Cyclic three-level-pulse-area theorem for enantioselective state transfer of chiral molecules

To describe our model, three rotational states of asymmetric top molecules, as shown in Fig. 1, are labeled as $|A\rangle$, $|B\rangle$ and $|C\rangle$ with a subscript $L/R$ for the left/right-handed enantiomer. The energies $E_{A}$, $E_{B}$ and $E_{C}$ of three rotational states are identical for enantiomers. We mark three microwave control fields as $\overrightarrow{\mathcal{E}}_{a/b/c}\left(t\right)$, which drive the three states with the transition dipole moments $\overrightarrow{\mu}_{a}$, $\overrightarrow{\mu}_{b}$ and $\overrightarrow{\mu}_{c}$. As demonstrated in Refs. Leibscher2019 ; kevin2018 , the ESST scheme with the use of linearly polarized control fields requires that a combination of control fields $\overrightarrow{\mathcal{E}}_{a}$, $\overrightarrow{\mathcal{E}}_{b}$ and $\overrightarrow{\mathcal{E}}_{c}$ with three orthogonal polarization directions along the directions of three dipole moment components $\overrightarrow{\mu}_{a}$, $\overrightarrow{\mu}_{b}$ and $\overrightarrow{\mu}_{c}$. To that end, we describe the time- and polarization-dependent electric fields of the three control fields by

where $\overrightarrow{e}_{a/b/c}$, $\mathcal{E}_{a/b/c}$, $f_{a/b/c}\left(t\right)$, $\omega_{a/b/c}$, $t_{a/b/c}$, and $\phi_{a/b/c}$ denote the polarization direction, the strength, envelope function, center frequency, center time, and phase of $\overrightarrow{\mathcal{E}}_{a/b/c}\left(t\right)$, respectively. The triple product $\overrightarrow{\mu}_{a}\cdot(\overrightarrow{\mu}_{b}\times\overrightarrow{\mu}_{c})$ is independent on the choice of the inertia principle axes $a$, $b$, and $c$, but is of opposite sign for the left- and right-handed enantiomers. For convenience, we specify that $\overrightarrow{\mu}_{b}$ and $\overrightarrow{\mu}_{c}$ are identical for two enantiomers, whereas $\overrightarrow{\mu}_{a}$ changes sign with the handedness. Thus, the Hamiltonian of two different handed-enantiomers in the presence of the control fields can be written as

where the three cyclic couplings read $\text{$\Omega_{a}\left(t\right)=-\mu_{a}$}\mathcal{E}_{a}\left(t\right)$, $\text{$\Omega_{b}\left(t\right)=-\mu_{b}$}\mathcal{E}_{b}\left(t\right)$ and $\text{$\Omega_{c}\left(t\right)=-\mu_{c}$}\mathcal{E}_{c}\left(t\right)$.
We now analyze how to achieve enantioselective excitation of a given target state from a given initial state. We assume that two enantiomers are initially in the state $|A\rangle$, and the control target can be either the excited state $|B\rangle$ or $|C\rangle$. For the choice of $|C\rangle$ as the target, there are a direct one-photon transition $|A\rangle\leftrightarrow|C\rangle$ and an indirect two-photon transition $|A\rangle\leftrightarrow|B\rangle\leftrightarrow|C\rangle$, which form the closed-loop interaction scheme. If we take $|B\rangle$ as the target, two transition paths correspond to a direct one $|A\rangle\leftrightarrow|B\rangle$ and an indirect one $|A\rangle\leftrightarrow|C\rangle\leftrightarrow|B\rangle$. Since it remains difficult to derive an analytical solution by directly using the Hamiltonian in Eq. (5), we use a strategy by dividing the excitation processes into two stages.

For ESST to $|B\rangle$, we apply the coupling $\Omega_{b}$ before the couplings $\Omega_{a}$ and $\Omega_{c}$, which results in a coherent superposition state of $|A\rangle$ and $|C\rangle$. As demonstrated above by involving the first-order Magnus expansion and further mathematical derivations, an analytic wave-function of the three-level $\Lambda$-type system can be given by

where $\xi\left(t\right)=\theta_{c}\left(t\right)\theta_{a}\left(t\right)\left[\cos\theta\left(t\right)-1\right]/\theta^{2}\left(t\right)$ and $\theta\left(t\right)=\sqrt{\left|\theta_{a}\left(t\right)\right|^{2}+\left|\theta_{c}\left(t\right)\right|^{2}}$.
To entirely transfer the left-handed enantiomer to the state $|B\rangle$ at the final time $t_{f}$, but the right-handed one is not populating, we have

Furthermore, we can obtain the amplitude condition for the three control fields

The ESST to $|B\rangle$ of the left-handed enantiomer can be reached by using the phase condition

The amplitude conditions by Eq. (27) in forms are the same as Eq. (21) with different orders. That is, the orders of the three pulses are interchangeable, dependent on the choice of the target state. From the phase condition in Eq. (28), we can find that a $\pi$ flip of the phase on one of three control fields can also lead to opposite ESST. Since our schemes that divide the closed-loop excitation schemes into two stages are different from previous works Wu2020 ; Torosov2020 ; Torosov2020-2 ; Wu2019 ; M3WM5 , which turned on the direct one-photon transition path before the two-photon one, these amplitude and phase conditions provide an alternative way to achieve ESST within a cyclic three-level systems in chiral molecules.
Note that the amplitude conditions of $|\theta_{a/b}(t_{f})|=\pi/4$ and $|\theta(t_{f})|=\sqrt{|\theta_{b/a}(t_{f})|^{2}+|\theta_{c}(t_{f})|^{2}}=\pi/2$ are equivalent to that with the use of $\pi/2$ and $\pi$ pulses, for which a scale of $1/2$ factor comes from the definition of the complex pulse-areas without using the rotating wave approximation and the resonant excitation conditions. To show the advantage of using the complex pulse-areas, we can have a frequency-domain analysis for the control fields

where $\mathcal{A}_{a/b/c}(\omega)$ and $\phi_{a/b/c}(\omega)$ are the spectral amplitude and phase, respectively. We can find that the values of $\theta_{a/b/c}(t_{f})$ depend only on $\mathcal{A}_{a/b/c}(\omega)$ and $\phi_{a/b/c}(\omega)$ of three control fields at transition frequencies $\omega_{AB}$, $\omega_{AC}$ and $\omega_{BC}$. Thus, our definitions of the complex pulse areas can also be applied to the pulsed control fields whose center frequencies are detuned away from the transition frequencies. In Sec. III, we present simulations to examine the amplitude and phase conditions for both center-frequency resonant and detuned microwave excitation schemes.

For the target state $|C\rangle$, we set the parameters $A_{a}=\pi/(4\mu_{a})$, $A_{b}=\pi/(2\sqrt{2}\mu_{b})$, and $A_{c}=\pi/(2\sqrt{2}\mu_{c})$. It is easily to verify that the three control fields defined by Eq. (30) with different values of $\tau_{0}$ exactly satisfy the amplitude conditions in Eq. (22) at transition frequencies by fixing the center frequencies $\omega_{a}=\omega_{AB}$, $\omega_{b}=\omega_{AC}$ and $\omega_{c}=\omega_{BC}$. Figure 2 shows the results of $P_{C}^{L,R}(t_{f})$ versus $\tau_{0}$ and $\phi_{a}$ with $\phi_{b}=\phi_{c}=0$. As expected, the ESST to the state $|C\rangle$ appears and depends highly on the phase values of $\phi_{a}$. The fidelity of $P_{C}^{L,R}(t_{f})>0.999$ can be reached for $\tau_{0}>35$ ns, indicating that the unwanted transition to the neighboring state $|B^{\prime}\rangle$ can be ignored. Figures 2 (b) and (e) plot the dependence of $P_{C}^{L,R}(t_{f})$ on the phase $\phi_{a}$ for the case of $\tau_{0}=35$ ns. There is no ESST at $\phi_{a}=0$ and $\pi$. The entire ESST to the left-handed molecule occurs at $\phi_{a}=\pi/2$ and a phase change to $\phi_{a}=3\pi/2$ results in an opposite transfer to the right-handed one. The similar features can be observed by changing the values of $\phi_{b}$ ( or $\phi_{c}$) while choosing $\phi_{a}=\phi_{c}=0$ ( or $\phi_{a}=\phi_{b}=0$). These results are in good agreement with the theoretical predication by the phase conditions as well as previous M3WM experiments.

To visualize the underlying quantum state transfer mechanism, Fig.3 shows the time-dependent populations of the four states induced by the control fields for the cases of $\tau_{0}=35$ ns and $\phi_{a}=\pi/2$. For the two enantiomers, there are no visible populations in the state $|B^{\prime}\rangle$ during the whole process. The four-level system is equivalent to the present closed-loop three-level model with the used pulse parameters. The control field $\mathcal{E}_{a}(t)$ with the pulse-areas $\theta_{a}(t_{1})=\pi/4$ drives the system to the maximal coherent superposition of $|A\rangle$ and $|B\rangle$ with $P_{A}^{L,R}(t_{1})=P_{B}^{L,R}(t_{1})=0.5$ for both enantiomers. Due to the sign difference of the transition from $|A\rangle$ to $|B\rangle$, $\mathcal{E}_{a}(t)$ with a phase $\phi_{a}=\pi/2$ will result in the phase of the state $|B\rangle$ in $0$ and $\pi$ for the left- and right-handedness, respectively, as described by Eq. (17). For the left-handedness, the transition path from $|A\rangle$ to $|C\rangle$ induced by $\mathcal{E}_{b}(t)$ will constructively with the path from $|B\rangle$ to $|C\rangle$ by $\mathcal{E}_{c}(t)$, leading to entire ESST to $|C\rangle$. For the right-handedness, however, the two paths are destructive, which keeps the molecules in the states $|A\rangle$ and $|C\rangle$, as shown in Fig. 3 (a) and (b).
As can be seen from our theoretical derivations, we divide the closed-loop excitation scheme into two time-separated stages. To see whether the amplitude and phase conditions can be applied to the overlapped cases, as an example, Fig. 4 plots the landscape of $P_{C}^{L}(t_{f})$ with respect to the time delays $t_{ba}=t_{b}-t_{a}$ and $t_{ca}=t_{c}-t_{a}$ while fixing the center time $t_{a}$ unchanged. $P_{C}^{L}(t_{f})$ strongly depends on the overlap between two control fields of the second stage. Interestingly, $P_{C}^{L}(t_{f})$ remains the value of $P_{C}^{L}(t_{f})>0.999$ for $t_{ba}=t_{ca}>2\tau_{0}$ when $\mathcal{E}_{b}(t)$ and $\mathcal{E}_{c}(t)$ are turned on simultaneously with $t_{ba}=t_{ca}$. Even the three control fields are applied without any delays, high fidelity of $P_{C}^{L}(t_{f})>0.90$ holds, as shown in Fig. 4 (b). The phenomena can also be observed for the right-handedness (not shown here).

Figure 5 examines the same simulations as Fig. 2 but for the target $|B\rangle$ with $\phi_{a}=\phi_{c}=0$. In our simulations, we choose the parameters $A_{b}=\pi/(4\mu_{a})$, $A_{a}=\pi/(2\sqrt{2}\mu_{b})$, and $A_{c}=\pi/(2\sqrt{2}\mu_{c})$ so as to the all fields satisfy the amplitude conditions. The influence of the state $|B^{\prime}\rangle$ looks more visible than that in Fig. 2 in the short duration regime, which becomes rather weak by increasing duration $\tau_{0}$. The final population $P_{B}^{L,R}(t_{f})$ can also reach high fidelity for $\tau_{0}>35$ns. As demonstrated in Fig. 2, the landscape of $P_{B}^{L,R}(t_{f})$ exhibits a chiral symmetry with respect to the phase $\phi_{b}$, for which the control field $\mathcal{E}_{b}(t)$ with $\phi_{b}=\pi/2$ leads to entire ESST to $|B\rangle$ of the left-handedness. For the right-handedness, however, it requires to $\phi_{b}=3\pi/2$. This dependence of $P_{B}^{L,R}(t_{f})$ on the phase is consistent with the theoretical predication.

Figure 6 plots the time-dependent populations of the system with $\tau_{0}=35$ and $\phi_{b}=\pi/2$. Since the transition moments $\mu_{b}$ are identical for the two enantiomers without a difference of sign, $\mathcal{E}_{b}(t)$ plays the same role in the first stage, generating the same maximal coherent superposition of $|A\rangle$ and $|C\rangle$. The constructive or destructive interference that depends on $\mu_{a}$ occurs between the transition paths from $|A\rangle$ and $|C\rangle$ to $|B\rangle$. As a result, the left-handed enantiomer is fully transferred to the state $|B\rangle$, whereas the right-handed one is still in the coherent states $|A\rangle$ and $|B\rangle$ at end of three pulses.

To see the robustness of the scheme on the time delays, Fig. 7 examines the dependence of $P_{B}^{L}(t_{f})$ on the time delays $t_{ab}=t_{a}-t_{b}$ and $t_{cb}=t_{c}-t_{b}$. Similar behaviors can be observed, indicating that both excitation schemes do not require strict separations between the control fields. The identical delays of the second stage fields are beneficial for the control. As a result, the amplitude and phase conditions can also be used for the overlapped control fields, leading to the high selectivity of handedness.

Finally, we examine the amplitude and phase conditions of control fields whose center frequencies $\omega_{a/b/c}$ are not exactly resonant with the transition center frequencies $\omega_{AB}$, $\omega_{BC}$ and $\omega_{AC}$. As can be seen from the definitions of the complex pulse areas $\theta_{a/b/c}(t_{f})$, when the center frequencies are detuned away from resonances, the values of $|\theta_{a/b/c}(t_{f})|$ will be decreased while keeping the parameters $A_{a/b/c}$ unchanged as used in resonant cases. We can increase the values of $A_{a/b/c}$ to revive the values of $\theta_{a/b/c}(t_{f})$ at the transition frequencies so as to satisfy the amplitude conditions. That is, ESST in principle could be achieved by using the center-frequency-detuned pulses, as long as they satisfy the amplitude and phase conditions $\mathcal{E}_{a/b/c}$ and $\phi_{a/b/c}$ at transition frequencies.

Figure 8 shows the dependence of $P_{B}^{L}(t_{f})$ on the detunings, for which the analytical simulations (left panels) are compared with the exact ones (right panels). We perform the simulations in Figs. 8 (a) and (b) to calculate $P_{B}^{L}(t_{f})$ for different values of $\Delta_{b}=\omega_{b}-\omega_{AC}$ while scaling the coupling $\Omega_{b}(t)$ with a factor $\alpha$, for which the center frequencies $\omega_{b}$ and $\omega_{c}$ are fixed at the resonant conditions. The simulations in Figs. 8 (c) and (d) are accomplished with different detunings $\Delta_{a}=\omega_{a}-\omega_{AB}$ and $\Delta_{c}=\omega_{c}-\omega_{BC}$ by scaling the couplings $\Omega_{a}(t)$ and $\Omega_{c}(t)$ with a factor $\beta$, for which we set $\Delta_{a}=\Delta_{c}=\Delta$ while fixing $\Delta_{b}=0$. For both analytical simulations, we can see that the detunings decrease $P_{B}^{L}(t_{f})$. By increasing the strengths of the control fields, the maximal value of $P_{B}^{L}(t_{f})$ can be revived to the same level as the resonant excitation, as shown in Figs. 8 (a) and (c). For the exact simulations, however, the maximal values can be increased but below the theoretical maximum. The differences can be attributed to the influence of high-order Magnus expansion terms, which are ignored in the analytical model. We also observe similar results for target state $|B\rangle$ (not shown here). Thus, the center-frequency detuned excitations with small detunings are also allowed by applying the corresponding amplitude and phase conditions at transition frequencies, whereas the larger detunings will reduce the ESST efficiency due to the optical processes via high-order high-order Magnus terms.