Enantiospecific Two-Photon Electric-Dipole Selection Rule of Chiral Molecules

TL;DR

The paper introduces an enantiospecific two-photon electric-dipole selection rule (TPSR) for chiral molecules under a static electric field, enabling enantiomer discrimination with only two drive beams without phase locking or resonance conditions. By analyzing the field-dressed Hamiltonian and exploiting symmetry under combined operations, it shows that the two-photon cascade transition can be forbidden for one handedness at a specific polarization angle $\theta_f^{(s)}(E_0)$, with the enantiospecificity quantified by $D(E_0)=\sin^2[(\theta_f^{(L)}-\theta_f^{(R)})/2]$. The theory provides explicit relations among intermediate-state amplitudes $a_\pm^{(s)}$ and $b_\pm^{(s)}$, and demonstrates how $\theta_f^{(L)}$ and $\theta_f^{(R)}$ diverge when $E_0>0$, enabling selective excitation or detection of a chosen enantiomer. Calculations for 1,2-propanediol indicate substantial enantiospecificity at modest fields and show potential for room-temperature enantiodetection and enantioseparation, significantly broadening the practicality of optical chiral differentiation.

Abstract

Distinguishing between enantiomers is crucial in the study of chiral molecules in chemistry and pharmacology. Many optical approaches rely on enantiospecific cyclic electric-dipole transitions induced by three microwave or laser beams. However, these approaches impose stringent requirements, including phase locking, three-photon resonance, and precise control over beam intensities and operation times, which enhance the complexity and restrict the applicability. In this letter, we present a novel optical method that {\it eliminates these constraints entirely.} Specifically, we demonstrate that in the presence of a static electric field, there is a selection rule for two-photon electric-dipole transitions that differs between enantiomers. This distinction arises because the static electric field breaks the symmetry associated with the combined action of a specific rotation and time-reversal transformation. Leveraging the enantiospecific two-photon selection rule, one can selectively excite a desired enantiomer using two beams, without the need for phase locking, resonance condition, and the precise control of their intensities and operation times. Our method significantly enhances the feasibility and applicability of optical approaches for enantiomer differentiation.

Figures (5)

• Figure 1: (a): The directions of the static E-field $\bm{E}$ (purple) and the polarizations of beams 1 (red) and 2 (blue). (b): The rovibrational levels and the circularly-polarized components of beams 1 (red) and 2 (blue) involved in the cascade transition of Eq. (\ref{['tp']}). (c): Enantiospecific transitions: when $\theta=\theta_{\rm f}^{(L/R)}$, the transition is allowed for the right-/left-handed enantiomer, but forbidden for the left-/right-handed one. (b) and (c) present the schematics for $\epsilon_{\alpha, 0}<\epsilon_{\beta,\pm 1}<\epsilon_{\gamma, 0}$. The ones for $\epsilon_{\alpha, 0}<\epsilon_{\gamma, 0}<\epsilon_{\beta,\pm 1}$ are given in Fig. \ref{['fig4']}.
• Figure 2: (a, b): The forbidden polarization angle $\theta_{\rm f}^{(L,R)}(E_0)$. (c, d): The degree $D(E_{0})$ of enantiospecificity of the TPSR. Here we show the results for transitions of Eq. (\ref{['tp']}) for 1,2-propanediol molecules, with relevant quantum numbers being $(\alpha=1;\beta=1;\gamma=4)$ (a, c), and $(\alpha=1;\beta=3;\gamma=4)$ (b, d).
• Figure 3: (a, b): Time evolution of the finial-state probability $P^{(L,R)}_\gamma(t)$ of the cascade transition (\ref{['tp']}) for $\theta=\theta_{\rm f}^{(L)}$ (a) and $\theta=\theta_{\rm f}^{(R)}$ (b). (c): The time-averaged finial-state probability $\bar{P}^{(L,R)}_\gamma$ as a function of $\theta$. In (a-c) we consider the cases with $\theta_{\rm f}^{(L)}=\pi/2$, $\theta_{\rm f}^{(R)}=-\pi/2$, $\epsilon_{\beta, \pm1}-\epsilon_{\alpha,0}-\omega_{1}={(2\pi)0.1}\,$MHz, $\epsilon_{\gamma,0}-\epsilon_{\beta,\pm 1}-\omega_2={(2\pi)0.4}\,$MHz, and $\Omega_{1}=\Omega_{2}=(2\pi)1\,$MHz, with $\omega_{1(2)}$ and $\Omega_{1(2)}$ being the angular frequency of beam 1(2) and the Rabi frequency of the transition induced by beam 1 (2), respectively. (d): Absorption $A(\theta)$ of beam 2, of left- and right-handed enantiomers. Here we show the results for the system with parameters (expect $\Omega_{2}$) being same as (c), and the spontaneous emission rates from $|\gamma,0,s\rangle$ to the lower states $|\alpha,0,s\rangle$ and $|\beta,\pm 1,s\rangle$ all being $\kappa=(2\pi)0.1\,$MHz. The details of the calculations for (a-d) are give in the SM SM.
• Figure 4: Schematics for $\epsilon_{\alpha, 0}<\epsilon_{\gamma, 0}<\epsilon_{\beta,\pm 1}$. (a): The rovibrational levels and the circularly-polarized components of beams 1 (red) and 2 (blue) involved in the cascade transition of Eq. (\ref{['tp']}). (b): Enantiospecific transitions.
• Figure S1: (a, b): The forbidden polarization angle $\theta_{\rm f}^{(L,R)}(E_0)$. (c, d): The degree $D(E_{0})$ of enantiospecificity of the TPSR. Here we show the results for transitions of Eq. (\ref{['tp']}) for 1,2-propanediol molecules, with relevant quantum numbers being $(\alpha=1;\beta=3;\gamma=2)$ (a, c), and $(\alpha=1;\beta=4;\gamma=2)$ (b, d).