molecules except for a difference $\pi$ between the overall phases of their closed-loop transition paths [13,14]. This chirality dependence of the overall phase makes the dynamics of enantiomers different, allowing for enantio-specific state transfer (ESST) of chiral molecules (i.e., the left- and right-handed chiral molecules can be transferred to different-energy target states from the same-energy initial ones, e.g. the ground ones) [13–29]. In experiments, ESSTs of three-level chiral molecules have been demonstrated in gaseous samples by applying three well-designed microwave fields [30–35]. In addition, a variety of theoretical methods of direct enantiodiscrimination [36–48] and enantioconversion [14,49–53] (together with spatial enantioseparation [54–58]) have also been proposed for chiral molecules with cyclic three-level (sub-) structures.

Among the above seminal works the method of coherently controlled adiabatic passage [13,14] appears to be a pathbreaking one, yet it typically leads to slow and complicated ESST processes due to the requirement of intrinsic adiabaticity. Subsequently, fast ESST methods based on resonant ultrashort pulses and the resulting nonadiabatic dynamics were proposed [15–18]. Moreover, Vitanov et al. proposed recently a dynamical method based on shortcuts to adiabaticity to realize the fast ESST [19]. Inspired by this, a series of theoretical extensions have been proposed and developed [20–29].

Most recently, a fast ESST method was proposed based on designing a desired population transfer process for only one enantiomer [26], in contrast to the previous methods [13,16–23] which focused on designing population transfer processes for both enantiomers simultaneously. The key element of the method in Ref. [26] is that the three-level systems of the chiral molecules possess specific SU(2) algebraic structures (where two of the Rabi frequencies are equal and the overall phase is fixed as $\pm \pi /2$). Such a structure leads to the high-efficiency ESST where the populations transferred to the two excited states are exchanged between the left- and right-handed chiral molecules if they are initially prepared in the ground states of the three-level systems.

Here we further investigate the ESST of cyclic three-level chiral molecules where all the Rabi frequencies have identical magnitude. This requirement makes it possible to realize the desired state transfer process from the ground state to the target excited one for one enantiomer (e.g., the left-handed chiral molecules) by using external fields with arbitrary pulse waveforms and appropriate values of pulse area of the Rabi frequency. Simultaneously, the other enantiomer (e.g., the right-handed chiral molecules) can be automatically transferred to the other excited state from the ground one. Therefore, by tuning the pulse area of the external fields, the left- and right-handed chiral molecules can evolve to different excited states from the initial ground states and thus exhibit fast ESST. Our current method shows the flexibility in controlling the waveform and overall phase of the pulses. In contrast, the method proposed in Ref. [26] requires precise control over these parameters. This flexibility makes our fast ESST method easier to implement and control, benefiting from its robustness and high efficiency.

In the previous ESST methods [15–18,31,35], the ESST processes of the cyclic three-level chiral molecules are typically accomplished by resorting to several successive pulses. In contrast, here we use three pulses which are applied simultaneously to the enantiomers to realize fast ESST. In addition, compared with the shortcuts-to-adiabaticity methods [19,27] where all the three electromagnetic fields are also applied simultaneously, our method shows the advantage in terms of the arbitrary pulse waveforms.

2. Cyclic three-level systems of enantiomers

The two enantiomers (i.e., the left- and right-handed chiral molecules) can be modeled as cyclic three-level systems based on the electric-dipole transitions when they are subject to three external electromagnetic fields [13,59]. Here, we only consider the case where all the three external fields are coupled resonantly to the corresponding electric-dipole transitions, respectively. Under the rotating-wave approximation, the Hamiltonian of the cyclic three-level systems can be written in the basis $\{|m\rangle _{Q}\}~(m=1,2,3)$ as ($\hbar =1$)

Here we only focus on the case

 

Fig. 1. Cyclic three-level systems of (a) left- and (b) right-handed chiral molecules. Three external electromagnetic fields are coupled resonantly to the electric-dipole transitions, respectively, with overall phase $\phi _{Q}$ and equal Rabi frequency $\Omega$.

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To illustrate our method, we use the 1,2-propanediol molecule as an example to construct the cyclic three-level system by selecting appropriate rotational states $|J,\tau,M\rangle$ [59] (with electronic and vibrational ground state). Here, $J$ represents the total angular momentum, $\tau$ ranges from $-J$ to $J$ in ascending energy order, and $M$ corresponds to degenerate magnetic sub-levels. The corresponding cyclic three-level system, as described in Refs. [35,59], consists of $|1\rangle =|0,0,0\rangle$, $|2\rangle =|1,-1,0\rangle$, and $|3\rangle =(|1,1,1\rangle +|1,1,-1\rangle )/\sqrt {2}$. Three microwave pulses are utilized to couple resonantly with the three transitions in the three-level system respectively and simultaneously. Specifically, the transition $|1\rangle \leftrightarrow |2\rangle$ (with frequencies $2\pi \times 6431.06\, \textrm {MHz}$) is couple resonantly to a Z-polarized microwave pulse. Meanwhile, the transitions $|1\rangle \leftrightarrow |3\rangle$ and $|2\rangle \leftrightarrow |3\rangle$ (with frequencies $2\pi \times 12212.15\, \textrm {MHz}$ and $2\pi \times 5781.09\, \textrm {MHz}$, respectively) are couple resonantly to Y- and X-polarized microwave pulses. Achieving equal Rabi frequencies in such a cyclic three-level system becomes very simple due to the comparable magnitudes of dipole moments associated with these three rotational transitions [59]. We would like to remark that the technique involving three microwaves has already been implemented in many experiments of cyclic three-level systems for chiral molecules [30–35].

3. Enantio-specific state transfer via dynamical processes

We assume that the two enantiomers stay in their ground states $|1\rangle _{Q}$ initially. In this section (as well as Sec. 4.), we consider the case of $\phi =\pi /2$ as an example. The dynamical evolutions of the two enantiomers can be captured by the Schrödinger equation $i\partial _{t}|\psi (t)\rangle _{Q}=\hat {H}^{Q}(t)|\psi (t)\rangle _{Q}$. For instance, one can assume the evolved state $|\psi (t)\rangle _{L}=\sum _{m=1,2,3}C^{L}_{m}(t)|m\rangle _{L}$ for the left-handed molecules, where $C_{m}^{L}$ represents the excitation amplitude of the state $|m\rangle _{L}$. Under the initial condition $C^{L}_{1}(0)=1, C^{L}_{2}(0)=0$, and $C^{L}_{3}(0)=0$, the evolved state can be obtained by solving the dynamical equations as

Likewise, the evolved state $|\psi (t)\rangle _{R}=\sum _{m=1,2,3}C^{R}_{m}(t)|m\rangle _{R}$ of the right-handed chiral molecules can be obtained by solving

4. Flexibility of the pulse waveforms

In Sec. 3, we have identified in Eq. (9) the condition for realizing ESST of the three-level chiral molecules. Such a condition is clearly irrelevant to the shape of the pulse waveform. Our scheme is thus flexible in that one can in principle use arbitrary types of pulse waveforms to realize fast ESST.

Here we assume that the characteristic length of the molecular sample is much smaller than the beam radii of the electromagnetic fields, so that the Rabi frequencies are almost uniform for the molecules. For the sake of simplicity, one can choose pulses of square waveforms with typical experimentally available Rabi frequency [31,32]

In order to achieve the ESST, the duration time of the pulse is set to be $T_{0}=19.2\,\text {ns}$ so that the related pulse area $A=\frac {2\sqrt {3}\pi }{9}$. Note that using square pulses typically requires switching the pulses on and off abruptly, which presents a challenge in precisely controlling the pulse area to achieve a specific value. In practical applications, it is preferable to smoothly turn on and off the applied pulses. In view of this, the waveform can be chosen as a single-period cosine-like one, i.e.,

Note that in this section we have assumed $\phi =\pi /2$ and identical magnitude for the three Rabi frequencies of the cyclic three-level systems. In Fig. 2, we plot the evolved populations in $|2\rangle _{Q}$ and $|3\rangle _{Q}$ for the two enantiomers in this case. It shows that the chiral molecules in the initial state $|1\rangle _{L}\,(|1\rangle _{R})$ are transferred to the state $|2\rangle _{L}\,(|3\rangle _{R})$ eventually, regardless of the explicit waveform of the pulses. All the three waveforms in Eqs. (10)–(12) enable the desired ESST, where the left-handed molecules are excited into $|2\rangle _{L}$ and the right-handed ones are excited into $|3\rangle _{R}$ from the initial ground state $|1\rangle _{Q}$ after the evolution with $A=\frac {2\sqrt {3}\pi }{9}$. Such robustness allows us to employ a large variety of pulses, thus offering the flexibility in implementing ESST.

 

Fig. 2. (a), (c), and (e) Pulse waveforms given in Eqs. (10), (11), and (12), respectively. (b), (d), and (f) Evolved populations for the left- and right-handed chiral molecules: $P_2^L(t)=P_3^R(t)$ (red solid line) and $P_3^L(t)=P_2^R(t)$ (blue dashed line), corresponding to the waveforms in panels (a), (c), and (e), respectively. Here the overall phase is $\phi =\pi /2$ and the initial states are assumed as $|1\rangle _{L,R}$.

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5. Flexibility of the overall phase and the pulse area

In Sec. 3 and Sec. 4, we focused on the case where the overall phase $\phi$ is fixed as $\pi /2$. In fact, the overall phase can be tuned by adjusting the initial phases of the three electromagnetic fields. Note that when the three Rabi frequencies have equal magnitude, the Hamiltonians $H^{Q}(t)$ in Eq. (1) at different time instants always commute with each other, i.e., $[H^{Q}(t),H^{Q}(t')]=0$, regardless of the overall phase. This allows to describe the evolution of the system using a time-evolution operator (instead of a general time-ordering operator)

To better illustrate the discussions, we now consider a general cyclic three-level system where the Rabi frequencies have equal magnitude. The corresponding Hamiltionian can be expressed, in the basis of $\{|1\rangle, |2\rangle, |3\rangle \}$, as

By solving the Schrödinger equation $i\partial _{t}|\psi (t)\rangle =\hat {H}(t)|\psi (t)\rangle$ with the initial condition $|\psi (0)\rangle =|1\rangle$, the evolved state can be formally obtained as

If we perform the transformation $|1'\rangle =ie^{-i\phi }|1\rangle, |2'\rangle =-i|3\rangle$, and $|3'\rangle =i|2\rangle$, the Hamiltonian in Eq. (14) can be re-written as

Since Eqs. (15) and (17) should give the same evolved states, one has

Here we note that the equation $-i\partial _{t}|\psi ^{*}(t)\rangle '=\hat {H}'^{*}(t)|\psi ^{*}(t)\rangle '$ should be satisfied via taking complex conjugate for the Schrödinger equation $i\partial _{t}|\psi (t)\rangle '=\hat {H}'(t)|\psi (t)\rangle '$. That means the complex conjugate $|\psi ^{*}(t)\rangle '$ of the state in Eq. (17), which follows

In this way, one can identify the relation between the populations $P_{m}(t)=|C_{m}(t)|^{2}$ for Hamiltonian (14) and $P'_{m}(t)=|C'_{m}(t)|^{2}$ for Hamiltonian (16,20), i.e.,

We would like to point out that when the Hamiltonian of the left-handed chiral molecules in the basis of $\{|m\rangle _{L}\}$ has the same form as Eq. (14), the Hamiltonian of the corresponding right-handed ones in the basis of $\{|m\rangle _{R}\}$ should have the same form as Eq. (20). According to Eq. (21), for left- and right-handed chiral molecules evolved from the initial ground states (i.e. $|1\rangle _{L,R}$, up to a constant phase factor), we have

This implies that the populations occupying the two excited states are always exchanged between the two enantiomers during the evolution, as long as the magnitude of the three Rabi frequencies are equal, regardless of the overall phase $\phi$.

Now we turn to study ESST for other general values of $\phi$ (i.e., $\phi \neq \pi /2$). We plot in Fig. 3(a) the populations of $|2\rangle _{L}$ or $|3\rangle _{R}$ after the evolution following Eq. (13) as a function of the pulse area $A$ and the overall phase $\phi$. As expected, one can find many regions where the efficiency of the ESST is very high (over 95%). In addition to the case of $\phi = \pi /2$, the nearly perfect ESST (with approximately 100% efficiency) can also be realized for many other values of $\phi$, such as $\phi = 0.18\pi$ (the black solid line) as shown in Fig. 3(b). This is accomplished by choosing an appropriate pulse area $A$ for a given $\phi$ (e.g., $A\simeq 3.2$ for $\phi = 0.18\pi$).

 

Fig. 3. (a) Populations of $|2\rangle _{L}$ or $|3\rangle _{R}$ in the final state versus the overall phase $\phi$ and the pulse area $A$. (b) Populations of $|2\rangle _{L}$ and $|3\rangle _{R}$ in the final state versus the pulse area $A$ with two chosen values of the overall phase $\phi =0.18\pi$ (the black solid line) and $\phi =0.5\pi$ (the green dashed line), corresponding to the black solid and green dashed lines in panel (a), respectively. The initial states are assumed as $|1\rangle _{L,R}$.

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Moreover, we conclude from Fig. 3(a) that our ESST scheme is robust against weak perturbations in both the overall phase and the pulse area. For example, a high-efficiency ESST (with the populations in the target states more than 95%) can still be achieved even if the overall phase deviates from the predetermined value of $\phi =0.5\pi$ by $\pm 0.1\pi$, or if the pulse area deviates from the value of $A\simeq 1.2$ by $\pm 0.15$. In the current scheme, the overall phase provides an alternative controllable knob, with which the high-efficiency ESST can be achieved despite some possible implementation imperfections.

We would like to point out that the previous method proposed in Ref. [26] also ensures the interesting exchange symmetry of the excited-state populations for the two enantiomers. It suffers from a limitation, however, that the overall phase is required to be $\pm \pi /2$ to protect such an exchange symmetry. In contrast, our current method enables the exchange symmetry regardless of the value of the overall phase. Moreover, the previous method [26] demands well-designed pulse waveform shapes to accomplish the ESST. As a result, it is extremely sensitive to variations in the pulse waveforms. However, our current method releases this limitation and thus simplifies the experimental implementation of ESST. Consequently, our current method shows advantages in terms of the robustness and experimental complexity.

6. Robustness

So far we have examined our ESST method with ideal parameters. In practice, however, the systems are inevitably influenced by various perturbations. Therefore, it is important to investigate the effects of possible perturbations which may arise from the implementation imperfections of ESST. Now let us consider the effect of the relaxation and dephasing to the ESST process under consideration.

Here we take pulses with single-period cosine-like waveforms as an example to realize ESST. In experiments, such pulses can be readily generated by using standard microwave techniques. However, random amplitude noises are typically unavoidable due to various implementation imperfections. Therefore, it is necessary to take into account such noises, which cause the Rabi frequencies to perturb randomly. Here we model the noises by using the random Rabi frequencies

As shown in Fig. 4, although the waveforms deviate from the ideal cosine-like shape due to the noises, the time evolutions of the populations in $|2\rangle _{Q}$ and $|3\rangle _{Q}$ are nearly unaffected in all three cases. The ESST can still show a nearly $100{\%}$ efficiency ESST as long as $t=T$ and $A=\frac {2\sqrt {3}\pi }{9}$. The exchange symmetry of the excited-state populations of the two enantiomers is also almost unaffected by the random amplitude noises, as shown in Fig. 4(b). This robustness is guaranteed by the zero mean value of the perturbations, with which the pulse areas remain almost unchanged.

 

Fig. 4. (a) Rabi frequencies of the three single-period cosine-like pulses with random amplitude perturbations [given in Eq. (23)], with perturbation coefficients $\eta _{12}=0.3$, $\eta _{23}=0.4$, and $\eta _{13}=0.5$, respectively. (b) Time evolutions of the corresponding populations in $|2\rangle _{Q}$ and $|3\rangle _{Q}$. Here the overall phase is $\phi =\pi /2$ and the initial states are assumed as $|1\rangle _{Q}$.

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We would also like to compare the robustness of our present scheme with the previous ESST scheme (e.g. Ref. [26]) where three driving fields with well-designed time-dependent amplitudes are applied simultaneously. We show that the ESST scheme in Ref. [26] is vulnerable to random perturbations in their waveforms, as seen in Appendix A. In contrast, our current method is highly robust against certain types of noises that do not change the pulse areas significantly.

In our scheme, the presumed initial state is the ro-vibrational ground state of an asymmetric-top molecule. In principle, the molecules are initially prepared naturally in the ro-vibrational ground state if their temperature is sufficiently low (e.g., below $0.01$ K for 1,2-propanediol molecule) [24,37,59]. Two possible techniques for cooling the molecular sample are buffer-gas cooling and supersonic expansion [31,37–40]. In recent experiments, the molecular samples can be cooled down to $1-2\,\text {K}$ by using cryogenic buffer gas cells [31,37,38] and cooled down to $1-2\,\text {K}$ by resorting to the supersonic expansion technique [39,40]. In current experiments, it is challenging to achieve the sufficiently low temperature to prepare these chiral molecules in their ground states directly. An innovative approach with an additional depletion process has been proposed in Ref. [35] to address this challenge, allowing for the preparation of chiral molecules in their rotational ground state without lowering the molecular temperature significantly. The chiral molecules are cooled to low temperatures (around 1-2 K) through a supersonic expansion technique by using Argon as the carrier gas. Subsequently, a continuous-wave ultraviolet (UV) laser is employed to selectively deplete the population of the target rotational state. By following UV excitation, the molecules predominantly emit toward higher vibrational levels and other rotational states that fall outside the selected cyclic three-level system. With the help of this approach, one can optically pump and deplete the target rotational states $J=1$ by the UV laser, thereby avoiding the influence of thermal populations and effectively preparing the chiral molecules in the rotational ground state $|0,0,0\rangle$.

In realistic scenarios, the ESST process of our scheme exists with relaxation and dephasing. In this case, the time evolution of the density matrix operator $\rho _{Q}$ is governed by the master equation

In Eq. (24), the population relaxation is described by the Lindblad superoperator $\mathcal {L}_{rl}$, which satisfies [52,60]

For the sake of simplicity, here we assume state-independent pure dephasing rate $\tilde {\gamma }$.

To quantify the efficiency of the ESST in the presence of the relaxation and dephasing effects, we define the enantiomeric excess of the ESST as [24]

According to state-of-the-art experiments, the typical pure dephasing rate is chosen as $\tilde {\gamma }=2\pi \times 0.1\,\textrm {MHz}$ [31,32] and the relaxation rates for the typical rotational transitions are chosen as $\gamma _{12}=\gamma _{23}=2\pi \times 0.1\,\textrm {MHz}$ and $\gamma _{13}=2\pi \times 0.15\,\textrm {MHz}$ [31,32]. We solve numerically the master Eq. (24) and plot the time evolutions of $P_2^L(t)$ and $P_2^R(t)$ in the presence of relaxation and dephasing effects in Fig. 5. Here we use pulses with a single-period cosine waveform given as $\Omega (t)$ in Eq. (11), the initial states are $|1\rangle _{Q}$. With appropriate parameters, one can see from Fig. 5 that the enantiomeric excess grows up to $\varepsilon = 99.15{\% }$ at $t=T=38.5\,\textrm {ns}$. Moreover, it shows that most left-handed chiral molecules remain in the state $|2\rangle _{L}$, while only a tiny fraction of the right-handed chiral molecules remain in the corresponding same-energy excited state $|2\rangle _{R}$. We thus conclude that the impact of the relaxation and dephasing is negligible in our method for typical experimental parameters.

 

Fig. 5. Time evolutions of the populations $P_2^L(t)$ (red solid line) and $P_2^R(t)$ (blue dashed line) and the corresponding enantiomeric excess $\varepsilon (t)$ (black dotted line). Here the systems are driven by external electromagnetic fields with pulse waveforms in Eq. (11) and also subject to the relaxation and dephasing effects. The initial states are $|1\rangle _{Q}$.

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7. Conclusion

In summary, we have proposed a robust and highly-efficient dynamical method to realize fast ESST for chiral molecules with cyclic three-level structures. This can be accomplished by using pulses of arbitrary waveforms, as long as the amplitudes of the three Rabi frequencies of the cyclic three-level systems are equal. The equal Rabi frequencies for all three transitions serve the purpose of ensuring the populations occupying the two excited states are always exchanged between the enantiomers during the evolution, regardless of the overall phase and the waveform. Based on this symmetry, we focus on designing the suitable pulse area of microwave fields with the equal Rabi frequencies to facilitate the population transfer evolution to the target excited state for one enantiomer (e.g., left-handed chiral molecules). Correspondingly, employing the same microwave fields allows the right-handed chiral molecules to automatically undergo population transfer to the other excited state simultaneously. Due to the fact that the exchange symmetry is independent of the waveform and overall phase of the pulses, it ensures the flexibility in controlling the waveform and overall phase of the pulses, facilitating the achievement of ESST over a broader parameter range.

We have demonstrated the robustness of our ESST method in the presence of specific types of noise. Additionally, we have shown that the potential impact of relaxation and dephasing on the ESST within our system is negligible under typical experimental parameters. Consequently, despite its simplicity, our current method provides a promising tool for implementing the fast ESST of chiral molecules.

Appendix

A. Discussion about the robustness of the ESST protocol in Ref. [26]

Here we consider the robustness of the ESST protocol in Ref. [26] by introducing the random perturbations with a similar manner as that in this work. In Ref. [26], three driving fields with well-designed time-dependent amplitudes are applied simultaneously to couple to the three transitions $|1\rangle \leftrightarrow |2\rangle$, $|2\rangle \leftrightarrow |3\rangle$, $|1\rangle \leftrightarrow |3\rangle$, respectively, with the corresponding coupling strengths (Rabi frequencies) “$\Omega _{x}$”, “$\pm \Omega _{y}$”, and “$\Omega _{z}e^{i\phi }$”. For the protocol 2 in Ref. [26], at the presence of the random perturbations, the coupling strengths are modified as

(28)$$\begin{aligned} \Omega_{x}(t)&=\dot{\theta}(t)\rightarrow\Omega^{\text{rand}}_{x}(t)=\dot{\theta}(t)[1+\text{rand}_{x}(t,\eta_{x})],\\ \pm\Omega_{y}(t)&={\mp}\dot{\theta}(t)\frac{\sin\theta(t)}{\cos[\theta(t)-\epsilon]}\rightarrow\pm\Omega^{\text{rand}}_{y}(t)={\mp}\dot{\theta}(t)\frac{\sin\theta(t)}{\cos[\theta(t)-\epsilon]}\ [1+\text{rand}_{y}(t,\eta_{y})],\\ \Omega_{z}(t)e^{i\phi}&=\dot{\theta}(t)e^{i\phi}\rightarrow\Omega^{\text{rand}}_{z}(t)e^{i\phi}=\dot{\theta}(t)[1+\text{rand}_{z}(t,\eta_{z})]e^{i\phi},\\ \end{aligned}$$

where $\theta (t)=\frac {\pi }{2}-\frac {3\pi }{2T^{2}}t^{2}+\frac {\pi }{T^{3}}t^{3}$, $\epsilon =0.04$ is a chosen small value, and the overall phase is taken as $\phi =\pi /2$. In Eq. (28), the random perturbations (i.e., certain noises) with $\eta _{x}=0.3$, $\eta _{y}=0.4$, and $\eta _{z}=0.5$ are introduced, similar to that in Eq. (23). And the “+” (“–”) sign before the Rabi frequency $\Omega _{y}(t)$ corresponds to left- (right-) handed chiral molecules, as shown in Ref. [26]. Without the presence of random perturbations, the Rabi frequencies in Eq. (28) had been shown in Ref. [26] to achieve fast ESST with $P^{L}_{2}(T)~[P^{R}_{3}(T)]=0.9946$ for $T=100~\text {ns}$ in the final target state.

The impact of the random perturbations in Eq. (28) has been plotted in Fig. 6. It shows such an ESST scheme in Ref. [26] is vulnerable. For example, the populations in $|2\rangle _{L}$ is $P^{L}_{2}(T)=0.9464$ and the populations in $|3\rangle _{R}$ is $P^{R}_{3}(T)=0.8502$. In contrast, the final populations in the target states ($|2\rangle _{L}$ and $|3\rangle _{R}$) are larger than 0.990 with the similar random amplitude perturbations of the three pulses in our current scheme. Thus our current method exhibits high robustness against random perturbations in the pulse waveforms.

 

Fig. 6. (a) The designed Rabi frequencies of the three pulses with random amplitude noises [given in Eq. (28)], with perturbation coefficients $\eta _{x}=0.3$, $\eta _{y}=0.4$, and $\eta _{z}=0.5$, respectively. (b) Time evolutions of the corresponding populations in $|2\rangle _{Q}$ and $|3\rangle _{Q}$. Here the overall phase is $\phi =\pi /2$, $\epsilon =0.04$, $T=100~\text {ns}$ and the initial states are assumed to be $|1\rangle _{Q}$.

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Funding

National Natural Science Foundation of China (12074030, 12088101, 12274107, U223040003); Hainan University (KYQD(ZR)23010).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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