We report on the development of a source of ultra-narrow-band photon pairs using the cavity-enhanced spontaneous parametric down conversion. The photon-pair source has a bandwidth of 265 ± 15 kHz at 606 nm and a spectral brightness of 216 ± 5 pairs/(s · mW · MHz) per longitudinal mode, which could be suitable for Pr3+ ion-based solid-state quantum memories.

Quantum memories have attracted considerable attention in recent years in view of their potential for use in quantum communication and quantum computing.1–4 One of the potential candidates for quantum memories is rare earth ion doped solids (REIDSs), such as Pr3+-doped Y2SiO5 (YSO:Pr) crystals, which have the advantages of long decoherence time, lack of atomic diffusion, and ease of integration.1 Heinze et al.5 have already achieved a storage time of 1 min for classical light pulses in YSO:Pr crystals. In turn, Zhong et al.6 demonstrated a decoherence time of 6 h in Eu3+-doped YSO crystals. Note that, for quantum memory based on the electromagnetically induced transparency (EIT) effect,5–10 the gradient echo memory,11 or the atomic frequency comb,12 the spectral bandwidth requirement for most REIDSs is, in general, of the order of MHz. For example, the spectral linewidth of the hyperfine energy level transition 3H41D2 of Pr3+ ions in YSO crystals is of the order of 1 MHz; especially, the bandwidth of the EIT window is much less than 1 MHz.7–10 Accordingly, for the storage of light pulses based on the EIT effect in YSO:Pr crystals, a sub-megahertz bandwidth is required. Indeed, a narrow-band photon source is compatible with all quantum memory protocols based on the hyperfine atomic transitions of REIDSs such as YSO:Pr. This bandwidth requirement is easy to meet in the case of classical light sources but not in the case of quantum light sources because the natural spectral broadening of quantum light sources, such as single-photon or photon pairs, generated by spontaneous parametric down conversion (SPDC) is in the GHz or THz range.13–15 Therefore, narrow-band classical light or strongly attenuated coherent light was used in the early light pulse storage experiments.16–18 

To realize quantum memories in REIDSs such as YSO:Pr crystals based on the EIT effect, one should first generate sub-megahertz narrow-band quantum light sources. Various methods have been proposed to generate quantum light sources, such as single photons and entangled photon pairs through SPDC,13 nitrogen-vacancy centers,19,20 trapped atoms,21 single ions in nanocrystals,22 four-wave mixing,23,24 rephased amplified spontaneous emission,25 and spin-wave storage,26 in which narrow-band quantum light sources were achieved through single ions in nanocrystals,22 four-wave mixing,24 rephased amplified spontaneous emission,25 and spin-wave storage.26 Among these methods, SPDC, which uses the second-order nonlinearity to convert one higher energy photon into two lower energy photons, provides highly bright entangled photon pairs, but the spectral bandwidth of the generated photon pairs is usually very broad and in the THz range. To narrow the spectral bandwidth of the photon pairs, one practical way is to use a narrow-band filter to filter out all unwanted spectral components.27 However, this would significantly reduce the brightness of the output photons. Cavity-enhanced SPDC was proposed by Ou et al.28,29 in 1991. The method solved the brightness problem based on the placement of a nonlinear crystal inside a high-finesse cavity in an effort to narrow the spectral bandwidth of the photons, while the photon generation rate is also enhanced. Using cavity-enhanced SPDC, researchers achieved narrow-band photon pair sources and single-photon sources at various wavelengths. For example, with a periodically poled KTP (PPKTP) crystal inside a standing-wave cavity, Bao et al.30 generated photon pairs with a bandwidth of 9.6 MHz at 780 nm that was suitable for Rb atomic vapor. With a PPKTP crystal inside a bow-tie cavity, Rambach et al.31,32 generated photon pairs with a bandwidth of 429 kHz at 795 nm and achieved a duty cycle of 100% by locking the optical parametric oscillator (OPO) cavity to the pump laser. Tsai and Chen33 used an electro-optic amplitude modulator to modulate the power of the pump laser so that the OPO cavity with the PPKTP crystal operated below and above the threshold. Nondegenerate photon pairs were generated at 780 nm/852 nm with a relatively high brightness. In all these experiments, the quasi-phase-matching technique was employed with periodically poled nonlinear crystals to improve the photon generation rate, and the majority of these studies were focused on the photon pair sources designed for atomic vapors in the near-infrared spectral regime.30,31,33–36 More recently, Moqanaki et al. reported a single-mode narrow-band photon source tuned to the cesium D2 line with a spectral bandwidth of 10.9 MHz and a spectral brightness of 436 pairs/(s · mW · MHz), where the mode-selection was achieved through the introduction of an additional birefringent element into the cavity.37 An overview of recent progress on cavity-enhanced narrow-band quantum light sources can be found in Ref. 38 and the references therein.

When REIDSs are used as the quantum storage media, the operating wavelength is in the visible (Pr3+)7 and the near-infrared (Tm3+, Nd3+, Yb3+, and Er3+)39–42 spectral regimes. For example, for YSO:Pr crystals, the operating wavelength for a quantum memory is at 606 nm, and a pump beam at 303 nm is required for the generation of degenerated photon pairs. However, the periodically poled nonlinear crystals, such as PPKTP and the periodically poled lithium niobate (PPLN), are practically not efficient for the photon pair sources with a pump beam in the UV spectral range, owing to the strong absorption. Fekete et al.43 avoided this problem with the use of a 426-nm pump laser to generate nondegenerate photon pairs with one photon matched to the energy level transition of Pr3+ ions in the YSO crystals, i.e., at 606 nm, and the other at the telecommunication wavelength of 1436 nm. Nevertheless, coating films with high reflectivities or transmissivities at three different wavelengths are difficult and costly.

To produce degenerate photon pairs at 606 nm, one can use β-BaB2O4 (BBO),44 CsLiB6O10 (CLBO),45 or BiB3O6 (BIBO)46 crystals. The nonlinear coefficient of BIBO is slightly larger than that of BBO, but BIBO is sensitive to temperature fluctuations. Therefore, this makes the realization of the phase-matching condition difficult. CLBO has a relatively small walk-off angle, but it is easy to deliquesce, and its nonlinear coefficient is small. In this study, we chose BBO crystals as the nonlinear crystals for the SPDC. BBO is transparent from 185 nm to 2600 nm and has a relatively large nonlinear coefficient in the UV spectral range. It has been widely used in broadband SPDC to generate entangled photon pairs.47–49 Only limited studies used the BBO crystal for cavity-enhanced SPDC. The reported works focused on the enhancement of the pump light and the use of non-collinear phase matching.50,51 We recently developed a high-finesse near-unstable cavity, which is suitable for OPO.52 Based on this progress, we achieved the cavity-enhanced SPDC at 606 nm in the BBO crystal with the collinear phase matching in the OPO below the threshold in this paper. The bandwidth and the spectral brightness of output degenerate photon pairs at 606 nm were measured to be 265 ± 15 kHz and 216 ± 5 pairs/(s · mW · MHz) per longitudinal mode. This sub-megahertz bandwidth matches the EIT window of the YSO:Pr crystal;7 therefore, it is suitable for EIT-based quantum storage in YSO:Pr crystals. In addition, the generated photon pair is theoretically frequency-entangled due to the requirement of phase-matching condition and energy conservation during the SPDC process. It may also be potentially used to achieve entanglement among different atomic ensembles because both the signal and the idler photons can be stored for quantum memories.53–55 This feature could be available for quantum network and entangling two remote, long lived YSO:Pr quantum memories for communication and computation. In addition, such narrow-band quantum sources are also desired to study the nonlinear light–matter interactions such as giant Kerr nonlinearity, cross-phase modulation, and four-wave mixing at the single-photon or few-photon level.1,56

The experimental setup used to generate the narrow-band photon pairs is shown in Fig. 1. A tunable, continuous-wave dye laser (Matisse 2 DX, Spectra-Physics) with a linewidth of 100 kHz and a central wavelength of 606 nm that matched the resonant hyperfine energy level transition 3H41D2 for Pr3+ ions doped in YSO crystals was used in the experiments. An isolator (712A, Conoptics Inc.) was used to block the reflected light from the optical elements into the dye laser cavity. The 606-nm laser beam was then split into two beams by using a beam splitter (BS1). One beam had a power of 70 mW and was used as a reference beam to stabilize an OPO cavity (located on the right-bottom corner of Fig. 1). The other beam had a power of 700 mW and served as a pump beam for second harmonic generation (SHG) to generate the UV laser beam through a Brewster-cut BBO crystal in a bow-tie cavity (MBD, Coherent Inc.) that was stabilized using the Hansch–Couilaud technique.57 The output laser power from the MBD bow-tie cavity was 80 mW at 303 nm.

FIG. 1.

Schematic diagram of experimental setup. BS1, BS2, BS3: beam splitter; PBS: polarization beam splitter; MBD: a bow-tie ring cavity for a second harmonic generation (SHG) with a BBO crystal; GT: Glan–Taylor polarizer; DM: dichroic mirror; EOM: electro-optic modulator; SPCM: single-photon counting module. The pinhole was used to block the unwanted stray light from entering the single-photon detectors.

FIG. 1.

Schematic diagram of experimental setup. BS1, BS2, BS3: beam splitter; PBS: polarization beam splitter; MBD: a bow-tie ring cavity for a second harmonic generation (SHG) with a BBO crystal; GT: Glan–Taylor polarizer; DM: dichroic mirror; EOM: electro-optic modulator; SPCM: single-photon counting module. The pinhole was used to block the unwanted stray light from entering the single-photon detectors.

Close modal

Owing to the birefringence of the BBO crystal, the transverse electric field distribution of the generated UV laser beam was not in a perfect Gaussian profile. Therefore, we used a spatial filter to clean the beam, and the residual power of the Gaussian beam was 3 mW. This Gaussian beam served as the pump beam to generate the photon pairs through the SPDC in the OPO cavity. Theoretically, the generated photon pair is frequency-entangled due to the requirement of phase-matching condition and energy conservation during the SPDC process. Because the pump beam did not resonate within the OPO cavity, and transmitted through the BBO crystal only once within the cavity, the mode-matching condition should be satisfied for the pump beam to improve the generation rate of photon pairs in the OPO cavity.29 To achieve this mode-matching condition, we used an additional three-mirror cavity as reference, as shown in Fig. 1. The UV pump beam was split by using a beam splitter (BS2) with a ratio of 9:1, and a small part of the pump beam was used to match the three-mirror cavity. The 606-nm laser beam split from the BS1 was also injected into the OPO cavity to generate a backward second harmonic field (303 nm). This backward second harmonic field from the BBO crystal in the OPO cavity was aligned to match the three-mirror cavity. In this way, when both UV beams, which propagated in opposite directions, were matched to the three-mirror cavity, the mode-matching condition was satisfied for the UV pump beam.29 The mode-matched UV pump beam was then injected into the OPO cavity to generate photon pairs based on the SPDC. Note that here the 606-nm laser beam split from the beam splitter BS1 was not only used to stabilize the OPO cavity but also served as the fundamental beam to generate the backward second harmonic field that was used to find the mode-matching condition for the UV pump beam.

The OPO cavity was a homemade plano-concave cavity made of invar and contained a type-I BBO crystal. Invar is a nickel–iron alloy with an ultra-low thermal expansion coefficient. The plane and the concave mirrors (with a curvature radius of 500 mm) were coated with an antireflection coating for 303 nm and a highly reflective coating for 606 nm. The reflectivity of the input mirror at 606 nm was 99.9841%, and the reflectivity of the output mirror at 606 nm was 99.955%. The BBO crystal in the OPO cavity was also coated with an antireflection coating for both 303 nm and 606 nm beams (Castech Inc.). We mounted the BBO crystal on a temperature controlled platform with a 0.01 °C control precision. The plano-concave cavity surrounding the BBO crystal had a finesse F0 of 13 995. Note that the OPO cavity has an ultra-high finesse and a length of 235 mm, which makes the locking of this cavity quite difficult due to mechanical vibrations and thermal expansion. We split the 606 nm laser beam (70 mW) from the BS1 at the very beginning of the setup, which served as a reference beam to stabilize the OPO cavity. An electro-optic phase modulator (EOM, PM7-VIS, Qubig GmbH) with a modulation frequency of 20 MHz was used to modulate the phase of this reference beam. With the help of a quarter-wave plate (QWP) and a polarizing beam splitter (PBS), the reflected light from the OPO cavity was detected and was used to stabilize the OPO cavity using the Pound–Drever–Hall (PDH) technique.58 A ring-shaped piezoelectric transducer was mounted on the posterior side of the concave mirror to achieve cavity stabilization. The reference beam and the generated photon pairs output from the OPO cavity were at the same wavelength of 606 nm and propagated in the same direction. To separate the generated photon pairs from the reference beam, the reference beam should be blocked, while the photon pairs were detected by the following single-photon detectors. We put a chopper before the OPO cavity in such a way that the incident reference beam was blocked by the chopper, while the generated photon pairs output from the OPO cavity transmitted through the chopper, or vice versa, as shown in Fig. 1. Although the OPO cavity was locked intermittently at only ∼40% of the duty cycle of the chopper, the OPO cavity was still stable during the experiments.

The photon pairs output from the OPO cavity were characterized with the use of a Hanbury Brown and Twiss (HBT) interferometer.59 The output photons were split by using a beam splitter (BS3) at a ratio of 1:1. The beam transmitted through a pinhole and a bandpass filter (FF01-605/15-25, Semrock Inc.) with a bandwidth of 15 nm to filter out unwanted stray light. It was then coupled to single-photon detectors (SPCM-AQRH-14-FC, Excelitas) through single-mode fibers with a coupling efficiency of 57% and 71%, respectively. The detection efficiency and the dark counting rate of the single-photon detector were ∼65% and 100 counts/s at 606 nm, respectively. The electronic signals from the single-photon detectors were sent to the photon counting system (DPC 230, Becker & Hickl, GmbH) with a counting accuracy of 0.1646 ns.

Compared to the type-II cavity-enhanced SPDC, the type-I cavity-enhanced SPDC does not induce cluster effects.60–62 Thus, the photon pairs leave the cavity in multilongitudinal modes. Owing to the dispersion of the BBO crystal, not all the longitudinal modes can exist in the cavity. The mode number m in a cavity can be estimated using the following formula:29 

ml+n(ω0)d+nωω0d2πcdF|2nω+2nω2ω0|,
(1)

where c is the speed of light in vacuum, l is twice the length of the cavity excluding the crystal, F is the finesse of the cavity with the nonlinear crystal inside, d is twice the length of the BBO crystal, and n is the refractive index of the BBO crystal at the operating frequency ω0. Correspondingly, nω and 2nω2 can be calculated based on the Sellmeier equation for the BBO crystal. By substituting the experimental parameters l = 470 mm, d = 14 mm, the measured cavity finesse F = 2289, the operating wavelength λ0 = 606 nm, and the BBO crystal parameters44 into Eq. (1), one can obtain the mode number m = 367 in our setup.

Owing to the existence of multilongitudinal modes, the temporal second-order correlation function of the entangled photon pairs has a comb-like structure, which can be expressed as28,29

Gs,i(2)(τ)=ϕ|Êi(t)Ês(t+τ)Ês(t+τ)Êi(t)|ϕ|ε|2F2F02e2πΔν|τ|sin((2m+1)ΔΩFSRτ/2)sin(ΔΩFSRτ/2)2,
(2)

where subscripts s and i denote the quantities related to the signal and idler photons, respectively; |ϕ⟩ is the wave function of the photon pairs; Ês and Êi are the field operators related to the signal and idler photons, respectively; ϵ is the gain parameter of the BBO crystal in a single pass; and Δν is the bandwidth of the photon pairs. In addition, τ is the time difference between the signal and idler photons. Equation (2) shows that the second-order correlation function of the photon pairs has a comb-like structure but is enveloped by e−2πΔν|τ| with a spectral bandwidth Δν. Moreover, the period of the comb-like structure is the time required by the photons to complete a round-trip 1/ΔΩFSR. In the experiments, the single-photon detector has a timing jitter that corresponds to the time resolution ΔT of the photon counting system and contributes to the measured second-order correlation function in the form of convolution.63,64 For simplicity but without loss of generality, we assumed that the time jitter caused by the single-photon detector has the form of a Gaussian function. The modified second-order correlation function can be written as

Gs,i(2)(τ)e2πΔν|T|sin(2m+12ΔΩFSRT)sin(12ΔΩFSRT)2e2(Tτ)2ΔT2dT.
(3)

The timing jitter can be measured by directly recording the coincidence counts of the SPDC photon pairs without the cavity, that is, by moving away all mirrors of the OPO cavity. Our experiment yielded a timing jitter of 1.3 ns in our setup.

The temporal second-order correlation function of the photon pairs was measured using the HBT interferometer, as shown in Fig. 1. Figure 2 is the measured correlation function Gs,i(2)(τ) of the photon pairs. The red squares are the coincidence counts with a temporal bin size of 100 ns. Fitting the measured data with an envelope formula exp(−2πΔν|τ|), as shown by the black curve, we obtained the spectral bandwidth of the photon pairs Δν = 265 ± 15 kHz. This spectral bandwidth is the narrowest bandwidth for the photon pair sources at 606 nm based on cavity-enhanced SPDC reported until now. The full width at half maximum (FWHM) of the correlation time was measured to be 1300 ± 20 ns and was consistent with the spectral bandwidth Δν. For comparison, Table I lists the parameters of the photon pair sources based on the cavity-enhanced SPDC reported recently by several groups.30,31,33,34,37,43 Besides, significant developments have been achieved by using REIDSs with other methods to generate entangled photon pairs. Ferguson et al. generated entangled photon pairs with a linewidth of 65 kHz using the process of rephased amplified spontaneous emission in YSO:Pr crystals.25 Kutluer et al. generated entangled photon pairs with a linewidth of 45 kHz through spin-wave storage in YSO:Pr crystals.26 The ultra-narrow bandwidth and the long correlation time will be beneficial to photon storage in the YSO:Pr crystal due to the narrow linewidth of the transition 3H41D2 of Pr3+ ions.7–10 Compared to the nondegenerate photon pairs,33,43 the coincidence count profile of the degenerate photon pairs is symmetric. This means that the two photons have the same spectral bandwidth. It is worth mentioning that the comb-like structure seems irresolvable in the coincidence curve even with a time bin size of 658.4 ps [as shown in Fig. 3(a)] because the round-trip time of the OPO cavity is too close to the time jitter. However, by a fast Fourier transform of the data in Fig. 3(a), it is still possible to get the inherent periodical characteristics of the comb-like structure of the coincidence data, corresponding to the free spectral range ΔΩFSR of the OPO cavity. Figure 3(b) shows the fast Fourier transform of the correlation function in Fig. 3(a). One sees that a narrow peak in blue color appears at 606.5 MHz, which is in good agreement with the theoretically predicted value of 607.7 MHz.

FIG. 2.

The correlation function Gs,i(2)(τ) measured with the HBT interferometer with 20-min integration time and 300-μW pump power. The time bin size is 100 ns. The red squares are experimental results, and the black curve is an exponential fit.

FIG. 2.

The correlation function Gs,i(2)(τ) measured with the HBT interferometer with 20-min integration time and 300-μW pump power. The time bin size is 100 ns. The red squares are experimental results, and the black curve is an exponential fit.

Close modal
TABLE I.

Comparison of several typical photon pair sources based on cavity-enhanced SPDC reported recently.

λ (nm)Δν (MHz)CrystalCavity typeaSpectral brightnessbAtomic speciesReferences
780 9.6 PPKTP SW Rb 30  
795 0.429 PPKTP BT 4400 Rb (31) and(65
795 15 PPKTP SW 2.98 Rb 34 and 38  
780/852 6.6/5.234 PPKTP SW 1.06 × 105 Rb/Cs 33  
606/1436 2.9/1.7 PPLN BT 1500 Pr3+ 43  
852 10.9 PPKTP SW 436 Cs 37  
606 0.265 BBO SW 216 Pr3+ This work 
λ (nm)Δν (MHz)CrystalCavity typeaSpectral brightnessbAtomic speciesReferences
780 9.6 PPKTP SW Rb 30  
795 0.429 PPKTP BT 4400 Rb (31) and(65
795 15 PPKTP SW 2.98 Rb 34 and 38  
780/852 6.6/5.234 PPKTP SW 1.06 × 105 Rb/Cs 33  
606/1436 2.9/1.7 PPLN BT 1500 Pr3+ 43  
852 10.9 PPKTP SW 436 Cs 37  
606 0.265 BBO SW 216 Pr3+ This work 
a

SW denotes the standing-wave cavity. BT denotes the bow-tie cavity.

b

Spectral brightness in units of pairs/(s · mW · MHz).

FIG. 3.

(a) The correlation function Gs,i(2)(τ) measured with the HBT interferometer with 20-min integration time and 1.5-mW pump power. The time bin size is 658.4 ps. The black curve is an exponential envelope fit. (b) The fast Fourier transform of the correlation function Gs,i(2)(τ).

FIG. 3.

(a) The correlation function Gs,i(2)(τ) measured with the HBT interferometer with 20-min integration time and 1.5-mW pump power. The time bin size is 658.4 ps. The black curve is an exponential envelope fit. (b) The fast Fourier transform of the correlation function Gs,i(2)(τ).

Close modal

Figure 4 shows the measured dependencies of the correlation bunching peak gs,i(2)(0) (blue triangles) on the pump power Ppump for photon pairs output from the OPO cavity when the OPO cavity is far below the lasing threshold to ensure the quantum nature of the generated photon-pair source.29 Here, the dark counts are already corrected to get the data in Fig. 4. As expected, the correlation bunching peak gs,i(2)(0) decreases as a function of the pump power.43,66 The measured dependence of the coincidence count rate (red dots) on the pump power Ppump is shown in Fig. 5. The spectral brightness is proportional to the slope of the coincidence count rate (red line). From Fig. 5, we estimated the slope of the coincidence count rate with respect to the pump power to be 81 ± 2 Hz/mW. Therefore, the spectral brightness of the photon pairs we detected was 306 pairs/(s · mW · MHz). Note that this spectral brightness is detected with a single-photon detector without correction to loss. Considering corrections based on the quantum efficiency of the single-photon detector, the coupling efficiencies of the single-mode fiber (57%, 71%), the chopper duty cycle (25%), and the transmission losses (60%), the spectral brightness of the photon pairs output from the OPO cavity was estimated to be ∼7.93 × 104 pairs/(s · mW · MHz). By further taking the mode number into consideration, the averaged spectral brightness per longitudinal mode is then ∼216 ± 5 pairs/(s · mW · MHz). The spectral brightness can be further improved by employing a triply resonant cavity, where all three beams, the signal, the idler, and the pump, are resonant in the cavity. In this case, the spectral brightness is proportional to the product of three cavity finesses at the signal, the idler, and the pump wavelengths. Therefore, an enhancement by two or three orders of magnitude, depending on the cavity finesse at the pump wavelength, is expected for the spectral brightness.

FIG. 4.

Dependence of the correlation bunching peak gs,i(2)(0) on the pump power Ppump. A linear-logarithmic coordinate system is employed.

FIG. 4.

Dependence of the correlation bunching peak gs,i(2)(0) on the pump power Ppump. A linear-logarithmic coordinate system is employed.

Close modal
FIG. 5.

Dependence of the coincidence count rate on the pump power Ppump. The red line is a linear fit to the coincidence count rate, which indicates the slope of the coincidence count rate with respect to the pump power.

FIG. 5.

Dependence of the coincidence count rate on the pump power Ppump. The red line is a linear fit to the coincidence count rate, which indicates the slope of the coincidence count rate with respect to the pump power.

Close modal

Another important parameter for the cavity-enhanced SPDC is the cavity enhancement factor B that is defined as the ratio between the spectral brightness values with and without the cavity.28,67 Theoretically, it is proportional to F3/πF0 and is calculated to be 2.73 × 105. By removing the cavity output mirror, we measured the spectral brightness of the single-pass SPDC to be 1.91 × 10−3 pairs/(s · mW · MHz). Therefore, the actual cavity enhancement factor B was determined to be 1.13 × 105, which is in the same order as the theoretical prediction.

In conclusion, we achieved degenerate ultra-narrow-band entangled photon pairs at 606 nm using cavity-enhanced SPDC. The spectral bandwidth of the photon pairs was measured to be ∼265 ± 15 kHz, which is the narrowest one based on the cavity-enhanced SPDC reported until now. The spectral brightness of the photon-pair source was 216 ± 5 pairs/(s · mW · MHz), as averaged for each longitudinal mode. The developed ultra-narrow bandwidth entangled photon pair source will be useful for quantum photon storage in YSO:Pr crystals and will have the potential to entangle different atomic ensembles based on quantum memories. Although the generation rate of quantum photons is relatively low at the current stage due to the small nonlinear coefficient of the BBO crystal, it is expected to be enhanced significantly by using a triply resonant cavity with the pump laser also resonant in the cavity.

The data that support the findings of this study are available within the article.

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 91750204, 61475077, and 11774182), the 111 Project (No. B07013), and the Fundamental Research Funds for the Central Universities. The authors thank Professor Z. Y. Ou from the Indiana University–Purdue University in Indiannapolis and Professor Weiguang Ma from Shanxi University, China, for helpful discussions.

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