Real single-loop cyclic three-level configuration of chiral molecules

Single-loop cyclic three-level ($\mathrm{\Delta }$-type) configuration of chiral molecules was used for enantio-separation in many theoretical works. Considering the effect of molecular rotation, this simple single-loop configuration is generally replaced by a complicated multiple-loop configuration containing multiple degenerate magnetic sublevels and the ability of the enantio-separation methods is suppressed. For chiral asymmetric top molecules, we propose a scheme to construct a real single-loop $\mathrm{\Delta }$-type configuration with no connections to other states by applying three microwave fields with appropriate polarization vectors and frequencies. With our scheme, the previous theoretical proposals for enantio-separation based on single-loop $\mathrm{\Delta }$-type configurations can be experimentally realized when the molecular rotation is considered.

Figure 1

$(J,\tau )$-level structure for the (multiple-loop) $\mathrm{\Delta }$-type configuration starting from the rotational ground state with cyclic transitions. (a) $|J=0,\tau =0\rangle \to |1,-1\rangle \to |1,1\rangle \to |0,0\rangle$; (b) $|0,0\rangle \to |1,-1\rangle \to |1,0\rangle \to |0,0\rangle$; (c) $|0,0\rangle \to |1,0\rangle \to |1,1\rangle \to |0,0\rangle$. The states are in the $|J,\tau \rangle$ notation. ${\mu }_x^m$, ${\mu }_y^m$, and ${\mu }_z^m$ are $nonzero$ electric dipoles in the molecule frame along the principal axis, and, respectively, proportional to the related reduced matrix elements (7).

Figure 2

Real single-loop $\mathrm{\Delta }$-type configuration starting from the rotational ground state for 1,2-propanediol. Three microwave fields with frequencies ${\nu }_1=6431.06\mathrm{MHz}$, ${\nu }_2=5781.09\mathrm{MHz}$, ${\nu }_3=12212.15\mathrm{MHz}$ and polarization vectors labeled with ${\sigma }_1=1$, ${\sigma }_2=-1$, ${\sigma }_3=0$ are applied to resonantly couple, respectively, with cyclic electric-dipole-allowed transitions $|a\rangle \to |b\rangle \to |c\rangle \to |a\rangle$. The dashed arrow denotes for the transition $|1,-1,0\rangle \to |1,1,-1\rangle$, which is coupled with the ${\sigma }_2=-1$ polarized electromagnetic field, and, however, is not involved in the single-loop $\mathrm{\Delta }$-type configuration based on $|a\rangle$, $|b\rangle$, and $|c\rangle$.

Figure 3

Real single-loop $\mathrm{\Delta }$-type configuration coupled with three linearly polarized electromagnetic fields for 1,2-propanediol. Their frequencies are ${\nu }_1=5781.09\mathrm{MHz}$, ${\nu }_2=6431.06\mathrm{MHz}$, ${\nu }_3=12212.15\mathrm{MHz}$. Their polarization vectors are $Z$, $X$, and $Y$, respectively. They are resonantly coupled, respectively, with cyclic electric-dipole-allowed transitions $|a\rangle \to |b\rangle \to |c\rangle \to |a\rangle$. Here $|a\rangle =|0,0,0\rangle$, $|b\rangle =|1,-1,0\rangle$, and $|c\rangle =\left(\right|1,1,1\rangle +|1,1,-1\rangle )/\sqrt{2}$. The dashed arrows denote for the transitions $|b^{\prime }\rangle \to |c^{\prime }\rangle$ and $|b^{\prime \prime }\rangle \to |c^{\prime }\rangle$, which are coupled with the $X$ polarized field, and, however, are not involved in the single-loop $\mathrm{\Delta }$-type configuration based on $|a\rangle$, $|b\rangle$, and $|c\rangle$. Here $|b^{\prime }\rangle =|1,-1,1\rangle$, $|b^{\prime \prime }\rangle =-|1,-1,-1\rangle$, $|c^{\prime }\rangle =-|1,1,0\rangle$, and $|c^{\prime \prime }\rangle =\left(\right|1,1,1\rangle -|1,1,-1\rangle )/\sqrt{2}$.