Single-photon sources based on plexcitonic systems are notable for their fast fluorescence rates, typically >100 GHz. Our investigations reveal that exceptional points (EPs) may unveil the quantum limit of fluorescence rates in plexcitonic single-photon sources. By employing a non-Hermitian Hamiltonian framework and field quantization model, we demonstrate how the fluorescence rate can be ingeniously designed in an exemplified plexcitonic system consisting of a nanocube-on-mirror cavity and a single quantum emitter. We predict the highest fluorescence rates of 11.0, 13.9, and 14.7 THz at the EPs with typical dipole moments of 25, 30, and 35 D, respectively.

Efficient single-photon sources are essential for quantum information processing,1 quantum communication,2 and quantum cryptography,3 whose efficiency is directly contingent upon the fluorescence rate, i.e., the generation rate of high-quality single-photons within the detection time. Over the years, solid-state systems built upon atom-like quantum emitters (QEs) have stood out as promising single-photon sources, offering distinct advantages in both photon purity and efficiency.4 The intrinsic emission rate of typical QEs is on the order of hundreds of MHz, e.g., 42 MHz for CdSe quantum dots (QDs),5 200 MHz for color centers in hexagonal boron nitride (hBN),6 and 260 MHz for terrylene molecules.7 In 1995, Purcell suggested cavity-enhanced fluorescence rates,8 showing that the fluorescence rate can be increased by a dielectric cavity compared to spontaneous emission in a vacuum through a ratio known as the Purcell factor. For instance, an InAs QD coupled to a distributed-Bragg-reflector (DBR) cavity with a Purcell factor of 5 exhibited a fluorescence rate of 11 GHz,9 and an electrically controlled micropillar achieved a near-optimal Purcell factor of 9.8, resulting in a fluorescence rate of 13 GHz for an InGaAs QD.10 The theoretical upper limit of fluorescence rates enhanced by dielectric cavities was predicted close to 60 GHz due to the diffraction limit.11 

To further enhance fluorescence rates, a plexcitonic system, i.e., a single QE coupled to a plasmonic nanostructure, was proposed.12,13 Over the years, plexcitonic systems have experimentally demonstrated their capability to further enhance photon generation rates, e.g., 80 GHz for a CdSe/ZnS QD in a nanocube-on-mirror (NCoM) cavity14 and 100 GHz for a single TPQDI dye molecule coupled to a bowtie antenna.15 Meanwhile, several theoretical advancements have been proposed to model the single-photon emission in plexcitonic systems, agreeably predicting that the fluorescence rates may hit the THz range. Classical models extend the Purcell formula to plasmonic cavities by reshaping the mode volume calculations, which predict fluorescence rates up to hundreds of THz.11,16 Semi-classically, Fermi’s golden rule describes the transition probability that depends on the number of states available to QE, quantified by the local density of states (LDOS), which replace the Purcell enhancement to predict the fluorescence rate—up to 100s- to 1000s-fold enhancement compared to intrinsic emission rate.17,18 These classical and semi-classical models focused on unidirectional energy transfer from QE to cavities, which may overestimate the limit of photon production rate, especially when the interaction enters a strong coupling regime where QE and plasmon exchange energies. In recent years, quantum models have been developed to reformulate the problem of single-photon emission, e.g., quantization of plasmonic cavities through quasi-static solutions to Maxwell’s equations19 and quantized quasinormal mode (QNM) master equation.20 In particular, Hugall reformulated the fluorescence rate as a function of the coupling rate between plasmon and QE, providing a comprehensive overview of fluorescence rate enhancement in the weak and strong coupling regimes.21 However, determining the precise maximum fluorescence rate in relation to the system’s design remains ambiguous, particularly in understanding the fluorescence rate in the strong coupling regime and the role of quenching in shaping the quantum limit.

Curious about this quantum limit and the corresponding optimal condition, we draw inspiration from the non-Hermitian nature of plexcitonic systems,22 which exhibit non-conservative dynamics and energy dissipation. One fascinating aspect of non-Hermitian systems is the presence of exceptional points (EPs), where the eigenvalues and eigenvectors of the system coalesce, and the eigenspace is abruptly reduced to lower dimensions. A recent study has shown that the photoluminescence of a CsPbBr3 layer in a photonic-crystal slab can be enhanced by EP in the background modal coupling by modifying LDOS.23 Our study will investigate the EP directly arising from the coupling between a QE and a background localized surface plasmon (LSP) mode in a plexcitonic single-photon source. We will develop a generalized non-Hermitian Hamiltonian framework combined with a field quantization model24 to describe the fluorescence excitation and emission in the QE–LSP system. Focusing on an exemplified plasmonic NCoM cavity,25 we will provide insights into the mechanisms underlying the manipulation of fluorescence rates and the system designs to attain EPs for the highest possible fluorescence rates. We hope to clarify the quantum limit for achieving highly efficient single-photon emission.

As illustrated in Fig. 1(a), our NCoM cavity consists of a lossless dielectric gap (refractive index n = 1.4; tunable gap thickness d) sandwiched between a silver nanocube of side length L = 75 nm and a gold substrate, supporting a few optical modes26 primarily confined by the cube facet.27 A single QE, as the source of single-photon fluorescence, is placed at the center of the gap and modeled as a point electric dipole excitation source, characterized by γ0=ωe3μe2/(3πε0c3)—the spontaneous decay rate in free space where ωe and μe represent the resonant frequency and the dipole moment of the QE, respectively. We simplify the coupling scenario between the QE and NCoM cavity by favorably choosing an isolated LSP mode α [a bright mode with a strong antenna effect28 featured in Fig. 1(a), inset] at a resonant frequency ωα and a decay rate κα. In addition, assume that the QE resonates with the α mode throughout the study: ωe = ωα = ω0. Simultaneously, the excited field Eex(ω) in the gap of the NCoM, in turn, serves as the localized excitation field for QE, determining the localized coupling strength ℏg(ω) = μe · Eex(ω). We can extract the coupling rate gα and decay rate κα based on a field quantization model,24 where we quantize the mode electromagnetic (EM) field by fitting the spectral density obtained from classical EM simulations. Taking an NCoM cavity with d = 3 nm as a case study, we apply the quantization model to obtain the characteristics of the α mode: ωα = 1.8 eV, κα = 167.7 meV, the coupling rate gα = 29.4 meV for a selected QE with μe = 30 D, ωe = 1.8 eV at resonant condition, and γ0 = 9.3 μeV. Notably, the coupling rate gα to the same α mode can be tuned by varying the property of QE: gα = 0.98μe (see the derivation in the supplementary material, SI-1).

FIG. 1.

Identifying EP of a representative plexcitonic system. (a) Schematic of an NCoM plasmonic cavity with a QE positioned at the center of the gap. The excited field distribution within the xy plane is depicted to visualize the characteristics of the bright LSP mode α. (b) Eigenfrequencies of the plexcitonic system as a function of the normalized coupling rate gα=gα/κα: real parts (solid lines) and imaginary parts (dotted lines). EP at gα=0.25 defines the boundary between the weak (purple) and strong (golden) coupling regimes. (c) Time evolution of fluorescence photon in the plexcitonic system in the weak coupling, EP, and strong coupling regimes at corresponding gα of 1/8, 1/4, and 1. The fluorescence intensity is normalized to the first peak to highlight the decay processes.

FIG. 1.

Identifying EP of a representative plexcitonic system. (a) Schematic of an NCoM plasmonic cavity with a QE positioned at the center of the gap. The excited field distribution within the xy plane is depicted to visualize the characteristics of the bright LSP mode α. (b) Eigenfrequencies of the plexcitonic system as a function of the normalized coupling rate gα=gα/κα: real parts (solid lines) and imaginary parts (dotted lines). EP at gα=0.25 defines the boundary between the weak (purple) and strong (golden) coupling regimes. (c) Time evolution of fluorescence photon in the plexcitonic system in the weak coupling, EP, and strong coupling regimes at corresponding gα of 1/8, 1/4, and 1. The fluorescence intensity is normalized to the first peak to highlight the decay processes.

Close modal
To identify the EP arising from the QE–LSP coupling, we employ a non-Hermitian Hamiltonian23,29 to describe the plexcitonic system (ℏ = 1 here and later),
H=ωeiγ0/2gαgαωαiκα/2,
(1)
where the uncoupled states with complex frequencies Ee = ωe0/2 and Eα = ωαα/2 are used as the basis to represent QE and α mode as a quasinormal mode (QNM)30,31 for this open system. Following the resonant condition ωe = ωα = ω0, the eigenfrequencies can be expressed as
E±=ω0iγ0+κα4±1416gα2(καγ0)2.
(2)
For the case study system with d = 3 nm, we plot the eigenfrequencies in Fig. 1(b) as a function of the normalized coupling rate gα=gα/(καγ0)gα/κα since γ0κα. For a fixed NCoM geometry with unchanged κα, the tuning of gα is implemented via changing the dipole moment of QE through gα = 0.98μe. As shown in Fig. 1(b), EP identified at gα=1/4 signifies the coalescence of the two eigenfrequencies and serves as a boundary to separate the purple and golden zones. When gα>1/4 (golden zone), the imaginary parts of the eigenfrequencies Im E± overlap, while the real parts Re E± deviate from each other, indicating strong coupling with Rabi splitting effects.32,33 In contrast, gα<1/4 (purple zone) corresponds to the weak coupling regime as Im E± split and Re E± degenerate.
We simulate the dynamics of the system based on the Jaynes–Cummings (JC) model,34 considering the Hamiltonian under the rotating-wave approximation,35,
Ĥ=ωeσ̂σ̂+ωαâαâα+gα(σ̂âα+âασ̂).
(3)
Here, α mode is quantized as a harmonic oscillator with the creation and annihilation operators âα and âα, and QE is treated as a two-level system by the transition operators σ̂ and σ̂. We assume that QE is initially excited and employ the master equation,19,
ρ̂t=i[ρ̂,Ĥ]+γ02L[σ̂]ρ̂+κα2L[âα]ρ̂,
(4)
where the Lindblad terms have the form of L[ô]ρ̂=2ôρ̂ôôôρ̂ρ̂ôô, indicating that the dynamics of each operator can be deduced by the evolution of the density matrix ρ̂. Considering the open nature of the plasmonic cavity, the fluorescence operator is defined as19,36
F̂=γ0σ̂+ηακαâα,
(5)
which encompasses the emission from both QE and α mode. The quantum yield ηα represents the probability that the excitation of α mode leads to emission.17 Consequently, the temporal evolution of the fluorescence intensity can be computed from the following relationship (see the derivation in the supplementary material, SI-2):
F̂F̂(t)=Tr(F̂F̂ρ̂(t)).
(6)
As depicted in Fig. 1(c), we showcase the fluorescence dynamics for three different gα: 1/8 (weak coupling), 1/4 (EP), and 1 (strong coupling). In general, the fluorescence intensity rises and falls when a single fluorescence photon is formed and emitted; that is, the completely decayed fluorescence intensity corresponds to a completely generated emission photon. Among the three cases, a fluorescence photon from weak coupling takes the longest time to exit the system, whereas a direct comparison between the remaining two is challenging due to the presence of Rabi oscillations for the strong coupling case.

To provide a quantitative comparison, we define a fluorescence rate γF to evaluate the photon generation rate, which equals the decay rate of fluorescence intensity in the system. Leveraging the exponential nature of the decay processes, we depict the same three dynamics in Fig. 1(c) on a natural logarithmic scale, as demonstrated by the solid lines in Fig. 2(a). Then, the fluorescence rates γF can be obtained through a linear fitting by assuming that the initial rising time is negligible compared to the entire decay process. This approach is widely used in studies of cavity quantum electrodynamics (cQED),37,38 particularly in the strong coupling regime where the peaks of Rabi oscillations intersect with the fitting line. We plot this γF fitted from the dynamics calculations for a wide range of normalized coupling rates gα via tuning μe in Fig. 2(b) (black circles). The fluorescence rates exhibited accelerated growth in the weak coupling regime (gα<0.25) and reached a saturated rate of 83.8 meV (or 20.3 THz) in the strong coupling regime (gα>0.25), where EP is seen clearly as a sharp turning point. We conclude that a maximized γF exists for any fixed NCoM geometry, and EP defines the minimum required coupling rate gα = 41.9 meV (correspondingly, μe = 42.8 D) for our case study (d = 3 nm) to achieve the maximized γF.

FIG. 2.

Fluorescence emission characteristics of a representative plexcitonic system. (a) Dynamics (in natural log scale) of the fluorescence intensity (solid lines) and the excited state population of the QE (dotted lines), which are fitted by the gray dashed lines, and the slopes are taken as the rates. (b) The fluorescence rates calculated from fitting dynamics in (a) (black circles) and deduced from an analytical model (gray line) as a function of the normalized coupling rate gα. The magenta dashed line highlights the EP. The results for gα=0.40.9 are shown in the supplementary material, SI-2.

FIG. 2.

Fluorescence emission characteristics of a representative plexcitonic system. (a) Dynamics (in natural log scale) of the fluorescence intensity (solid lines) and the excited state population of the QE (dotted lines), which are fitted by the gray dashed lines, and the slopes are taken as the rates. (b) The fluorescence rates calculated from fitting dynamics in (a) (black circles) and deduced from an analytical model (gray line) as a function of the normalized coupling rate gα. The magenta dashed line highlights the EP. The results for gα=0.40.9 are shown in the supplementary material, SI-2.

Close modal
To find out the underlying mechanism, we plot the dynamics of QE’s excited state σ̂σ̂(t) in Fig. 2(a) (dashed lines), which are also fitted to obtain their decay rates γe. All three cases indicate γF = γe, which holds for any non-zero gα in the quantum dynamics because Eq. (5) indicates the linear relationship between the fluorescence operator F̂ and the transition operator σ̂. This relationship leads to an analytical model for fluorescence rates. As marked in Fig. 1(b), the eigenfrequency E+ corresponds to the curves where E+ = Ee = ωe0/2 when gα = 0. This implies that E+ = ω++/2 characterizes the coupling-dependent behavior of the QE, with Re E+ reflecting frequency detuning and Im E+ indicating the variation in decay rate. From Eq. (2), we have
γ+=2|ImE+|=γ0+κα2καγ021(4gα)2
(7)
in the weak coupling regime and
γ+=2|ImE+|=γ0+κα2
(8)
at EP and in the strong coupling regime. Both agree with another analytical approximation through the decomposition of the master equation (see the supplementary material, SI-3). Hence, the fluorescence rate can be analytically deduced as γF = γe = γ+ = 2|Im E+|. We plot this analytical solution of γF = 2|Im E+| in Fig. 2(b) (gray solid line) and find a good match to the dynamics calculations. The value of the saturation rate of 83.8 meV nicely corresponds to (γ0 + κα)/2 from Eq. (8). This analytic solution offers a simplified method to directly deduce the fluorescence rate from the imaginary part of the eigenfrequency 2|Im E+| (see more cases in the supplementary material, SI-4).

Up to this point, we have demonstrated the generalized procedure to identify EP and calculate the maximized fluorescence rate γF for any given plasmonic nanocavity, exemplified using an NCoM cavity with d = 3 nm. We can now analyze how the geometric control (in particular, gap thickness d) of the NCoM influences γF39 by designing the LDOS of the EM environment of QE.40 In this study, we fix the QE dipole moment μe = 30 D and alternate d of the NCoM. The normalized LDOS (see the supplementary material, SI-1, for the method) is presented in Fig. 3(a), corresponding to an increase in d from 1.0 to 4.4 nm with 0.2 nm intervals. This range is chosen carefully due to practical fabrication limitations41 at the lower limit of 1.0 nm, while the blue shift of the mode α determines the upper limit of 4.4 nm. The quantization of the bright α mode can be notably affected by the overlap between α and its neighboring modes, encompassing a dark mode β and various high-order modes (further elaborated in the supplementary material, SI-5). From the spectra, we trace the α mode and obtain a set of parameters (ωα, κα, gα, ηα) for each d and constantly set ωe = ωα to ensure resonance between QE and α mode.

FIG. 3.

Geometric control of NCoM cavities to tune normalized coupling rate. (a) Normalized LDOS spectra for NCoM cavities with d from 1.0 to 4.4 nm at 0.2 nm intervals. The gray dashed curves trace two neighboring modes: bright α and dark β. (b) Tunable normalized coupling rates gα from values of coupling rates gα and decay rates κα by varying d of NCoM cavities. The magenta lines indicate EP to separate the weak (purple) and strong (golden) coupling regimes, corresponding to d ≃ 2.5 nm.

FIG. 3.

Geometric control of NCoM cavities to tune normalized coupling rate. (a) Normalized LDOS spectra for NCoM cavities with d from 1.0 to 4.4 nm at 0.2 nm intervals. The gray dashed curves trace two neighboring modes: bright α and dark β. (b) Tunable normalized coupling rates gα from values of coupling rates gα and decay rates κα by varying d of NCoM cavities. The magenta lines indicate EP to separate the weak (purple) and strong (golden) coupling regimes, corresponding to d ≃ 2.5 nm.

Close modal

We present the quantization results in Fig. 3(b), focusing on how the geometric control on d of the NCoM cavity can practically tune the normalized coupling rate gα. As d increases, gα=gα/κα decreases monotonically, but the decrease slows down. First, the coupling rate gα decreases as d increases, reflecting the expansion of the mode volume.21 Second, the decay rate κα, comprising radiative emission and nonradiative loss, initially increases with d. This rise is primarily driven by an enhanced radiative emission resulting from a more “opened” cavity as the gap size expands, which outweighs the reduction in metallic absorption. Subsequently, as d continues to increase, the increase in κα slows down, primarily due to the saturation of radiative emission and the resurgence of metallic absorption caused by the overlap with the dark mode β, particularly evident beyond 3.6 nm. In short, we show a sensitive tuning of d from 1 to 4.4 nm to realize the variation of gα from 0 to 1 for a specific μe = 30 D. At gα= 0.25, corresponding to d ≃ 2.5 nm, we draw magenta lines to highlight EP.

To reveal how EP influences the fluorescence rate, we extend our calculations of γF for our case study d = 3 nm shown in Fig. 2 based on Eqs. (7) and (8) to a wide range of gap thickness with a fixed μe = 30 D and show our results in Fig. 4. First, Fig. 4(a) plots the real and imaginary parts of the eigenfrequencies for different cases of d (corresponding to different gα), where EP (marked by a dotted magenta line) represents the unique singular point at which both the real and imaginary parts coalesce occurring at gα= 0.25. Second, we plot all possible values of γF using the rainbow curves (each rainbow color corresponds to a gap thickness d) with a free choice of QE dipole moment for different d, following the same procedure illustrated in Fig. 2(b). Next, by specifying the QE with fixed dipole moment, we can determine the pure geometric control over γF of the plexcitonic systems. For example, we show γF (black circles) for μe = 30 D in Fig. 4(b), which exhibits a maximized fluorescence rate of 57.5 meV (or 13.9 THz) at EP with d ≃ 2.5 nm and ω0 = 1.66 eV. Additional results for dipole moments of μe = 25 D and μe = 35 D, presented in the supplementary material, SI-6, yield the highest fluorescence rates of 45.3 meV (or 11.0 THz) for d ≃ 2.3 nm and 60.6 meV (or 14.7 THz) for d ≃ 2.6 nm.

FIG. 4.

Unveiling the quantum limit of fluorescence rate at EP by geometric control. (a) Eigenfrequencies and (b) fluorescence rates γF as a function of the normalized coupling rate gα for a wide range of gap thickness d. Each rainbow curve in (b) depicts the γF tunable with the dipole moment μe of the QE for one d, similar to Fig. 2(b). The black circles indicate the γF for a specified QE with μe = 30 D on the background rainbow curves. The magenta line highlights EP, where the eigenfrequencies completely coalesce and the maximum fluorescence rate is achieved.

FIG. 4.

Unveiling the quantum limit of fluorescence rate at EP by geometric control. (a) Eigenfrequencies and (b) fluorescence rates γF as a function of the normalized coupling rate gα for a wide range of gap thickness d. Each rainbow curve in (b) depicts the γF tunable with the dipole moment μe of the QE for one d, similar to Fig. 2(b). The black circles indicate the γF for a specified QE with μe = 30 D on the background rainbow curves. The magenta line highlights EP, where the eigenfrequencies completely coalesce and the maximum fluorescence rate is achieved.

Close modal

According to our quantum model, non-Hermiticity is key to understanding the turning trend of γF near EP. For weak coupling, γF follows a nonlinear increasing trend (square root dependence) of gα suggested by Eq. (7). Such a nonlinearity near EP has been widely reported in non-Hermitian systems for sensing,42 luminescence,23 and laser gyroscope.43 For strong coupling regime where each d holds a saturated γF = (κα + γ0)/2 ≃ κα/2 from Eq. (8), γF drops from 57.5 to 20.6 meV as d decreases from 2.5 to 1.0 nm due to the reduction in κα. EP, the intersecting point by both sides holding opposite trends, naturally emerges as the upper limit of γF. A more intuitive description can be found in the supplementary material, SI-7.

The mechanism can be explained in terms of energy transfer. In the weak coupling regime, the QE transfers its energy to α instead of emitting it independently, while α emits more energy than those returned to the QE. Consequently, the photon, originating from the QE transition, is rapidly emitted by the plasmonic cavity, known as the Purcell effect. Typically, the coupling rate gα1/V, where V represents the effective mode volume,21 so that the nonlinearly increasing trend aligns with the stronger Purcell effect in the weak coupling regime. Conversely, in the strong coupling regime, gα becomes sufficiently large for α to return energy to the QE, and the QE subsequently transfers energy back to α. Now, both the QE and α can emit, yet with a substantial portion of the energy cycling between them. As a result, the energy-storing effect arising from strong coupling diminishes the emission rate, and this reduction intensifies as gα continues to rise. Finally, the opposite trends naturally converge at the EP, where the fluorescence rate enhancement reaches its maximum value.

In conclusion, EP in a non-Hermitian plexcitonic system leads to the quantum limit of fluorescence rates. We demonstrate a maximized fluorescence rate of 14.7 THz from a QE with a dipole moment of μe = 35 D coupled to an experimentally accessible NCoM cavity by controlling the gap below 5 nm to reach the EP condition. Compared to previous experimental investigations,14–17 our predicted maximum fluorescence rate on the order of 10 THz seems achievable to the yet-to-be-optimized practical single-photon sources (around 100 GHz to 1 THz). An earlier classical model44 has drawn similar conclusions regarding an “upper bound” positioned at the boundary between the weak and strong coupling regimes, aligning cohesively with our quantum limit. Nevertheless, the previously predicted upper limit of about 300 THz16 from the classical model may be an overestimated value, given that the model relies on effective mode volume estimations that lack a careful consideration of QE–LSP coupling. Here, our approach involves the quantization of the field and a detailed exploration of non-Hermitian physics, offering a “quantum” perspective into light–matter interactions in a plasmonic single photon source.

Our designing procedure to realize EP in plexcitonic coupling and maximize the fluorescence rate should work for a wide range of plexcitonic systems with isolated bright modes via geometric control. First, one should select QE that can potentially resonate with the bright mode of the plasmonic cavity and possess a large μe, which ensures efficient coupling between QE and plasmon. Second, inverse designing40,45 of the plasmonic cavity is done to match the bright mode with the transition frequency of QE (ωα = ωe) and simultaneously satisfy the EP condition (gα=gα/κα= 0.25). Following this generalized procedure, one can maximize the fluorescence rate and enhance the efficiency of single-photon emission in various plexcitonic systems.

See the supplementary material for more details on field quantization model (SI-1), derivation of fluorescence operator (SI-2), fluorescence rates deduced from the decomposition of the master equation (SI-3), additional cases of analytical solutions vs dynamic calculations (SI-4), impact of mode overlapping (SI-5), additional cases for the maximum fluorescence rates at EPs (SI-6), and intuitive description of the fluorescence rate near EPs.

This work was supported by the Singapore University of Technology and Design for the Start-Up Research Grant No. SRG SMT 2021 169, Kickstarter Initiative (SKI) Grant Nos. SKI 2021-02-14 and SKI 2021-04-12, and National Research Foundation Singapore via Grant Nos. NRF2021-QEP2-02-P03, NRF2021-QEP2-03-P09, and NRF-CRP26-2021-0004. W.Z. acknowledges the support of Ph.D. RSS. J.L. acknowledges the support from the National Natural Science Foundation of China (Grant No. 11874438) and the China Scholarship Council (CSC).

The authors have no conflicts to disclose.

Wenjie Zhou: Investigation (lead); Methodology (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Jingfeng Liu: Investigation (supporting); Methodology (equal); Project administration (supporting); Supervision (supporting); Writing – review & editing (supporting). Juanfeng Zhu: Visualization (supporting); Writing – review & editing (supporting). Dmitrii Gromyko: Investigation (equal); Writing – review & editing (equal). Chengwei Qiu: Supervision (equal); Writing – review & editing (equal). Lin Wu: Funding acquisition (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material