# Phase-matched locally chiral light for global control of chiral light-matter
interaction

Chong Ye  Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic
Systems, School of Physics, Beijing Institute of Technology, Beijing 100081,
China<EMAIL_ADDRESS>Yifan Sun  Beijing Key Laboratory of Nanophotonics
and Ultrafine Optoelectronic Systems, School of Physics, Beijing Institute of
Technology, Beijing 100081, China  Libin Fu  Graduate School of China Academy
of Engineering Physics, Beijing 100193, China  Xiangdong Zhang  Beijing Key
Laboratory of Nanophotonics and Ultrafine Optoelectronic Systems, School of
Physics, Beijing Institute of Technology, Beijing 100081, China

###### Abstract

Locally chiral light is an emerging tool for probing and controlling molecular
chirality. It can generate large and freely adjustable enantioselectivities in
purely electric-dipole effects, offering its major advantages over traditional
chiral light. However, the existing types of locally chiral light are phase-
mismatched, and thus the global efficiencies are greatly reduced compared with
the maximum single-point efficiencies or even vanish. Here, we propose a
scheme to generate phase-matched locally chiral light. To confirm this
advantage, we numerically show the robust highly efficient global control of
enantiospecific electronic state transfer of methyloxirane at nanoseconds. Our
work potentially constitutes the starting point for developing more efficient
chiroptical techniques for the studies of chiral molecules.

††journal: opticajournal

Molecular chirality plays a central role in chemical reactions, biological
activities, drug industries, and life’s chemistry, which makes probing and
controlling molecular chirality vitally important and challenging tasks across
natural science. Using single circularly light (i.e., traditional chiral
light) to probe [1, 2, 3, 4] molecular chirality began with the first
discovery of molecular chirality. It was also proposed to control enantiomeric
excess in an inner state [5], i.e., enantiospecific state transfer. The
enantioselectivities generated by the traditional chiral light come from the
interference between electric- and magnetic-dipole light-matter interactions.
The relative strength between them is almost a fixed parameter at a ratio of
($\simeq 1/137$) [5], so the generated enantioselectivities are usually tiny
and not freely adjustable.

Locally chiral light (LCL) [6, 7, 8, 9, 10, 11, 12] is an emerging chiral
light source that can generate large and freely adjustable
enantioselectivities in purely electric-dipole effects. LCLs are multi-chrome,
composed of carrier waves with well-designed polarizations and frequencies.
While the traditional chiral light’s degree of chirality (DOC) at a single
point in space is determined by the electric and magnetic fields at that point
and nearby [13], LCLs’ DOC at a single point only depends on the electric
field at that point (or equivalently the local electric field). Specifically,
at each fixed point in space, the tip of the electric field vector draws a
chiral 3D Lissajous curve when it evolves in time. Beyond probing and
controlling chiral molecules, the LCLs were also proposed to imprint chirality
on achiral matter [14], offering new opportunities to realize laser-driven
achiral–chiral phase transitions in matter [15, 16, 17].

With the benefit of hindsight, the studies of all-electric-dipole chiroptical
techniques by using LCLs began almost two decades ago in the framework of the
cyclic three-level model [18, 19, 20, 21], where the driving electromagnetic
fields are tricolor LCLs. The three carrier waves couple with chiral molecules
in the one-photon (near-)resonance, and thus the dynamics of chiral molecules
can be described by the cyclic three-level model [22, 23, 24, 25, 26, 27, 28].
Experimentally, the tricolor LCLs were used in the well-demonstrated microwave
techniques of enantiodetection [29, 30, 31, 32, 33, 34] and enantio-specific
state transfer [35, 36, 37]. The equivalent chiroptical techniques in the UV-
IR region [18] are long pursued but never realized. One key problem is that
tricolor LCLs are phase-mismatched [38], such that their DOCs and the
generated enantioselectivities change periodically in space and even vanish by
integrating over the whole spatial period (or equivalently globally achiral).
This problem becomes grave in the UV-IR region because of the short
wavelengths [7].

To solve this problem, chirality polarized light (CPL) [11] and locally-and-
globally chiral light (LGCL) [6] were proposed. CPL is globally achiral but
has the polarization of chirality, just like the neutral and polarized one-
dimensional medium of alternating negative and positive charges. This property
makes the emission directions highly enantioselective for CPL. LGCL is more
efficient because it is globally chiral. However, they are still phase-
mismatched, such that the enantiospecific electronic state transfer in the UV-
IR region is still out of reach. Moreover, CPL and LGCL worked in the strong-
field region, which is usually accompanied by the destruction of the molecule
or the change of the molecular conformer (not chirality).

In this Letter, we propose a scheme to generate a novel type of LCL without
the phase-mismatch problem. Thus, we name it the phase-matched LCL. This
feature offers our scheme advantages over existing LCLs in global control of
chiral light-molecule interactions. To illustrate this, we use a four-color
example of our scheme and numerically show its ability in the robust highly
efficient global control of enantiospecific electronic state transfer of
methyloxirane at nanoseconds in the UV-IR region.

Our scheme consists of two non-collinear sub-beams. The first sub-beam
[labeled with $c=1$ in Fig.1 (a)] is linearly polarized and multi-chrome. The
frequencies of carrier waves therein are in the cyclic odd-photon resonance.
For the case of a four-color example [see Fig. 1 (a)], such a condition is
$\omega_{1}+\omega_{2}-\omega_{3}=0$, yielding
$\bm{k}_{1}+\bm{k}_{2}-\bm{k}_{3}=0$ [see the left lower corner of Fig. 1
(a)]. The second sub-beam [labeled with $c=2$ in Fig.1 (a)] is monochrome and
exhibits circular polarization in the plane perpendicular to the polarization
of the first sub-beam.

Figure 1: (a) Four-color phase-matched LCL: the $c=1$ sub-beam is a
$z$-polarized three-color field propagating in the $x$-direction, and the
$c=2$ sub-beam is a one-color circularly polarized field propagating in the
$z$-direction. The carrier waves are Gaussian beams with waists of
$w_{0,0}=10^{3}\lambda_{0}$ and $w_{0,1}=w_{0,2}=w_{0,3}=10^{3}\lambda_{1}$.
The wavelengths $\lambda_{0}$ and $\lambda_{1}$ are chosen according to the
example of methyloxirane in Fig. 2. The lowest-order CCFs [$h^{(5)}$] in the
$y-z$ plane at $z=0$ clearly show its phase-matched feature: (b) the
corresponding amplitudes (in arbitrary unit) and (c) phases (in the unit of
$\pi$). (d) Scheme of enantioselective light-molecule interaction for our
four-color LCL in the one-photon (near-)resonance. The enantioselectivity is
reflected in the transition electric dipoles with
$\bm{\mu}^{L}_{ij}=-\bm{\mu}^{R}_{ij}=\bm{\mu}_{ij}$.

The overall field of the four-color example is

$\displaystyle\bm{E}(\bm{r},t)=$
$\displaystyle\sum^{N}_{m=1}\frac{{E}_{m}}{2}\bm{e}_{z}e^{-i(\omega_{m}t-k_{m}x)}+\frac{{E}_{0}}{2}e^{-i(\omega_{0}t-k_{0}z)}\bm{e}_{\sigma}+c.c.,$
(1)

where $\sigma(=\pm 1)$ denotes the handedness of the circularly polarized
field and the corresponding polarization vector is
$\bm{e}_{\sigma}=(\sigma\bm{e}_{x}+i\bm{e}_{y})/\sqrt{2}$. The amplitudes and
phases of ${E}_{i}$ are $\mathcal{E}_{i}$ and $\phi_{i}$. For the four-color
example of Fig. 1, we have $N=3$. We choose the carrier waves to be Gaussian
beams [see Fig. 1 (a)]. Their amplitudes and phases are
$\mathcal{E}_{i}=u_{i}w_{0,i}e^{-r^{2}_{\perp_{i}}/w^{2}_{i}}/w_{i}$ and
$\phi_{i}=k_{i}r^{2}_{\perp_{i}}/[4r^{2}_{\parallel_{i}}+(k_{i}w^{2}_{0,i})^{2}]-\arctan[2r_{\parallel_{i}}/(k_{i}w^{2}_{0,i})]+\varphi_{i}$
with $w_{i}=w_{0,i}\sqrt{1+[2r_{\perp_{i}}/(k_{i}w^{2}_{0,i})]^{2}}$, where
$r_{\parallel_{i}}$ and $r_{\perp_{i}}$ are projections of the position
vectors parallel and vertical to the corresponding propagation directions.
Their waists are chosen as $w_{0,0}=10^{3}\lambda_{0}$ and
$w_{0,1}=w_{0,2}=w_{0,3}=10^{3}\lambda_{1}$. The initial phases are
$\varphi_{i}$. Here, $\lambda_{0}$ and $\lambda_{1}$ are wavelengths of
$E_{0}$ and $E_{1}$, whose values are chosen according to the example of
methyloxirane in Fig. 2.

The DOC of locally chiral light at a single point in space is characterized by
the $n$th order chiral correlation functions (CCFs) [7] defined as
$h^{(n)}(\bar{\omega}_{0},...,\bar{\omega}_{n-1})\equiv\\{\bm{F}(\bar{\omega}_{0})\cdot[\bm{F}(\bar{\omega}_{1})\times\bm{F}(\bar{\omega}_{2})]\\}[\bm{F}(\bar{\omega}_{3})\cdot\bm{F}(\bar{\omega}_{4})]...[\bm{F}(\bar{\omega}_{n-2})\cdot\bm{F}(\bar{\omega}_{n-1})]$
with the cyclic odd-photon resonance condition of $\bar{\omega}_{i}$. Here,
$\bm{F}(\bar{\omega}_{i})$ is a Fourier transform of the electric field vector
at the angular frequency $\bar{\omega}_{i}$. We can prove that all the nonzero
CCFs of the overall field in Eq. (1) are irrelevant to the spatially
oscillated phase of each carrier wave $\bm{k}_{i}\cdot\bm{r}$. That is to say,
the proposed LCLs in Eq. (1) are phase-matched. For more details on the proof,
see Sec. I A of Supplement.

Specifically, we show the amplitude [Fig. 1 (b)] and phase [Fig. 1 (c)] of
lowest-order CCFs [$h^{(5)}$] in the $y-z$ plane at $z=0$ for the four-color
example. The results clearly show that the four-color example is phase-
matched. In contrast, the nonzero CCFs of the tricolor LCL and LGCL are
functions of $\bm{k}_{i}\cdot\bm{r}$ (see Sec. I C and Sec. I D of
Supplement), i.e., they are phase-mismatched. We note that Eq. (1) provides a
general scheme to form phase-matched LCLs, including the two-color phase-
matched LCL (see Sec. I B of Supplement).

For the four-color example, we are interested in the case of the one-photon
(near-)resonance. Because the residual electronic transitions are coupled with
the overall field off-resonantly, they are negligible in the weak-field
region, yielding the simplified working model as shown in Fig. 1 (d). The
molecules also have vibrational and rotational degrees of freedom. We deal
with them in the Born-Oppenheimer approximation. According to the
Franck–Condon principle [39], which says that an electronic transition occurs
within a stationary nuclear framework, the vibrational degrees of freedom can
be assumed frozen, i.e., the electronic transitions are vertical transitions.
In our discussions, we are interested in the gaseous phase chiral molecules.
For them, the rotational degree of freedom is described by
$\hat{H}_{\mathrm{rot}}=\hbar(A\hat{J}^{2}_{a}+B\hat{J}^{2}_{b}+C\hat{J}^{2}_{c})$
[40]. $\hat{J}_{a,b,c}$ are angular momentum operators along the principal
axes of the moment of inertia, respectively. $A$, $B$, and $C$ are the
corresponding rotational constants. Because the nuclear locations are
preserved in electronic vertical transitions, the rotational constants are
preserved in our consideration.

In the rotating-wave approximation, the concrete Hamiltonian of the chiral
light-molecule interaction is

$\displaystyle\hat{H}=$
$\displaystyle\sum_{\alpha,\beta}[\frac{1}{2}(\Omega^{\alpha\beta}_{10}\hat{s}_{10}+\sum^{3}_{i>j=1}\Omega^{\alpha\beta}_{ij}\hat{s}_{ij})\hat{\sigma}_{\alpha\beta}+h.c.]+\sum_{i\neq
1}\Delta_{i1}\hat{s}_{ii}+\hat{H}_{\mathrm{rot}}.$ (2)

The detunings are $\Delta_{01}=\omega_{0}-(v_{1}-v_{0})$,
$\Delta_{21}=v_{2}-v_{1}-\omega_{1}$, $\Delta_{31}=v_{3}-v_{1}-\omega_{3}$,
and $\Delta_{32}=v_{3}-v_{2}-\omega_{2}$.
$\hat{\sigma}_{\alpha\beta}\equiv|{\alpha}\rangle\langle{\beta}|$ are
operators in the rotational degree of freedom with
$|\alpha\rangle\equiv|J^{\alpha}_{K^{\alpha}_{a},K^{\alpha}_{c},M^{\alpha}}\rangle$
being the asymmetric-top rotational eigenstates [40]. We note that
$\sum_{\alpha}$ means that the summation is taken over all $J^{\alpha}$,
$K^{\alpha}_{a}$, $K^{\alpha}_{c}$, and $M^{\alpha}$. $\hat{s}_{ij}$ are
operators in the electronic degree of freedom. Because the transition electric
dipoles change signs with enantiomers [see Fig.1 (a)], the interaction
Hamiltonian (2) is enantioselective as indicated by
$\Omega^{\alpha\beta}_{ij,L}=-\Omega^{\alpha\beta}_{ij,R}$ with the subscripts
“$L$" and “$R$" denoting the chirality. We note that the operators
$\hat{s}_{ij}\equiv|\tilde{i}\rangle\langle\tilde{j}|$ are defined in the new
basis with
$|\tilde{0}\rangle=e^{-\mathrm{i}[(v_{1}-\omega_{0})t+\bm{k}_{0}\cdot\bm{r}]}|0\rangle$,
$|\tilde{1}\rangle=e^{-\mathrm{i}v_{1}t}|1\rangle$,
$|\tilde{2}\rangle=e^{-\mathrm{i}[(v_{1}+\omega_{1})t-\bm{k}_{1}\cdot\bm{r}]}|2\rangle$,
and
$|\tilde{3}\rangle=e^{-\mathrm{i}[(v_{1}+\omega_{3})t-\bm{k}_{3}\cdot\bm{r}]}|3\rangle$.
Then, the spatially and temporally oscillated phases
$(\bm{k}_{i}\cdot\bm{r}-\omega_{i}t)$ are transferred to
$\Omega^{\alpha\beta}_{32}e^{i(\delta\omega t-\delta\bm{k}\cdot\bm{r})}$ with
$\delta\omega\equiv\omega_{3}-\omega_{2}-\omega_{1}=0$ and
$\delta\bm{k}\equiv\bm{k}_{3}-\bm{k}_{2}-\bm{k}_{1}=0$. This clearly indicates
that the Hamiltonian (2) is phase-matched, such that the molecules at
different positions evolve identically in the plane-wave limit.

To illustrate the advantages of our four-color phase-matched LCL, we use
methyloxirane as an example, which is a typical chiral molecule [see Fig. 1
(d)]. The working electronic states $|0\rangle$, $|1\rangle$, $|2\rangle$, and
$|3\rangle$ are chosen as the electronic ground state, the $3p_{y}$-state, the
$3p_{x}$-state, and the $3d_{z^{2}-x^{2}}$-state [12]. The carrier waves with
angular frequencies $\omega_{0}$, $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$
are chosen to resonantly couple with transitions
$|0\rangle|0_{0,0}\rangle\leftrightarrow|1\rangle|1_{0,1}\rangle$,
$|1\rangle|1_{0,1}\rangle\leftrightarrow|2\rangle|2_{1,1}\rangle$,
$|2\rangle|2_{1,1}\rangle\leftrightarrow|3\rangle|2_{0,2}\rangle$, and
$|1\rangle|1_{0,1}\rangle\leftrightarrow|3\rangle|2_{0,2}\rangle$. The
corresponding wavelengths are $\lambda_{0}\simeq 162.2633$ nm,
$\lambda_{1}\simeq 6241.4404$ nm, $\lambda_{2}\simeq 1541.6517$ nm, and
$\lambda_{3}\simeq 2047.3531$ nm [12]. The spatial profiles of carrier waves
are shown in Fig. 1 (a).

To achieve enantiomeric excess in the state $|3\rangle$, we take the
temporally square pulses, whose amplitudes at $r=0$ evolve as shown in Fig. 2
(a). To justify the validity of using the classical description of light in
our discussions, we give the average photon number of the pulses, which are
about $6.2\times 10^{15}$, $7.7\times 10^{18}$, $8.7\times 10^{17}$, and
$1.2\times 10^{18}$. They are large enough to make the photon-number phase
uncertainty negligible [41]. The two enantiomers are assumed initially in the
state $|0\rangle|0_{0,0,0}\rangle$. This is a good approximation at the
rotational temperate $T_{\mathrm{rot}}\lesssim 10$ mK because the relative
populations in the first excited rotational state with respect to the ground
rotational state is $\lesssim 7\times 10^{-6}$.

Figure 2: Global control of enantiospecific electronic state transfer of
methyloxirane by using our four-color phase-matched LCL. (a) Temporal
envelopes of the applied fields at $r=0$. (b) Time evolution of the population
in each working electronic state for molecules at $r=0$, yielding the
enantiomeric excess of $-99.27\%$ at $r=0$. (c-f) $\log_{10}(\rho^{L}_{3})$
and $\log_{10}(\rho^{R}_{3})$ after the four pulses in the $y-z$ plane for
$x=0$ and $x=2w_{0,0}\simeq 0.32$ mm. (g) $P^{Q}_{2D}(x)\equiv\int
dydz\rho^{Q}_{3}(\bm{r})$ with $Q(=L,R)$ denoting the handedness of the chiral
molecules. The global enantiomeric excess is
$\varepsilon_{\mathrm{global}}\simeq-93.74\%$.

The evolution of the population in each electronic state at $r=0$ is shown in
Fig. 2 (b), which is given by summing over its rotational sub-states. We note
that the light-molecule interactions in sufficiently high $J$ are negligible
because these transitions are off-resonantly coupled with light. That is to
say, although the number of rotational levels is infinite, the calculations
can be done in the subspace with $J\leq J_{c}$. We numerically find that the
calculations converge at $J_{c}=3$ (see Sec. II A of Supplement). When all
pulses end, while almost all the population of the right-handed molecule is
transferred to the electronic state $|3\rangle$, most of the left-handed
molecule’s population is transferred to the electronic state $|2\rangle$. We
focus on the enantiomeric excess in the electronic state $|3\rangle$ defined
as
$\varepsilon\equiv({\rho^{L}_{3}-\rho^{R}_{3}})/({\rho^{L}_{3}+\rho^{R}_{3}})$,
where $\rho^{L}_{3}$ and $\rho^{R}_{3}$ are the final populations in the
electronic state $|3\rangle$ for the two enantiomers. The achieved
enantiomeric excess corresponding to Fig. 2 (b) is about $\varepsilon\simeq
99.27\%$.

Usually, the $M$-degeneracy problem and the broad frequency features of
applied fields significantly reduce enantioselectivity. In our calculations,
without their influences, the applied pulses will form perfect
“$\pi-{\pi}/{2}-\pi-\pi/2$" pulses, yielding the final enantiomeric excess of
$100\%$. Here, we carefully design the polarization and the center frequency
of each field, these problems are largely resolved at nanoseconds, and thus
these pulses did not deviate too much from perfect $\pi$ and $\pi/2$ pulses.
Then, the final high enantiomeric excess is considerably high.

In Fig. 2 (c-f), we give $\log_{10}(\rho^{L}_{3})$ and
$\log_{10}(\rho^{R}_{3})$ in the $y-z$ plane at $x=0$ and $x=2w_{0,0}$. In the
case of $x=0$, $\rho^{L}_{3}$ [see Fig. 2 (c)] are much smaller than
$\rho^{R}_{3}$ [see Fig. 2 (d)] near the origin of the plane. When the
molecules are away from the origin of the plane, $\rho^{L}_{3}$ and
$\rho^{R}_{3}$ decrease to $zero$. Because the waist of the $\omega_{0}$ field
is much smaller than those of others, the decreasing rate in the $y$-direction
is much larger than that in the $z$-direction. In the case of $x=2w_{0,0}$
[see Fig. 2 (e) and Fig. 2 (f)], similar phenomena happen in the $y-z$ plane.

Further, we sum $\rho^{L}_{3}$ and $\rho^{R}_{3}$ over the $y-z$ plane at
different $x$, yielding $P^{Q}_{2D}(x)\equiv\int dydz\rho^{Q}_{3}(\bm{r})$.
The results in Fig. 2 (g) show that $P^{R}_{2D}$ and $P^{L}_{2D}$ decrease
when the plane is moved away from $x=0$ in the $x$-direction. To evaluate
global efficiency, we define the global enantiomeric excess

$\displaystyle\varepsilon_{\mathrm{global}}\equiv\frac{P^{L}_{3D}-P^{R}_{3D}}{P^{L}_{3D}+P^{R}_{3D}}$
(3)

with $P^{Q}_{3D}\equiv\int d\bm{r}^{3}\rho^{Q}_{3}(\bm{r})$. The corresponding
global enantiomeric excess is $\varepsilon_{\mathrm{global}}\simeq-93.74\%$.
This shows considerably high efficiency in the global control of chiral light-
molecule interactions. For the tricolor LCLs, $\varepsilon_{\mathrm{global}}$
quickly decreases to $zero$ when the size of the sample becomes comparable to
the wavelengths (see Sec. II B of Supplement). Thus, our four-color LCLs show
a considerable advantage in global control of chiral light-molecule
interaction. At typical supersonic beam temperature ($T_{\mathrm{rot}}=1$ K)
[37] and with the same four-color LCLs, the enantiomeric excess at the origin
is about $93\%$ and the global enantiomeric excess is about $91\%$ (see Sec.
II A of Supplement).

Figure 3: Robustness of highly efficient global control of enantiomeric excess
($|\varepsilon_{\mathrm{global}}|\geq 0.9$) against relative control errors
with respect to their corresponding values in Fig. 2: (a) the waists $w_{0,0}$
and $w_{0,1}$ ($w_{0,2}=w_{0,3}=w_{0,1}$), (b) the phases $\delta\varphi$, (c)
the duration of each pulse, and (d) the field amplitude of each pulse.

We also explore the robustness against control errors at $T_{\mathrm{rot}}\leq
10$ mK (see Fig. 3). When the relative variations of $w_{0,0}$ and $w_{0,1}$
are limited to $\pm 50\%$, the worst result is
$\varepsilon_{\mathrm{global}}\simeq-87\%$ [see Fig. 3 (a)]. That is to say,
the high global efficiency is robust against the variations of the waists of
the beams. As shown in Fig. 3 (b), the variation of
$\delta\varphi\equiv\varphi_{3}-\varphi_{2}-\varphi_{1}$ can affect the
enantiomeric excess. The highly efficient global control
($|\varepsilon_{\mathrm{global}}|\geq 90\%$) can be achieved around
$\delta\varphi=0$ and $\delta\varphi=\pi$. We note that the variation of
$\phi_{0}$ will not affect the enantiomeric excess. As shown in Fig. 3 (c),
the high global efficiency is robust against the relative variations of the
pulses’ duration ($dt/t$). For $E_{0}$, $E_{1}$, $E_{3}$, and $E_{2}$ pulses,
$|\varepsilon_{\mathrm{global}}|\geq 90\%$ can be obtained with the relative
variations about $-50\%\sim 100\%$, $-15\%\sim 30\%$, $-25\%\sim 75\%$, and
$-25\%\sim 15\%$, respectively. Each line in Fig. 3 (c) is obtained by
changing the duration of the corresponding pulse around its value in Fig. 2
and fixing the other pulses’ duration to their values in Fig. 2.
Correspondingly, the pulses’ duration for the four carrier waves can be tuned
in the regions about $0.37\sim 1.10$ ns, $1.35\sim 2.06$ ns, $1.38\sim 3.22$
ns, and $1.65\sim 2.52$ ns, respectively. In Fig. 3 (d), we show
$\varepsilon_{\mathrm{global}}$ as the function of the variation of each
pulse’s amplitude around the corresponding value in Fig. 2, which also show
considerable robustness.

We believe our scheme provides a promising chiral light source in the UV-IR
region. It purely depends on strong electric-dipole effects and is phase-
matched. Our results clearly show its advantage in global control of
enantiospecific state transfer. It constitutes the starting point for the
applications of other UV-IR spectroscopic techniques in the studies of chiral
molecules.

We note that there are other factors that limit the enantiospecific state
transfer in the UV-IR region, including the involvement of multiple rotational
levels due to the broad-frequency features of the applied fields and the short
lifetime of electronic excited states. These problems exist for all the chiral
light source, and should be addressed in further research.

Acknowledgments This work is supported by the National Natural Science
Foundation of China (No. 12105011, No. 12234004, No. 12088101, No. U2330401,
and No. 11904022).

Disclosures The authors declare no conflicts of interest.

Data availability No data were generated or analyzed in the presented
research.

Supplemental document See Supplement for supporting content.

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