\section{\label{sec:intro}Introduction}

The axion is a hypothetical elementary particle induced by the Peccei-Quinn mechanism that was proposed to solve the strong CP problem~\cite{PRL1977PQ,PRL1978Weinberg,PRL1978Wilczek}, one of the major fine-tuning problems in particle physics.
Symmetry breaking at a sufficiently high energy scale could yield a very light axion that couples feebly to the standard particles~\cite{PRL1979Kim,NPB1980SVZ,YF1980Zhitnitsky,PLB1981DFS}.
Because the {\it invisible} axion would maintain a stable non-relativistic state, it has been considered as a strong candidate for cold dark matter~\cite{PLB1983Wilczek,PLB1983Abbott,PLB1983Dine}. 
The most sensitive method to search for the dark matter axion in the microwave regime is the cavity haloscope which is designed to probe events of the axion-to-photon conversion inside a microwave cavity under a strong magnetic field~\cite{PRD1985Sikivie}.
The tiny signals of axion-induced photons are enhanced to an observable level by resonating with the cavity at the axion mass. 

Because the mass of axion dark matter is unknown {\it a priori}, all possible masses should be probed by tuning the resonant frequency of the cavity.
How fast a given mass range can be scanned is quantified by the scan rate, which has the following proportionality within the limit of the cavity quality factor, which is much smaller than the axion quality factor $(Q_{c} \ll Q_{a})$~\cite{JCAP2020Kim}:
\begin{equation}
\label{eq:scan_rate}
    \frac{d\nu}{dt} \propto \left(\frac{g_{a\gamma\gamma}^{2}(\rho_{a}/m_{a})\langle \mathbf{B}_{0}^{2}\rangle V_{c} C}{k_{B} T} \right)^{2} Q_{c}Q_{a},
\end{equation}
where $g_{a\gamma\gamma}$ is the axion-photon coupling constant, $\rho_{a}$ and $m_{a}$ are the local axion density and axion mass, $\langle \mathbf{B}_{0}^{2} \rangle$ is the square average of the external magnetic field over the cavity volume $V_{c}$, and $Q_{c}$ and $Q_{a}$ are the cavity and axion quality factors, respectively.
The form factor $C$, defined below, describes how well the electric field of the resonant mode ($\mathbf{E}_{r}$) is aligned with the applied magnetic field ($\mathbf{B}_{0}$):
\begin{equation}
\label{eq:form_factor}
    C = \frac{|\int \mathbf{B}_{0}\cdot \mathbf{E}_{r}dV_{c}|^{2}}{\int |\mathbf{B}_{0}|^{2} dV_{c} \int \varepsilon |\mathbf{E}_{r}|^{2}dV_{c}},
\end{equation}
where $\varepsilon$ is the dielectric constant inside the cavity.
A cavity haloscope usually employs a solenoid magnet because it can form a very strong magnetic field uniformly in a localized space.
Accordingly, the TM$_{010}$ mode of a cylindrical cavity is adopted for the resonant mode of a detector cavity to take advantage of the highest form factor with the largest volume usage. 
The haloscope technique is almost the only experimental method that can reach the sensitivity suggested by plausible theories in the microwave regime, but it has the disadvantage of rapidly decreasing its efficiency in higher-mass regions.
This is mainly because the resonant frequency of the TM$_{010}$ mode is inversely proportional to the cavity radius, giving rise to substantial loss in the detection volume.
In addition, at higher frequencies, the quality factor decreases due to the anomalous skin effect while the fundamental noise level, which is set by quantum mechanics, increases.
Taking these effects into account, the scan rate decreases with the search frequency $\omega_{c}$ by
\begin{equation}
\label{eq:scan_rate_for_omega_c}
    \frac{d\nu}{dt} \propto \omega_{c}^{-20/3},
\end{equation}
which indicates, for example, that doubling the search frequency reduces the scanning speed by a factor of 100.

Various ideas have been proposed to address these issues.
One of the intuitive ways to recover the detection volume is to configure the system with multiple identical small cavities filling a given magnet bore~\cite{UCD2001Kinion,MCD2020Yang}.
In such a multiple-cavity system, a major challenge is to match the resonant frequency of the individual cavities to an acceptable level that does not broaden the resonance width of the combined system~\cite{AP2018Jeong}.
Another approach exploits higher-order modes of cylindrical cavities with a precisely positioned hollow-shaped dielectric segments~\cite{JPG2020Kim,PRA2020Quiskamp}.
This benefits high-$Q$ values but results in poor frequency tunability.
Some other interesting proposals were made for searches in even higher mass regions by introducing a periodic structure of dielectric discs~\cite{PRL2017MADMAX} or metallic wires~\cite{PRL2019Lawson}.
However, handling a large number of base materials involves engineering challenges to achieve accurate alignment.

A new cavity concept, called a multiple-cell cavity, was proposed by the Center for Axion and Precision Physics Research of the Institute for Basic Science (IBS-CAPP).
It was designed to minimize the volume loss for high-mass axion searches and to avoid the necessity for frequency matching, which is inevitable in multiple cavity systems~\cite{PLB2018Jeong}.
The effectiveness of the cavity design was successfully demonstrated by conducting an axion search experiment using a double-cell cavity, which set a new exclusion limit around 13.5~$\mu$eV~\cite{PRL2020Jeong}.
The result verified that the double-cell cavity boosts the scan rate by a factor of four compared to that of a conventional single cavity at the same mass.
In other experiments at CAPP, this new design was adopted with higher-cell multiplicities for efficient searches around 25\,$\mu$eV and 30\,$\mu$eV.

In this report, we review the multiple-cell cavity as an effective haloscope for high-mass axion searches.
Section~\ref{sec:pizza} describes the basic concept of the cavity design and discusses its various advantages over other options.
In Section~\ref{sec:chara}, major features of this new concept are presented based on analytical calculations and numerical simulations.
Section~\ref{sec:local} provides a methodology for obtaining the optimal geometry to minimize the field localization owing to fabrication tolerance.
Finally, in Section~\ref{sec:discussion}, we briefly discuss the impact of this cavity design on axion search experiments in high-mass regions.

\section{\label{sec:pizza}Multiple-cell cavity}

The multiple cavity system is an array of smaller identical cavities fitting inside a magnet bore and is one of the most intuitive ways to minimize the detection volume loss when extending the search range to high mass regions~\cite{UCD2001Kinion,MCD2020Yang}.
However, because there is unavoidable packing loss due to the cavity thickness and the shape of the cavity, it is difficult to make full use of a given magnet bore.
Moreover, this requires the resonant frequency of individual cavities to be synchronized within the cavity bandwidth to enhance the sensitivity effectively, which will be more challenging to achieve for systems with a larger number of cavities and higher quality factors.
The receiver chain also becomes complicated because multiple antennae and an RF combiner are required to extract and combine the signals from individual cavities.

The multiple-cell cavity design, on the other hand, addresses these shortcomings, improving the detection efficiency for high-mass axion searches.
A multiple-cell cavity is constructed by vertically dividing a cylindrical cavity into pizza slice-shaped cells using equally spaced thin metal partitions~\cite{PLB2018Jeong}.
The insertion of thin partitions is a key idea of the design for minimizing the loss of volume while increasing the resonant frequency.
The partitions distort the original field of the TM$_{010}$ mode, eventually forming azimuthal nodes at the position of the partitions.
The configuration produces higher resonant modes resembling TM$_{n10}$, where $n$ is half the number of the divided cells.
Those TM$_{n10}$ modes are degenerated at the same frequency.
The degeneracy is broken by introducing a partition gap in the middle of the cavity, as shown in Table~\ref{tab:multicell_comparison}.
The lowest resonant mode, TM$_{010}$-like, which is distinguishable from the others, is our desired mode as the EM fields of the individual cells oscillate in phase, yielding the maximal form factors under a uniform magnetic field.
Another important benefit of the partition space is that all the cells become spatially interconnected, enabling the signal to be extracted from the entire cavity volume using a single antenna, and thus eliminating the necessity of a signal combiner.
Such features considerably simplify the structure of the receiver chain and make experiments simpler.
Table~\ref{tab:multicell_comparison} summarizes the pros and cons of the multiple cavity system and the multiple-cell cavity design.
The robustness against fabrication tolerance issues can also be enhanced by optimal determination of the size of the partition gap, which is described in Section~\ref{sec:local} in more detail.

\begin{table}
\caption{\label{tab:multicell_comparison}Comparison between the multiple cavity system and the multiple-cell cavity design. The thick brown lines represent the metallic cavity surface and the black dashed lines refer to the magnet bore.}
\begin{ruledtabular}
\begin{tabular}{lcc}
        & Multiple cavity system & Multiple-cell cavity \\ \colrule
 Sketch & \parbox[c][0.205\linewidth]{0.2\linewidth}{\includegraphics[width=\linewidth]{multiple-cavity_system.png}} & \parbox[c][0.205\linewidth]{0.2\linewidth}{\includegraphics[width=\linewidth]{multiple_cell_cavity.png}} \\ \colrule
 \multirow{3}{*}{Pros} & & Minimal volume loss \\
 & Intuitive & Single antenna \\
 & & Robust against tolerance \\ \colrule
 \multirow{3}{*}{Cons} & Inefficient in volume & \\
 & Multiple antennae & Difficult to fabricate \\
 & Frequency matching & \\
\end{tabular}
\end{ruledtabular}
\end{table}


\section{\label{sec:chara}Characteristics}

The characteristics of multiple-cell cavities are determined by the cavity radius, the number of cells, and the size of the partition gap.
For a small center gap, the radius and central angle have a leading effect on the field solution in each cell such that analytical calculations are possible to determine the resonance frequency, quality factor, and form factor.
However, the field solution significantly varies as the center gap becomes large, requiring numerical approaches to study the characteristics.

\begin{figure}
\includegraphics[width=0.9\linewidth]{multiple-cell_field_solution.png}
\caption{Norm of the electromagnetic fields for the lowest TM mode of a circular sector cavity. The color and height indicate the field strength.} 
\label{fig:solution}
\end{figure}

The electromagnetic field solution for the lowest TM mode of a closed circular sector with radius $R_c$ is given by the Bessel functions using cylindrical coordinates $\rho$, $\varphi$, and $z$:
\begin{equation}
\begin{split}
    \mathbf{E}_{r} &= E_{0} J_{m}\left( \chi_{m1}\frac{\rho}{R_{c}} \right) \cos(m\varphi) e^{-i \omega_{c} t} \hat{z}, \\
    \mathbf{B}_{r} &= \frac{1}{i\omega_{c}} \nabla \times \mathbf{E}_{r},
\end{split}
\end{equation}
where $E_{0}$ is the amplitude of the electric field, $J_{m}$ is the $m$-th order Bessel function of the first kind, and $\chi_{m1}$ is the first root of $J_{m}$.
The resonant frequency $\omega_{c}$ satisfies $\omega_{c} = c \chi_{m1} / R_{c}$ where $c$ is the speed of light.
The subscript $m$ represents the azimuth node number, which is determined by the central angle of the sector $\phi$ and which yields $m = \pi / \phi$ from the boundary condition of $\mathbf{E}_{r}(\varphi=\pm\phi/2) = 0$.
The root of the Bessel function increases with decreasing $\phi$, so the more a cavity is divided, the higher the resonant frequency.
An example of the field distributions for $m=2$ are shown in Fig.~\ref{fig:solution}.

The quality factor is defined as the ratio of the stored energy per resonance cycle to the dissipation power, and can be exactly calculated from the field solution.
For a circular sector cavity with radius $R_{c}$, length $L_{c}$, and azimuth node number $m$, the quality factor is:
\begin{equation}
    Q_{c} = \frac{1}{1 / \mathcal{Q} + R_{c} / L_{c}} \frac{R_{c}}{\delta_{c}},
\end{equation}
where $\delta_{c}$ is the skin depth of the cavity surface and $\mathcal{Q}$ is the geometrical factor that can be obtained from,
\begin{equation}
\label{eq:q_coeff}
\begin{split}
    \frac{1}{\mathcal{Q}} = \frac{1}{4\pi J_{m+1}(\chi_{m1})^{2}}\bigg[ & \pi\left(J_{m-1}(\chi_{m1})-J_{m+1}(\chi_{m1})\right)^{2} \\
    & + \frac{32 m^{3}}{4 m^{2}-1}J_{m+1}(\chi_{m1})^{2} \bigg].
\end{split}
\end{equation}
Likewise, the form factor is also exactly calculable from the field solution using the definition, Eq.~\ref{eq:form_factor}.
Assuming a uniform magnetic field, $\mathbf{B}_{0} = B_{0}\hat{z}$, the form factor is expressed as 
\begin{equation}
\begin{split}
    C =& \frac{2^{3 - 2m}(\chi_{m1})^{2m}}{\pi^{2}J_{m+1}(\chi_{m1})^{2}} \left[\Gamma \left(\frac{m}{2} + 1 \right) \right]^{2} \\
    & \left[ {}_{1}\tilde{F}_{2}\left(m/2 + 1; m/2 + 1, m + 1; -\frac{\chi_{m1}^{2}}{4}\right) \right]^{2},
    \label{eq:form_cal}
\end{split}
\end{equation}
where $\Gamma$ is the Euler Gamma function and ${}_{p}\tilde{F}_{q}$ is a generalized hypergeometric function with argument lengths of $p$ and $q$.

Table~\ref{tab:multicell_charac} summarizes the characteristic parameters of a circular sector cavity for various values of $m$.
The resonant frequency increases with increasing number of partitions as expected.
The geometrical factor, defined in Eq.~\ref{eq:q_coeff}, reflects the surface-to-volume ratio to cavity quality -- for higher frequencies, the ratio increases and thus the quality factor decreases.
The normalized quality factor for a fixed resonant frequency and the form factor remain at reasonably high levels.
Therefore, the multiple-cell cavity design can increase the resonant frequency with improved performance, providing a useful tool for high-mass axion searches.

\begin{table}
\begin{threeparttable}
\caption{\label{tab:multicell_charac}Characteristic parameters of a circular sector cavity with different values of $m$ with $m=0$ representing a conventional cylindrical cavity.
The quantities are normalized to those of a cylindrical cavity ($m=0$) except for the form factor.
}
\begin{ruledtabular}
\begin{tabular}{p{0.03\textwidth}|ccccc}
$m$ & 0 & 1 & 2 & 3 & 4 \\
$\phi$ &  & 180$^{\circ}$ & 90$^{\circ}$ & 60$^{\circ}$ & 45$^{\circ}$ \\
\colrule
$\omega_{c}{}$ & 1 & 1.5933 & 2.1356 & 2.6531 & 3.1555 \\
$\mathcal{Q}{}$ & 1 & 0.5409 & 0.4241 & 0.3373 & 0.2788 \\
$Q_{c}$\tnote{*} & 1 & 0.8618 & 0.9057 & 0.8949 & 0.8797 \\
$C$ & 0.6917 & 0.6440 & 0.6493 & 0.6341 & 0.6132 \\
\end{tabular}
\end{ruledtabular}
\begin{tablenotes}
\item[*] Calculated for a fixed $\omega_{c}$.
\end{tablenotes}
\end{threeparttable}
\end{table}

For frequency tuning, the same mechanism as for the conventional multiple cavity system can be applied using a set of dielectric or conducting rods individually inserted into each cell~\cite{AP2018Jeong}.
Because the multiple-cell design has a discrete symmetry in the azimuthal direction, the resonant frequency can be tuned by moving the rods along the same direction.
This can be achieved by rotating the set of tuning rods (which are attached to a single structure, typically a rotational actuator) simultaneously with respect to the center of the cavity.
The tuning path is determined to maximize the tunable frequency range such that a rod passes through the point where the highest electric field of the resonant mode is formed.

The size of the tuning rod affects the tunable range $\Delta \nu$ -- in general, the larger the rod is, the greater the field variation and thus the wider the tunable range.
However, for dielectric rods, as the size increases, the form factor decreases and so does the average scan rate $\langle d\nu / dt \rangle$.
Therefore, it is necessary to optimize the rod dimensions, typically its radius $R_r$, to obtain a high scan rate over a reasonable tuning range.
In earlier work~\cite{PATRAS2016Jeong}, it was found that for a cylindrical cavity with a dielectric tuning rod with $\varepsilon=10$, $R_r/R_c\sim0.1$ maximizes the product of the average scan rate and the tunable range.
A similar simulation study was repeated for various circular sectors and the results
are shown in Fig.~\ref{fig:rod_radius}.
In the multiple-cell design, the resonant frequency depends on the size of the sector, equivalently the number of cells.
For a given cavity size, the relative rod size needs to account for the frequency dependence on the cell multiplicity.
The effect can be reflected by multiplying the physical size of a rod by the normalization factor, $\chi_{m1}/\chi_{01}$, which appears on the $x$-axis in Fig.~\ref{fig:rod_radius}.
From the figure, it is observed that the optimal rod dimensions are consistent with that of a cylindrical cavity regardless of the cell multiplicity.

\begin{figure}
\includegraphics[width=0.9\linewidth]{multiple-cell_rod_radius_vs_fom.png}
\caption{(a) Average scan rate (solid line) over tunable frequency range (dashed line) for different cell multiplicities as a function of size of a dielectric tuning rod relative to the cavity size. 
The quantities are normalized to those of the empty cavity without the rod.
(b) Product of scan rate and tunable range.
The relative radius of the rod maximizing the quantity is found to be approximately the same regardless of the number of cells.
}
\label{fig:rod_radius}
\end{figure}


\section{\label{sec:local}Field localization}
In the multiple-cell design, geometric asymmetry between cells will induce field localization to specific cells, which eventually degrades the form factor.
Naturally, the greater the asymmetry is, the greater the degree of localization will be.
A major cause of asymmetry is unavoidable tolerances in cavity fabrication, including machining and assembly.
The tolerance effect can be estimated based on numerical simulation studies by slightly changing the dimensions of individual cells.
An example is shown in Fig.~\ref{fig:localization} using COMSOL Multiphysics software~\cite{COMSOL}.
Inspired by the fact that no considerable localization issue appears in the cylindrical cavities, it is considered that the degree of localization could be reduced for a multiple-cell design by increasing the partition space in the middle of the cavity.
The effect was assessed by repeating the aforementioned simulation with different values of partition gap.
Figure~\ref{fig:localization} illustrates how the field localization is being alleviated, i.e., the field becomes more evenly distributed over the cavity with increase in the central space.

\begin{figure}
\includegraphics[width=0.32\linewidth]{4cell_localization_CG0.15.png}
\includegraphics[width=0.32\linewidth]{4cell_localization_CG0.19.png}
\includegraphics[width=0.32\linewidth]{4cell_localization_CG0.25.png}
\caption{Simulated electric field distributions for the TM$_{010}$-like mode of a 4-cell cavity with the dimensions altered by 0.05\%. The partition space increases from left to right.} 
\label{fig:localization}
\end{figure}

\begin{figure}[b]
\includegraphics[width=0.9\linewidth]{2cell_tolerance_FOM_CG.png}
\caption{Relative scan rate for a 2-cell cavity as a function of partition gap for various tolerances.
Different colors represent different tolerance levels.
The error bar at each point reflect the statistical deviation from the 100 simulations depicted in the text.
The black dashed line represents the ideally modeled cavity.
} \label{fig:2cell_tolerance_FOM}
\end{figure}

Increasing the partition space, however, also reduces the resonant frequency, thereby losing the desired benefits of the multiple-cell design for a high-frequency approach.
Therefore, an optimal center gap needs to be determined to compromise these two aspects under circumstances that fabrication tolerances are taken into account.
Monte Carlo simulations based on COMSOL were carried out for this purpose.
Beginning with an ideal cavity model, the cavity dimensions were allowed to randomly vary within 0.01\% to 0.05\% which are typical tolerances depicted in the international standard~\cite{ITgrade}.
Such perturbations were modeled 100 times and each time a quantity $V_c^2C^2Q_c$, a cavity-associated parameters in the scan rate Eq.~\ref{eq:scan_rate}, is calculated to measure performance.
This procedure is repeated for different sizes of partition gap to find the optimal space.
The results for a 2-cell cavity are shown in Fig.~\ref{fig:2cell_tolerance_FOM}, where the computed scan rates are normalized to that of a cavity without a partition gap.
Several points were observed: 1) a larger tolerance value degrades performance for a fixed gap, $R_g$, 2) the overall behavior converges to the ideal case with increasing gap size, and 3) for a larger tolerance level, a larger gap size is required to maximize the performance.
In general, precision fabrication will require smaller partition spacing to avoid field localization and to achieve better performance.

The same simulation study was extended to cavities with higher-cell multiplicities.
Figure~\ref{fig:ncell_tolerance_FOM} shows the results for 4-cell and 6-cell cavities assuming a 0.05\% tolerance level, and compares them with those for the 2-cell cavity.
It is found that higher-cell multiplicities require larger openings in the middle of the cavity for maximal performance.
It is also seen that the scan rates relative to those of the ideal designs are comparable to each other regardless of cell multiplicity.
The numerical values of the optimal gap and the corresponding scan rate from Figs.~\ref{fig:2cell_tolerance_FOM} and~\ref{fig:ncell_tolerance_FOM} are summarized in Table~\ref{tab:optimal_cg}.

\begin{figure}
\includegraphics[width=0.9\linewidth]{ncell_tolerance_FOM_CG.png}
\caption{Extended version of Fig.~\ref{fig:2cell_tolerance_FOM} to cavities with higher-cell multiplicities, 4-cell and 6-cell, represented in different colors. Only 0.05\% tolerance level is considered.} \label{fig:ncell_tolerance_FOM}
\end{figure}

\begin{table}[b]
\caption{\label{tab:optimal_cg}Optimal center gap for various multiple-cell cavities and tolerance levels.}
\begin{ruledtabular}
\begin{tabular}{cc|cc}
$m$ & Error [\%] & $R_{g}^{\rm opt.}/R_{c}$ & Relative scan rate \\
\colrule
\multirow{5}{*}{1} & 0.01 & 0.032 & 0.90 \\
 & 0.02 & 0.040 & 0.87 \\
 & 0.03 & 0.047 & 0.85 \\
 & 0.04 & 0.052 & 0.83 \\
 & 0.05 & 0.054 & 0.82 \\
2 & 0.05 & 0.20 & 0.84 \\
3 & 0.05 & 0.26 & 0.90 \\
\end{tabular}
\end{ruledtabular}
\end{table}

\section{\label{sec:discussion}Discussion}

The multiple-cell cavity haloscope is expected to provide an efficient way to search for high mass dark matter axions.
The feasibility was successfully tested by performing a proof-of-concept experiment using a double-cell cavity at IBS-CAPP~\cite{PRL2020Jeong}.
This unique cavity concept has been adopted by various axion experiments.
KLASH proposed an experiment with multiple-cell cavities inside a large solenoid magnet to increase the search frequency~\cite{ARXIV2018KLASH}.
CAST-RADES utilizes a similar concept for efficient axion searches within a long dipole magnet~\cite{JHEP2021RADES}.
At IBS-CAPP, a conventional cylindrical cavity of an existing experiment, of which the natural frequency was about 1.7\,GHz, was divided into eight cells to increase the target frequency around 6\,GHz.

Because the resonant frequency is determined by the cell number (i.e., the more cells the higher the resonant frequency), a multiple-cell haloscope, in principle, can be efficiently designed for any arbitrary search frequency. 
This provides a useful feature, particularly when multiple experiments with different detector sizes are intended to utilize the same search frequencies.
For example, IBS-CAPP plans to install 4-cell, 6-cell, and 8-cell cavities in separate magnets with different dimensions, all of them aiming for near 6\,GHz in order to take advantage of phase-matched signal combination, which effectively enhances the experimental sensitivity~\cite{AP2018Jeong}.

As mentioned earlier, there are other interesting ideas for accessing high-mass regions, such as exploiting higher-order resonant modes~\cite{JPG2020Kim,PRA2020Quiskamp} or employing the principle of metamaterials~\cite{PRL2019Lawson}.
These would offer orthogonal approaches to our multiple-cell concept.
This eventually indicates that if these additional ideas could be strategically implemented in each cell of a multiple-cell cavity, it would push the search regions into even higher frequency domains than any individual approach could access.

\section{\label{sec:conclusion}Summary}

We reviewed the concept of multiple-cell cavities and discussed their characteristics for use as an axion haloscope in high-mass regions.
The major advantages over traditional designs at high frequencies include 1) minimal volume loss, 2) simple experimental design, and 3) comparable cavity properties (form factor and quality factor).
Frequency tuning is achieved by simultaneously rotating a set of tuning rods, each introduced into a single cell, with respect to the cavity center. 
The potential field localization owing to machining tolerance is mitigated by increasing the gap in the middle of the cavity.
The experimental feasibility of this cavity concept was successfully demonstrated by employing a double-cell cavity into an axion search experiment at CAPP.
This verifies that the multiple-cell haloscope can provide an efficient way to search for high-mass axions.


\begin{acknowledgments}
J. Jeong and S. Youn are supported by the Institute for Basic Science (IBS-R017-D1-2022-a00).
J. E. Kim is supported in part by the National Research Foundation grants NRF-2018R1A2A3074631.
\end{acknowledgments}

