Optically-Induced Highly-Efficient Detection and Separation of Chiral Molecules through Shortcuts to Adiabaticity

TL;DR

This paper introduces a highly-efficient optical method for detecting and separating chiral molecules by exploiting phase-sensitive population dynamics in a three-state system, achieving perfect contrast using shortcuts to adiabaticity.

Contribution

It presents a novel phase-sensitive scheme utilizing shortcuts to adiabaticity for 100% contrast in chiral molecule detection and separation.

Findings

Achieves 100% population transfer contrast between enantiomers.

Uses phase-dependent population dynamics for chirality discrimination.

Applicable at both ensemble and single-molecule levels.

Abstract

A highly-efficient method for optical detection and separation of left- and right-handed chiral molecules is presented. The method utilizes a closed-loop three-state system in which the population dynamics depends on the phases of the three couplings. Due to the different signs of the coupling between two of the states for the opposite chiralities and the phase sensitivity of the scheme, the population dynamics is chirality-dependent. By using concepts from the `shortcuts to adiabaticity' approach, one can achieve 100\% contrast in state population transfer between the two enantiomers, which can be probed by light-induced fluorescence in the case of large ensembles or through resonantly-enhanced multiphoton ionization at the single molecular level.

Equations

$${\Omega }_p(t)$$

$${\Omega }_s(t)$$

$${\Omega }_q(t)$$

$$\mathbf{H}_{a}^{L,R}=\left[\begin{array}[]{ccc}-\frac{1}{2}\Omega&\frac{i}{\sqrt{2}}(\dot{\theta}\mp\frac{1}{2}\Omega_{q})&0\\ -\frac{i}{\sqrt{2}}(\dot{\theta}\mp\frac{1}{2}\Omega_{q})&0&-\frac{i}{\sqrt{2}}(\dot{\theta}\mp\frac{1}{2}\Omega_{q})\\ 0&\frac{i}{\sqrt{2}}(\dot{\theta}\mp\frac{1}{2}\Omega_{q})&\frac{1}{2}\Omega\end{array}\right],$$

$$P_3^L$$

$$P_3^R$$

Full text

Supplementary information for the article “Optically-Induced Highly-Efficient Detection and Separation of Chiral Molecules through Shortcuts to Adiabaticity”

by Nikolay V. Vitanov and Michael Drewsen

In addition to the Gaussian pulse shapes considered in the main text, we present here an exactly soluble analytic model. This analytic solution presents an explicit expression for the transition probability, which allows one to easily analyze its properties. This model demonstrates that the proposed method for chirality resolution is not limited to the Gaussian pulse shapes but is generally applicable.

The pulse shapes in the analytic model are Vitanov1997

[TABLE]

with $\Omega(t)=\Omega_{0}\text{sech}\,^{2}(t/T)$ and $\theta(t)=\arctan[\Omega_{p}(t)/\Omega_{s}(t)]=[\tanh(t/T)+1]\pi/4$. Then $\dot{\theta}(t)=(\pi/4T)\text{sech}\,^{2}(t/T)$ and $\Omega_{q}(t)=2\dot{\theta}(t)=(\pi/2T)\text{sech}\,^{2}(t/T)$. These pulse shapes are plotted in Fig. 1. The pulse area of each of the P and S pulses is $A=(4/\pi)\Omega_{0}T$ and the pulse area of the Q-field is $\pi$. This model has been solved in Ref. Vitanov1997 in the absence of the Q-field, $\Omega_{q}(t)=0$. Here we extend the solution to nonzero $\Omega_{q}(t)$.

The Hamiltonian in the adiabatic basis

[TABLE]

has SU(2) symmetry and can be reduced to an effective two-state Hamiltonian Vitanov1997bec ; Randall2018 . However, this is not necessary here because for the pulse shapes of Eq. (1), $\mathbf{H}_{a}^{L,R}$ contains the same time-dependent function $\text{sech}\,^{2}(t/T)$ in all of its elements. After the change of the independent variable $t\rightarrow\theta(t)$, $\mathbf{H}_{a}^{L,R}$ becomes constant and readily solved. The propagator reads $\mathbf{U}_{a}=\exp[-i\int_{-\infty}^{\infty}\mathbf{H}_{a}^{L,R}(t)\mathrm{d}t]$. Its explicit calculation is straightforward but the final expression is too cumbersome to be presented here. The probability for transition $|1\rangle\rightarrow|3\rangle$ is equal to the probability to remain in the dark state $|\chi_{0}\rangle=\cos\theta|1\rangle-\sin\theta|3\rangle$ because $|\chi_{0}\rangle\rightarrow|1\rangle$ as $t\rightarrow-\infty$ and $|\chi_{0}\rangle\rightarrow-|3\rangle$ as $t\rightarrow\infty$. Therefore the population of the bare molecular state $|3\rangle$ reads $P_{3}=P_{0}=|(\mathbf{U}_{a})_{22}|^{2}$, or explicitly,

[TABLE]

The population $P_{3}^{R}$ vanishes when the P and S pulse area is $A\approx 1.017\pi$. Then the contrast between L and R handedness is maximal.

The population (3b) of state $|3\rangle$ for the analytic model (1) is plotted vs the P and S pulse area $A$ in Fig. 2. For the L handedness, $P_{3}=1$ due to “shortcut to adiabaticity” effect, while it oscillates for the R handedness. The largest contrast between the L and R signals occurs for $A\approx 1.017\pi$, where the R signal vanishes. At this value of the pulse areas the chiral resolution is ideal: a nonzero signal from state $|3\rangle$ unambiguously indicates a certain handedness while no signal indicates the opposite one.