# Enantioselective Topological Frequency Conversion

Kai Schwennicke and Joel Yuen-Zhou  Department of Chemistry and Biochemistry,
University of California, San Diego, La Jolla, California 92093, United States
<EMAIL_ADDRESS>
###### Abstract

Two molecules are enantiomers if they are non-superimposable mirror images of
each other. Electric dipole-allowed cyclic transitions
$|1\rangle\to|2\rangle\to|3\rangle\to|1\rangle$ obey the symmetry relation
$\mathcal{O}^{R}=-\mathcal{O}^{S}$, where
$\mathcal{O}^{R,S}=(\mu_{21}^{R,S}E_{21})(\mu_{13}^{R,S}E_{13})(\mu_{32}^{R,S}E_{32})$,
and $R,S$ label the two enantiomers. Here we generalize the concept of
topological frequency conversion to an ensemble of enantiomers. We show that,
within a rotating-frame, the pumping power between fields of frequency
$\omega_{1}$ and $\omega_{2}$ is sensitive to enantiomeric excess,
$\mathcal{P}_{2\to
1}=\hbar\frac{\omega_{1}\omega_{2}C_{L}^{R}}{2\pi}(N_{R}-N_{S})$, where
$N_{i}$ is the number of enantiomers $i$ and $C_{L}^{R}$ is an enantiomer-
dependent Chern number. Connections with chiroptical microwave spectroscopy
are made. Our work provides an underexplored and fertile connection between
topological physics and molecular chirality.

In the mid-nineteenth century, Louis Pasteur discovered that molecules can
possess handedness, or chirality, an attribute that influences how they
interact with their surroundings 1. More generally, the two species of a
chiral molecule, referred to as enantiomers, are nonsuperimposable mirror
images of each other and, while they feature many identical physicochemical
properties (up to very small parity violation corrections 2), they can also
exhibit drastically different behavior when exposed to chiral environments or
stimuli. Thus, enantioselectivity plays a crucial role in biological activity
as well as in the synthesis, purification, and characterization of
pharmaceuticals 3, 4, 5. Traditionally, optical rotation and circular
dichroism have served as optical tools to obtain enantioselective information;
however, these techniques rely on the weak interaction between molecules and
the magnetic component of the optical field. A very active effort in chirality
research consists of spatially shaping electromagnetic fields 6, 7, 8 to
enhance these weak interactions. Other techniques that rely solely on electric
dipole interactions 9 have been recently advocated. For instance, many efforts
are currently invested in photoelectron circular dichroism (PECD) 10, 11, 12,
13. Yet, others focus on nonlinear optical signals that depend on the sign of
the electric fields with which the molecules interact 14, 15, including
photoexcitation circular dichroism 16, the use of synthetic chiral fields 17,
18, 19, 20, and microwave three-wave mixing 21, 22, 23, 24, 25. More
precisely, the latter technique can be understood through cyclic three-level
models 26, 27, 28, 29, 30, 31, 32 where the product of three light-matter
couplings [hereafter referred to as the Král-Shapiro (KS) product] differs by
a phase of $\pi$ between the two enantiomers. This remarkable symmetry has
been exploited to propose cyclic population transfer schemes 26, 32 or the use
of cross-polarized terahertz pulses 33 to prepare the enantiomers in different
energy configurations or orientations for separation. This symmetry has also
been utilized to suggest an enantioselective generalization of the Stern-
Gerlach 34 or spin Hall 35 experiments, where spatial separation of
enantiomers, rather than spins, is achieved using artificial gauge fields 36,
37, 38. The analogy between enantiomer and spin labels is intriguing and
surprisingly underexplored, and serves as the motivation of our present work.
More specifically, we wish to demonstrate an enantioselective analogue to the
Quantum Spin Hall Effect (QSHE) 39.

On the other hand, since the pioneering work of Thouless, Kohmoto,
Nightingale, and den Nijs in relation to the Quantum Hall Effect (QHE) 40,
notions of symmetry-protected topological phases (SPTPs) have been at the
heart of condensed matter research, and have only been exacerbated in the past
fifteen years with the discovery of topological insulators 41. These notions
guarantee that certain response properties of so-called topologically
nontrivial systems are largely independent of material specification, instead
depending only on products of universal constants and integer quantities known
as topological invariants. The discrete nature of these properties implies
that they are robust against material imperfections, thus making them
attractive for metrology, among other applications. While topological
protection was originally identified in translationally invariant 2D systems,
its scope has been enlarged through the use of Floquet engineering in systems
of different dimensionality 42, 43, 44, 45 and the consideration of the 2D
phase space of 1D systems 46, 47. Of particular interest is an elegant
construction due to Martin, Refael, and Halperin 48 called topological
frequency conversion (TFC), where quantized "current" is observed. In this
Letter, we design a novel spectroscopic scheme that generalizes TFC to the
microwave spectroscopy of an ensemble of chiral molecules. The very first link
between chiroptical spectroscopy and topology was suggested recently in work
by Ordoñez and Smirnova 49 within the context of PECD. These authors showed
that the propensity field (a pseudoscalar) as a function of ejected
photoelectron direction (Berry curvature) can be integrated over all solid
angles to yield a quantized enantiosensitive flux which is proportional to a
Chern number. Similarly, the authors showed that a microwave three-wave mixing
signals can be interpreted in terms of an analogous quantity to the propensity
field 9. However, it is not clear from that work if there exists a parameter
space upon which integration of the signals lead to topological invariants, so
geometric and topological consequences of these nonlinear spectroscopies were
not explored. In this Letter, we use TFC to identify time as the missing
parameter space and for simplicity, restrict our attention to frequency
conversion rather than three-wave mixing. The result is a signal that is
proportional to enantiomeric excess (EE), with a simple prefactor containing
the sign of the KS product. Owing to the topological nature of the signal, it
should also serve as a very sensitive detection of EE. As far as we are aware,
our work provides the first connection between topological physics,
chiroptical spectroscopy, and nonlinear spectroscopy, and anticipates a
fertile ground for further exploration.

(a)  (b)

Figure 1: _The model._ (a) Cyclic three-level transitions for an asymmetric
top, such as enantiomers. Three near-resonant, linear polarized lasers with
modulated field amplitudes $\mathcal{E}_{ij}(t)$ interact with these
transitions. (b) The principal axes components of the dipole moments for the
$R-$ and $S-$1,2-propanediol enantiomers. Note that
$(\boldsymbol{\mu}_{a}^{R}\cdot\hat{\boldsymbol{a}})(\boldsymbol{\mu}_{b}^{R}\cdot\hat{\boldsymbol{b}})(\boldsymbol{\mu}_{c}^{R}\cdot\hat{\boldsymbol{c}})=-(\boldsymbol{\mu}_{a}^{S}\cdot\hat{\boldsymbol{a}})(\boldsymbol{\mu}_{b}^{S}\cdot\hat{\boldsymbol{b}})(\boldsymbol{\mu}_{c}^{S}\cdot\hat{\boldsymbol{c}})$

Following the principles of enantioselective microwave three wave-mixing 50,
51, we treat the enantiomers as asymmetric tops whose Hamiltonian is

$H_{0}=AJ_{a}^{2}+BJ_{b}^{2}+CJ_{c}^{2},$ (1)

where $J_{a}$, $J_{b}$, $J_{c}$ are the angular momentum operators with
respect to the principal axes $\hat{\boldsymbol{a}}$, $\hat{\boldsymbol{b}}$,
$\hat{\boldsymbol{c}}$, and $A>B>C$ are the corresponding rotational
constants. The eigenstates are labeled as $|J,\tau,M\rangle$, where
$J=0,1,2...$ is the rotational quantum number, $M=-J,-J+1,-J+2,...,J$ is the
quantum number that characterize the projection of the total angular momentum
along the $z$-laboratory-fixed axis, and $\tau$ serves as the quantum number
to differentiate between states with the same $J$ and $M$. We consider the the
following low angular momentum eigenstates of Eq. 1 with a rotational quantum
number of $J=0$ or $J=1$

$\displaystyle|0,\tau=1,0\rangle$ , $\displaystyle|1,\tau=2,M\rangle$ ,
$\displaystyle|1,\tau=3,M\rangle$ , (2)

where $M=-1,0,1$ 52 (see Supporting Information Section 1, SI-1). The ground
state $|0,\tau=1,0\rangle$, with energy $\hbar\epsilon_{1}$, and the excited
states $|1,\tau=2,M\rangle$ and $|1,\tau=3,M\rangle$, with energies
$\hbar\epsilon_{2}$, $\hbar\epsilon_{3}$, respectively, are coupled to each
other using a set of three orthogonally-polarized time-dependent electric
fields

$\displaystyle\boldsymbol{\mathbb{E}}_{21}(t)$ $\displaystyle={\cal
E}_{21}(t)\sin(\Omega_{21}t)\hat{\boldsymbol{y}},$
$\displaystyle\boldsymbol{\mathbb{E}}_{32}(t)$
$\displaystyle=\mathcal{E}_{32}(t)\cos(\Omega_{32}t)\hat{\boldsymbol{x}},$
$\displaystyle\boldsymbol{\mathbb{E}}_{31}(t)$
$\displaystyle=\mathcal{E}_{31}(t)\cos(\Omega_{31}t)\boldsymbol{\hat{\boldsymbol{z}}},$
(3)

where $\hat{\boldsymbol{x}}$, $\hat{\boldsymbol{y}}$, $\hat{\boldsymbol{z}}$
denote the three laboratory-fixed axes, the frequencies
$\Omega_{21}=\epsilon_{2}-\epsilon_{1}-\delta$,
$\Omega_{32}=\epsilon_{3}-\epsilon_{2}-\delta$,
$\Omega_{31}=\epsilon_{3}-\epsilon_{1}-2\delta$, are slightly detuned from the
system’s natural frequencies, and the field amplitudes $\mathcal{E}_{21}(t)$,
$\mathcal{E}_{32}(t)$, $\mathcal{E}_{31}(t)$ are slowly modulated. Note from
the selection rules for electric dipole interactions 51, that $\Delta M=0$ for
the $z$ polarized field and $\Delta M=\pm 1$ for the $x$ and $y$ polarized
field (see Fig. 1). Ignoring all states that are not coupled through the
driving electric fields, and assuming that
$|\mu_{i,M^{\prime};j,M}^{R,S}\mathcal{E}_{ij}(t)|/2\ll\hbar\Omega_{ij}$, the
Hamiltonian for the laser dressed $R-$ and $S-$enantiomer, after making the
rotating wave approximation, is

$\displaystyle H^{R,S}(t)$
$\displaystyle=\sum_{i=1,3}\hbar\epsilon_{i}|i,0\rangle\langle
i,0|+\hbar\epsilon_{2}\sum_{M=\pm 1}|2,M\rangle\langle 2,M|$
$\displaystyle-\mathcal{E}_{21}(t)\sum_{M=\pm
1}\Big{(}\frac{i\mu_{2,M;1,0}^{R,S}e^{-i\Omega_{21}t}}{2}|2,M\rangle\langle
1,0|+\text{h.c.}\Big{)}$ $\displaystyle-\mathcal{E}_{32}(t)\sum_{M=\pm
1}\Big{(}\frac{\mu_{3,0;2,M}^{R,S}e^{-i\Omega_{32}t}}{2}|3,0\rangle\langle
2,M|+\text{h.c.}\Big{)}$
$\displaystyle-\mathcal{E}_{31}(t)\Big{(}\frac{\mu_{3,0;1,0}^{R,S}e^{-i\Omega_{31}t}}{2}|3,0\rangle\langle
1,0|+\text{h.c.}\Big{)},$ (4)

where for simplicity we have introduced the notation
$|1,0\rangle\equiv|0,\tau=1,0\rangle$, $|2,M\rangle\equiv|1,\tau=2,M\rangle$,
$|3,0\rangle\equiv|1,\tau=3,0\rangle$. In Eq. 4 $\mu_{i,M^{\prime};j,M}^{R,S}$
is the component of the transition-dipole moment for the
$|j,M\rangle\to|i,M^{\prime}\rangle$ transition that is projected along the
polarization axis of $\boldsymbol{\mathbb{E}}_{ij}(t)$ . Following the
procedure of Refs. 50, 51, the values of $\mu_{i,M^{\prime};j,M}^{R,S}$ are

$\displaystyle\mu_{2,\pm 1;1,0}^{R,S}$
$\displaystyle=-\frac{i\mu_{b}^{R,S}}{\sqrt{6}},$ $\displaystyle\mu_{3,0;2,\pm
1}^{R,S}$ $\displaystyle=\frac{\mu_{a}^{R,S}}{2\sqrt{2}},$
$\displaystyle\mu_{3,0;1,0}^{R,S}$
$\displaystyle=-\frac{i\mu_{c}^{R,S}}{\sqrt{3}},$ (5)

where $\mu_{a}^{R,S}$,$\mu_{b}^{R,S}$,$\mu_{c}^{R,S}$ are the components of
the dipole moment along the principal molecular axes. These components are
real valued and $|\mu_{a}^{R}|=|\mu_{b}^{S}|$. Note that for the chosen
polarizations for the three electric fields (see Eq. 3) and studied energy
levels, $\mu_{i,M^{\prime};j,M}^{R,S}$ does not depend on the quantum number
$M$. The associated time-dependent wavefunction for the $R-$ and $S-$
enantiomer the system is $|\psi^{R,S}(t)\rangle$.

Next, we consider the rotating frame

$U(t)=e^{-i(\epsilon_{2}-\Omega_{21})t}|1,0\rangle\langle 1,0|+\sum_{M=\pm
1}e^{-i\epsilon_{2}t}|2,M\rangle\langle
2,M|+e^{-i(\epsilon_{2}+\Omega_{32})t}|3,0\rangle\langle 3,0|,$ (6)

such that $|\psi^{R,S}(t)\rangle=U(t)|\tilde{\psi}^{R,S}(t)\rangle$, in order
to remove the central frequencies $\Omega_{ij}$. In this frame,
$i\hbar\partial_{t}|\tilde{\psi}{}^{R,S}(t)\rangle=\mathcal{H}^{R,S}(t)|\tilde{\psi}^{R,S}(t)\rangle$,
with the effective Hamiltonian:

$\mathcal{H}^{R,S}(t)=\frac{1}{2}\begin{pmatrix}-2\hbar\delta&-\frac{\mu_{b}^{R,S}}{\sqrt{6}}\mathcal{E}_{21}(t)&-\frac{\mu_{b}^{R,S}}{\sqrt{6}}\mathcal{E}_{21}(t)&-\frac{i\mu_{c}^{R,S}}{\sqrt{3}}\mathcal{E}_{31}(t)\\\
-\frac{\mu_{b}^{R,S}}{\sqrt{6}}\mathcal{E}_{21}(t)&0&0&-\frac{\mu_{a}^{R,S}}{2\sqrt{2}}\mathcal{E}_{32}(t)\\\
-\frac{\mu_{b}^{R,S}}{\sqrt{6}}\mathcal{E}_{21}(t)&0&0&-\frac{\mu_{a}^{R,S}}{2\sqrt{2}}\mathcal{E}_{32}(t)\\\
\frac{i\mu_{c}^{R,S}}{\sqrt{3}}\mathcal{E}_{31}(t)&-\frac{\mu_{a}^{R,S}}{2\sqrt{2}}\mathcal{E}_{32}(t)&-\frac{\mu_{a}^{R,S}}{2\sqrt{2}}\mathcal{E}_{32}(t)&2\hbar\delta\end{pmatrix}.$
(7)

After a change of basis (see SI-2), we arrive at the following effective
Hamiltonian

$\displaystyle\mathcal{H}^{R,S}(t)$
$\displaystyle=-\frac{\mu_{b}^{R,S}\mathcal{E}_{21}(t)}{2\sqrt{3}\hbar}L_{x}-\frac{\mu_{a}^{R,S}\mathcal{E}_{32}(t)}{4\hbar}L_{y}$
$\displaystyle+\frac{\mu_{c}^{R,S}\mathcal{E}_{31}(t)}{2\sqrt{3}\hbar}L_{z}-\frac{\delta}{2\hbar}(L_{+}^{2}+L_{-}^{2})$
(8)

where $L_{x}=\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix}$, $L_{y}=\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{pmatrix}$, $L_{z}=\hbar\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}$ are the angular momentum operators for a spin-1 particle
and $L_{+}=\sqrt{2}\hbar\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\end{pmatrix}$,
$L_{-}$$=\sqrt{2}\hbar\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&1&0\end{pmatrix}$ are
the corresponding ladder operators. We use the form of the effective
Hamiltonian in Eq. 8 to calculate the toplogy of the system. Hereafter, we
will assume that the slowly-modulated electric field amplitudes are

$\displaystyle\mathcal{E}_{21}(t)$ $\displaystyle=E_{21}\sin(\omega_{1}t),$
$\displaystyle\mathcal{E}_{32}(t)$ $\displaystyle=E_{32}\sin(\omega_{2}t),$
$\displaystyle\mathcal{E}_{31}(t)$
$\displaystyle=E_{31}[m-\cos(\omega_{1}t)-\cos(\omega_{2}t)],$ (9)

where $\omega_{1}$, $\omega_{2}$ are two modulation frequencies, and $m$ is a
scalar that characterizes a non-modulated component of the electric field.
These functional forms are inspired from the TFC scheme reported in Ref. 53.

TFC. —For completeness, we briefly rederive the TFC formalism using adiabatic
perturbation theory (the original paper does so within Floquet theory 48). In
the rotating frame, the rate of the system’s energy absorption for the
enantiomers is given by
$\partial_{t}E^{R,S}(t)=\langle\tilde{\psi}^{R,S}(t)|\partial_{t}\mathcal{H}^{R,S}(t)|\tilde{\psi}^{R,S}(t)\rangle$.
In the long time limit, $t\to\infty,$ the time-averaged energy-absorption
rate, or average power, is

$\displaystyle\mathcal{P}_{av}^{R,S}$
$\displaystyle=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}\partial_{t^{\prime}}E^{R,S}(t^{\prime})=\sum_{\omega_{i}}\mathcal{P}_{av}^{R,S}(\omega_{i}),$
(10a) $\displaystyle\mathcal{P}_{av}^{R,S}(\omega_{i})$
$\displaystyle=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}\omega_{i}\langle\partial_{\omega_{i}t^{\prime}}\mathcal{H}^{R,S}(t^{\prime})\rangle,$
(10b)

where $\mathcal{P}_{av}^{R,S}(\omega_{i})$ is the average power at the
modulation frequency $\omega_{i}$.

Let $|\epsilon_{l}^{R,S}(t)\rangle$ denote the $l-$th adiabatic state of
$\mathcal{H}^{R,S}(t),$ where
$\mathcal{H}^{R,S}(t)|\epsilon_{l}^{R,S}(t)\rangle=\epsilon_{l}^{R,S}(t)|\epsilon_{l}^{R,S}(t)\rangle$
(Fig. 2). If $\omega_{1},\omega_{2}$ are incommensurate, _i.e._ ,
$\omega_{1}/\omega_{2}$ is irrational, $\mathcal{H}^{R,S}(t)$ is not periodic.
However, if we write
$\mathcal{H}^{R,S}(t)=\mathcal{H}^{R,S}(\boldsymbol{\theta})=\mathcal{H}^{R,S}(\theta_{1},\theta_{2})$
with $\theta_{i}=\omega_{i}t\,(\text{mod}\,2\pi)$, we notice that
$\mathcal{H}^{R,S}(\boldsymbol{\theta})$ is quasiperiodic,
$\mathcal{\mathcal{H}}^{R,S}(\theta_{1}+2\pi,\theta_{2})=\mathcal{H}^{R,S}(\theta_{1},\theta_{2}+2\pi)=\mathcal{H}^{R,S}(\theta_{1},\theta_{2})$,
and the domain of $\mathcal{H}^{R,S}(\theta_{1},\theta_{2})$ is a two-
dimensional torus $\mathbb{T}=[0,2\pi)\otimes[0,2\pi)$.Near the adiabatic
limit where $\omega_{1}$, $\omega_{2}$ are much smaller than the instantaneous
energy gap of $\mathcal{H}^{R,S}(t)$, and if the system is initiated in the
$l-$th adiabatic state, _i.e.,_
$|\tilde{\psi}^{R,S}(0)\rangle=|\epsilon_{l}^{R,S}(0)\rangle$, the expected
quantities $\langle\partial_{\omega_{1}t}\mathcal{H}^{R,S}(t)\rangle$ and
$\langle\partial_{\omega_{2}t}\mathcal{H}^{R,S}(t)\rangle$ for
$|\tilde{\psi}^{R,S}(t)\rangle$, to first order in $\omega_{1}$, $\omega_{2}$
are

$\displaystyle\langle\partial_{\omega_{1}t}\mathcal{H}^{R,S}(t)\rangle=\langle\partial_{\theta_{1}}\mathcal{H}^{R,S}(\boldsymbol{\theta})\rangle$
$\displaystyle=\partial_{\theta_{1}}\epsilon_{l}^{R,S}(\boldsymbol{\theta})-\hbar\omega_{2}F_{l}^{R,S}(\boldsymbol{\theta})$
(11a)
$\displaystyle\langle\partial_{\omega_{2}t}\mathcal{H}^{R,S}(t)\rangle=\langle\partial_{\theta_{2}}\mathcal{H}^{R,S}(\boldsymbol{\theta})\rangle$
$\displaystyle=\partial_{\theta_{2}}\epsilon_{l}^{R,S}(\boldsymbol{\theta})+\hbar\omega_{1}F_{l}^{R,S}(\boldsymbol{\theta})$
(11b)

where
$F_{l}^{R,S}(\boldsymbol{\theta})=i\langle\partial_{\theta_{1}}\epsilon_{l}^{R,S}(\boldsymbol{\theta})|\partial_{\theta_{2}}\epsilon_{l}^{R,S}(\boldsymbol{\theta})\rangle+\text{h.c.}$
is the Berry curvature of the $l-$th adiabatic band (see, SI-3).

According to the mean-value theorem for incommensurate $\omega_{1}$,
$\omega_{2}$ 54, the linear flow of $\boldsymbol{\theta}$ covers the torus
densely for long enough times. Thus, the time average of $F_{l}^{R,S}(t)$ is
the same as the average of $F_{l}^{R,S}(\boldsymbol{\theta})$ over the entire
torus $\mathbb{T}$:

$\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}F_{l}^{R,S}(t^{\prime})=\frac{1}{4\pi^{2}}\int_{\mathbb{T}}d\boldsymbol{\theta}F_{l}^{R,S}(\boldsymbol{\theta}).$
(12)

From a practical standpoint, $t\to\infty$ means
$t>p\frac{2\pi}{\omega_{1}}=q\frac{2\pi}{\omega_{2}}$, where
$\frac{\omega_{1}}{\omega_{2}}\approx\frac{p}{q}$ for $p,q\in\mathbb{Z}^{+}$.
Substituting Eqs. 11a-11b into Eq. 10b gives rise to the average power lost by
the fields at $\omega_{1},$ $\omega_{2}$ when the system is initiated in the
$l-$th band,
$\mathcal{P}_{av}^{R,S}(\omega_{1})=-\mathcal{P}_{av}^{R,S}(\omega_{2})=-\frac{\hbar\omega_{1}\omega_{2}C_{l}^{R,S}}{2\pi}$.
Here the average of
$\partial_{\theta_{i}}\epsilon_{l}^{R,S}(\boldsymbol{\theta})$ is zero since
$\epsilon_{l}^{R,S}(\boldsymbol{\theta})$ is quasiperiodic in
$\boldsymbol{\theta}$, and
$C_{l}^{R,S}=\frac{1}{2\pi}\int_{\mathbb{T}}d\boldsymbol{\theta}F_{l}^{R,S}(\boldsymbol{\theta})$
is the Chern number of the $l-$th band for the corresponding enantiomer. Thus,
the enantiomer dependent average energy-pumping rate between the two
modulation fields $\mathcal{P}_{2\to
1}^{R,S}=[\mathcal{P}_{av}^{R,S}(\omega_{2})-\mathcal{P}_{av}^{R,S}(\omega_{1})]/2$
is quantized,

$\displaystyle\mathcal{P}_{2\to 1}^{R,S}$
$\displaystyle=\frac{\hbar\omega_{1}\omega_{2}C_{l}^{R,S}}{2\pi},$ (13)

or in other words, after one period of the $\omega_{2}$ modulation,
$C_{l}^{R,S}$ photons with frequency $\omega_{1}$ are produced. The photons
produced are in the same spatial modes as the incoming electric fields. The
very off-resonant nature of this process guarantees that the molecule does not
retain energy and the energy transfer process occurs only between the fields.

Figure 2: _Example adiabatic bands giving rise to enantioselective TFC._ The
color gradient is a visual aid for the band dispersion.

For $\delta=0,$ $\mathcal{H}^{R,S}(\boldsymbol{\theta})$ (see Eq. 8),
resembles half of the Bernevig-Hughes-Zhang Hamiltonian 55, except that the
Pauli matrices are replaced with the spin-1 angular momentum operators. As
expected, $\mathcal{H}^{R,S}(\boldsymbol{\theta})$ is topologically non-
trivial for $|m|<2$, where the Chern numbers for the upper ($U$) and lower
($L$) adiabatic bands remarkably acquire the value,

$C_{U}^{R,S}=2\text{sgn}(m)\text{sgn}(\mathcal{O}^{R,S})=-C_{L}^{R,S},$ (14)

and that for the middle ($M$) band $C_{M}=0$ (for an analytical proof, see
SI-4 ).
$\mathcal{O}^{R,S}=(\mu_{b}^{R,S}E_{12})(\mu_{a}^{R,S}E_{23})(\mu_{c}^{R,S}E_{31})$
is the KS product which obeys the enantioselective symmetry relation
$\mathcal{O}^{R}=-\mathcal{O}^{S}$, since
$\mu_{a}^{R}\mu_{b}^{R}\mu_{c}^{R}=-\mu_{a}^{S}\mu_{b}^{S}\mu_{c}^{S}$, and we
have assumed that $E_{ij}=E_{ji}$. Therefore $C_{L}^{R}=-C_{L}^{S}$, and the
TFC for the two enantiomers initiated in the lower (upper) adiabatic band at
$t=0$ is expected to have the same magnitude but opposite sign, _i.e._ ,
$\mathcal{P}_{2\to 1}^{R}=-\mathcal{P}_{2\to 1}^{S}$. This results begs us to
consider the fruitful analogy between enantiomer label and spin degrees of
freedom. Just like in the QSHE, where the transverse conductivity for opposite
spins bears opposite signs, so does the TFC for opposite enantiomers. Eq. 14
is the central result of this letter and relates a fundamental topological
invariant from chiroptical spectroscopy ($\text{sgn}\mathcal{O}=\pm 1$) with
the notions of SPTPs. Fig. 3 shows the computed value of $C_{L}^{R}$ for
different values of $m$ when $\delta\neq 0.$

Figure 3: _Topological phase diagram._ The value of $C_{L}^{R}$ is calculated
taking the magnitudes of the light-matter couplings to be equal, _i.e.,_
$|\mu_{2,M^{\prime};1M}^{R}E_{21}|=|\mu_{3,M^{\prime};2,M}^{R}E_{32}|=|\mu_{3,M^{\prime};1,M}^{R}E_{31}|=\hbar
D$, while the laser-driving parameters $m$ and $\delta$ are varied. We obtain
$C_{L}^{R}=-2\text{sgn}(m)\text{sgn}(\mathcal{O}^{R})$ at the vicinity of
$\delta=0$, where $\mathcal{O}^{R}=-\mathcal{O}^{S}$ is the Král-Shapiro
product, which is enantioselective.

By analogy with Eq. 10, we can compute the enantiomer dependent average power
absorbed in the original frame as
$P_{av}^{R,S}(\Omega)=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}\Omega\langle\partial_{\Omega
t^{\prime}}H^{R,S}(t^{\prime})\rangle$, obtaining:

$\displaystyle\frac{\mathcal{P}_{av}^{R,S}(\omega_{1})}{\hbar\omega_{1}}$
$\displaystyle=\frac{P_{av}^{R,S}(\Omega_{21+1})}{\hbar\Omega_{21+1}}-\frac{P_{av}^{R,S}(\Omega_{21-1})}{\hbar\Omega_{21-1}}$
(15a)
$\displaystyle+\frac{P_{av}^{R,S}(\Omega_{31+1})}{\hbar\Omega_{31+1}}-\frac{P_{av}^{R,S}(\Omega_{31-1})}{\hbar\Omega_{31-1}},$
$\displaystyle\frac{\mathcal{P}_{av}^{R,S}(\omega_{2})}{\hbar\omega_{1}}$
$\displaystyle=\frac{P_{av}^{R,S}(\Omega_{32+2})}{\hbar\Omega_{32+2}}-\frac{P_{av}^{R,S}(\Omega_{32-2})}{\hbar\Omega_{32-2}}$
$\displaystyle+\frac{P_{av}^{R,S}(\Omega_{31+2})}{\hbar\Omega_{31+2}}-\frac{P_{av}^{R,S}(\Omega_{31-2})}{\hbar\Omega_{31-2}},$
(15b)

where $\Omega_{ij\pm 1,2}=\Omega_{ij}\pm\omega_{1,2}$. Thus, the quantization
due to the enantioselective TFC can be extracted from an experimentally
detected difference power spectrum of the fields interacting with the
molecule. Notice that the topology is preserved for $\delta\neq 0$ as long as
$\hbar|\delta|<|\mu_{i,M^{\prime};j,M}^{R,S}E_{ij}|/2$. In general, we expect
our scheme to maintain the nontrivial topology with respect to changes in
experimental conditions (such as laser spot size or collection efficiency) so
long as adiabaticity still holds and the necessary peaks in the power spectrum
can be resolved.

Figure 4: _Adiabatic state preparation._ Presented are the plots for the
functions $\alpha(t)$ and $\beta(t)$. We also feature the populations
$|\tilde{c}_{L}^{R}|^{2}$, $|\tilde{c}_{L}^{S}|^{2}$ (shifted vertically
slightly to be visible) of the lower adiabatic state for each enantiomer. As
shown, the system is effectively prepared in the lower adiabatic bands for
both enantiomers.

The dynamics of the system is calculated by numerically integrating the
Schrödinger equation in the rotating frame (Eq. 7), and the power spectrum is
obtained by returning to the original frame. In atomic units ($\hbar=1$), the
electric field amplitudes are taken to be $E_{21}=5\sqrt{3}E_{0}$,
$E_{32}=6E$, $E_{31}=\sqrt{3}E_{0}$, where $E_{0}=4.0\times
10^{-9}\text{\;a.u.}$, the dipole moment principal axes components are
$\mu_{a}^{R}=\text{$\mu_{a}^{S}$}=0.47\text{\;a.u.}$,
$\mu_{b}^{R}=\mu_{b}^{S}=0.75\text{\;a.u.}$,
$\mu_{c}^{R}=-\text{$\mu_{c}^{S}$}=0.14\text{\;a.u.}$, and the molecular
transition energies are $\epsilon_{2}-\epsilon_{1}=4.4\times
10^{-8}\;\text{a.u.}$ and $\epsilon_{3}-\epsilon_{1}=4.7\times
10^{-8}\;\text{a.u.}$ The dipole moment components and molecular energies are
extracted from a microwave three-wave-mixing model for $R-$ and
$S-$1,2-propanediol 21. Using these parameters, it is true that
$|\mu_{i,M^{\prime};j,M}^{R,S}E_{ij}|/2\ll\hbar\Omega_{ij}$, so the rotating
wave approximation holds. The slow incommensurate modulation frequencies and
laser detuning are taken to be $\omega_{1}=\omega_{2}/\phi=\delta=1\times
10^{-11}\text{\;a.u.}$, where we take $\phi=\frac{\sqrt{5}-1}{2}$ as in Ref.
48, satisfying the perturbative condition
$\hbar|\delta|,\hbar\omega_{1},\hbar\omega_{2}\ll|\mu_{i,M;j,M^{\prime}}^{R,S}\cdot
E_{ij}|/2$. Setting $m=1.4$, the system is in the topologically nontrivial
regime.

To obtain the desired enantioselective TFC, both enantiomers need to be
prepared in the lowest adiabatic bands in the rotating frame at $t=0$. Suppose
that before fields are turned on
($\mu_{i,M^{\prime};j,M}^{R,S}\mathcal{E}_{ij}(t)\to 0$ as $t\to-\infty$), the
molecules start in the ground state $|1,0\rangle$. Under those circumstances,
the eigenstates of Eq. 7 are the states $|1,0\rangle,|2,M\rangle,|3,0\rangle$
with eigenenergies $\epsilon_{L,M,U}^{R,S}(-\infty)=-\delta,0,\delta$, and the
state of each molecule is $|\epsilon_{L}^{R,S}(-\infty)\rangle$. If the
electric fields are slowly turned on at a rate $\omega_{r}$ that is much
smaller than the instantaneous band gaps
$|\epsilon_{l}^{R,S}(t)-\epsilon_{l^{\prime}}^{R,S}(t)|$, both enantiomers are
prepared in the lower band, _i.e._ $|\epsilon_{L}^{R,S}(0)\rangle$. Note that
the modulating frequencies $\omega_{1},\>\omega_{2}$ must also be much smaller
than $|\epsilon_{l}^{R,S}(t)-\epsilon_{l^{\prime}}^{R,S}(t)|$ at all times.
Chirped microwave fields for $t<0$ satisfy this constraint. The adiabatic
protocol we choose is $E_{ij}\to E_{ij}\alpha(t)$ and
$\omega_{1,2}\to\omega_{1,2}\beta(t)$, where the ramp-up functions slowly vary
at the rate $\omega_{r}=2\times 10^{-13}\text{\;a.u.}$ (see Fig. 4),

$\displaystyle\alpha(t)=$
$\displaystyle\begin{cases}0&t<-\frac{2\pi}{\omega_{r}},\\\
\frac{1-\text{cos}\omega_{r}t}{2}&-\frac{2\pi}{\omega_{r}}<t<-\frac{\pi}{\omega_{r}},\\\
1&-\frac{\pi}{\omega_{r}}<t,\end{cases}$ (16a) $\displaystyle\beta(t)=$
$\displaystyle\begin{cases}0&t<-\frac{\pi}{\omega_{r}},\\\
\frac{1+\text{cos}\omega_{r}t}{2}&-\frac{\pi}{\omega_{r}}<t<0,\\\
1&0<t.\end{cases}$ (16b)

After a sufficiently long time (we choose $t^{*}=2000\times 2\pi/\omega_{2}$),
the frequency-resolved time-averaged power spectrum $P_{av}(\Omega)$ lost by
the fields is numerically calculated considering only $t\geq 0$. This quantity
is indeed enantioselective, and using Eqs. 15a and 15b, each enantiomer Chern
number for the lower band $C_{L}^{R}=-2=-C_{L}^{S}$ is extracted, revealing
the topological nature of this nonlinear optical phenomenon (Fig. 5).

Figure 5: _Enantioselective TFC._ Plotted is the difference power spectrum for
the driving electric field when coupled to a single $R-$ (red) and $S-$ (cyan)
1,2-propanediol enantiomers. In terms of intensity, assuming the laser beam
waist area is $\sim 1$ $\text{cm}^{2}$, the change observed is $\sim
10^{-15}\,\text{W}\cdot\text{m}^{-2}$ per molecule. This spectrum is
enantioselective, and using Eqs. 15a and 15b, we can see that the frequency
conversion in the rotating frame is topological,
$\frac{2\pi}{\hbar\omega_{1}\omega_{2}}\mathcal{P}_{2\to
1}^{R}=-\frac{2\pi}{\hbar\omega_{1}\omega_{2}}\mathcal{P}_{1\to 2}^{S}=-2$.

For an ensemble containing $N_{R}$ $R-$molecules and $N_{S}$ $S-$molecules,
which are all prepared in the ground state $|1,0\rangle$, the expected pumping
rate is

$\mathcal{P}_{2\to
1}=\frac{\hbar\omega_{1}\omega_{2}C_{L}^{R}}{2\pi}(N_{R}-N_{S}).$ (17)

which is zero for a racemic mixture, but otherwise, reveals the EE
$|N_{R}-N_{S}|$ and chirality $\text{sgn}(N_{R}-N_{S})$. Notice that in line
with other nonlinear chiroptical signals that depend on electric but not
magnetic dipole contributions 9, Eq. 17 contains no background achiral signal,
unlike traditional circular dichroism, where both enantiomers have the same
electric dipole and magnetic dipole absorption strengths for circularly
polarized light 56.

Let us briefly discuss the limits of enantioselective TFC. First, the
linewidths of microwave transitions are on the order of 10-100 kHz 57, which
are smaller than the adiabatic state preparation gap $\delta\approx
1\,\text{MHz}$, as well as the light-matter interactions
$|\mu_{ij}E_{ij}|/\hbar\approx 10\,\text{MHz}$ inducing the topological gap,
or even the smallest difference in energies in the power spectrum (see, Fig.
5, $\Omega_{31\pm 1}-\Omega_{31\pm 2}\approx 1\,\text{MHz}$). Thus, the
described protocol should be resilient to the finite linewidths of these
transitions. Second, another source of imperfections stems from laser shot
noise. Assuming that the laser beam waist area is $\sim 1\>\text{cm}^{2}$ and
considering the field strength above, the shot noise for a time interval
$t^{*}$ is $\sqrt{N}\sim 10^{9}$ (where $N$ is the expected number of photons
produced by the field, see SI-5). From the power spectrum (Fig. 5), we find
that for the same time interval, that the minimal magnitude of the change in
the photon number due to the TFC is
$\min\Big{(}\left|\frac{P_{av}(\Omega_{ij\pm 1,2})t^{*}}{\hbar\Omega_{ij\pm
1,2}}\right|\Big{)}\approx 100\times|N_{R}-N_{S}|.$ Therefore, as long as the
magnitude of the enantiomer excess $|N_{R}-N_{S}|$ is much larger than $\sim
10^{7}$ molecules, the signal should be detectable above the shot noise. In
terms of percentage of the total molecule count $N_{R}+N_{S}$, the lower end
of the EE detection limit for $1$ mL of a $1$ $\mu$M solution is $10^{-6}\%$.
We conclude with a few comments on the observability of our predictions.
First, this study has assumed the ideal limit that the molecules are at $0$ K.
Under typical experimental conditions for microwave-three wave mixing at 7 K
21 all three rotational energy levels used in our model are substantially
thermally occupied. In this scenario, enantioselective frequency conversion
still survives; however, the integer Chern number will be replaced by a
thermal average of the Chern numbers $C_{L}^{R},C_{M}^{R},C_{U}^{R}$. Second,
the excited-state thermal populations can be bypassed by working in a
different energy range, such as the UV-visible one involving electronic
transitions and the infrared one involving vibrations 58, 59; the price to pay
in the first case is the complication introduced by electron-vibration
coupling. These complications will be addressed in future works.

In summary, we have presented an enantioselective TFC setup for an ensemble of
chiral molecules. Owing to the dependence of the topological invariant on the
sign of the KS product (Eq. 14), which differs by a phase of $\pi$ for the two
enantiomers, the quantized time-averaged energy-pumping rate is of opposite
sign for the $R-$ and $S-$ molecules, just like transverse conductivity is of
opposite sign for up and down spins in the QSHE. We show that the computed
signal is non-zero for any sample with EE and vanishes for a racemic mixture.
An intriguing consequence of Eq. 13 is that as long as the timescale
separations required by the model are fulfilled, the chemical identity of the
probed molecules (_e.g._ , through the strengths of the transition dipole
moments) in the rotating frame is erased by the signal, leading to a universal
nonlinear optical response which acknowledges the enantiomeric excess only.
This characteristic is reminiscent to the very accurate determination of the
quantum of conductance with a wide range of QHE systems. Thus, from a
metrological standpoint, the generality of the enantioselective TFC can be
exploited to accurately measure EE by running a linear fit of the pumping rate
$\mathcal{P}_{2\rightarrow 1}$ for a series of experiments where $\omega_{1}$
(or $\omega_{2}$) is varied. Furthermore, if one is only concerned with
$|\text{EE}|$, a practical asset of the presented methodology is that there is
no need to calibrate the signal with an enantiopure sample beforehand. We
believe that the removal of calibration counterbalances the complexity of the
experimental setup proposed in this Letter.

While concepts of topology have been very productive in the exploration of new
condensed matter physics phenomena, most of them are restricted to periodic
solids (see Ref. 60, 61 for a few molecular exceptions). TFC 48, 44 is a
powerful tool that opens doors to the application of those concepts to 0D
systems such as finite molecular and non-periodic nanoscale systems. In
particular, this work reveals that laser-dressed chiral molecules support
SPTPs that are not adiabatically connected to their non-laser-dressed
counterparts. It also provides a fruitful playground to explore further
conceptual connections between topological physics and molecular chirality 49,
62

## Acknowledgments

K.S. and J.Y.- Z. were supported by NSF CAREER CHE 1654732. This work employed
computational resources of the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Science Foundation Grant
No. ACI-1548562, under allocation No. TG-ASC150024. K.S. acknowledges helpful
discussions with Matthew Du, Stephan van den Wildenberg, and Jorge A. Campos-
González-Angulo. Both K.S. and J.Y.-Z. acknowledge the help of insightful
comments from previous Reviewers.

## Supporting Information Available:

Presentation of rotational eigenstates of an asymmetric top; discussion of the
change of basis applied to obtain the necessary effective Hamiltonian, review
of adiabatic perturbation theory; derivation of the expected laser power in
the rotating frame; analytical evaluation of enantioselective Chern number for
zero detuning; calculation of expected laser shot noise.

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