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Abstract
We used precision spectroscopy to analyze the R(53)24-1, P(49)24-1, and R(95)25-1 lines of molecular iodine (127I2) to establish optical frequency references for the laser cooling of Yb atoms using the 1S0 – 3P1 intercombination transition at 556 nm. A laser frequency instability of < 2 × 10−12 (for 0.01 s < τ < 3000 s, τ is the average time of the measurement) was attained using the observed Doppler-free hyperfine transitions of the iodine lines. The absolute frequencies of the observed 63 hyperfine transitions were determined with an uncertainty of 7 kHz (fractional uncertainty of 1.3 × 10−11). Highly accurate hyperfine constants were determined by fitting the measured hyperfine splittings to a four-term Hamiltonian that includes the electric quadrupole, spin-rotation, tensor spin-spin, and scalar spin-spin interactions with an uncertainty of approximately 1 kHz. The observed hyperfine transitions of molecular iodine provide new frequency references for research using atomic Yb, because these transitions are close to the intercombination transition of Yb at 556 nm.
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1. Introduction
Optical frequency metrology [1], which is concerned with the measurement of laser frequencies and the development of standards, is of great interest for a wide range of applications. For example, in practice, optical frequency metrology is the foundation of time and length standards that support precision measurements and broadband communication networks [2]. On the other hand, the measurement and control of laser frequencies are the key technology for research on cold atoms, which is applicable to optical clocks [3,4], quantum simulation [5,6], and quantum information processing [7,8]. In the case of alkaline earth and alkaline earth-like atoms such as Sr and Yb, laser cooling can be accomplished using the narrow 1S0 – 3P1 intercombination transition, which results in cold atoms at the µK level [9,10]. Because the linewidth of the intercombination transition is narrow, laser frequency control is highly important in these experiments.
For Yb, the 1S0 – 3P1 intercombination transition at 556 nm has a linewidth of 182 kHz [10]. The 556 nm light source is usually realized with the second-harmonic generation (SHG) of a 1112 nm Yb:fiber laser or external cavity diode laser (ECDL). The frequency of the 1112 nm laser can be stabilized using an ultralow-expansion (ULE) cavity [11] or Yb atomic beam [12,13]. For laser sources with a relatively large linewidth, both the ULE cavity (for pre-stabilization) and atomic beam are necessary. Laser frequency stabilization was also performed using an optical frequency comb for the 556 nm magneto-optical trap (MOT) [14]. The setup of a ULE cavity with an atomic beam or optical frequency comb can be bulky. Recently, a light source was prepared for the 556 nm MOT using frequency stabilization based on molecular iodine spectroscopy [15]. However, detailed information on the spectroscopic method and laser frequency measurement was not reported [15].
The absorption lines of molecular iodine are spread across the visible wavelength region from 500 nm to the near infrared. In particular, the iodine absorption lines in the green wavelength region are characterized by their high intensity and relatively narrow natural linewidth (a few hundred kHz). For example, iodine-stabilized Nd:YAG lasers at 532 nm [16–19] and iodine-stabilized lasers at 515 nm [20–22] demonstrated excellent laser frequency stability and reproducibility. The absorption of iodine lines weakens at larger wavelengths. The well-known iodine-stabilized He-Ne lasers at 633 nm contain an iodine cell inside the laser cavity to enhance the absorption signal. The iodine lines in the yellow-green wavelength region (for example, 556 nm, the wavelength of the Yb intercombination line) are weaker than those at 532 nm and 515 nm but are still sufficiently strong to be observed without an enhancement cavity. The Doppler-free spectroscopy and hyperfine structure of the iodine lines in the yellow-green wavelength region have been studied using frequency-stabilized lasers at 556 nm [15], 560 nm [23], 561 nm [24], 565 nm [25], 576 nm [25], and 578 nm [26,27]. However, the report of a frequency-stabilized laser at 556 nm [15] does not contain any information on the iodine line, such as the line assignment, absolute frequency, and hyperfine structure.
Studies with iodine-stabilized lasers have provided information on the hyperfine structure of iodine lines. A theoretical fit of the hyperfine structure yielded the hyperfine constants of molecular iodine [28], which contributes to molecular physics studies. Empirical formulae that describe the dependence of the hyperfine structure on the vibrational and rotational quantum numbers with an uncertainty at the kHz level were derived based on precision spectroscopy of molecular iodine mainly near 532 nm [29–32]. Across a wide range of spectra, from 514 to 820 nm, interpolation formulae of hyperfine constants were derived with an uncertainty of ≤ 30 kHz [33]. To date, approximately 25 iodine lines with a lower vibrational quantum number v$^{\prime\prime}$ = 1 have been investigated, mainly in the yellow-green wavelength region [24–27,33–36]. However, only four of their hyperfine structures have been fitted with deviations of 1-2 kHz [27]. The fitting deviations of the other lines were mostly at the level of tens or hundreds of kHz. According to the iodine atlas [37], the strong iodine lines at 556 nm have a lower vibrational quantum number, v$^{\prime\prime}$ = 1. The 556 nm iodine lines near the 1S0 – 3P1 intercombination transition of Yb should also be good candidates for molecular physics studies.
In this study, we demonstrate the Doppler-free spectroscopy of three relatively strong iodine lines, R(53)24-1, P(49)24-1, and R(95)25-1, at 556 nm, near the 1S0 – 3P1 intercombination transition of Yb [38,39] (see Fig. 1). This is the first work in which Doppler-free spectroscopy is applied to these iodine lines. The iodine line that formed the topic of a previous study [15] might have been the R(95)25-1 line although a clear assignment of this line was not reported. Frequency stabilization was demonstrated using the observed Doppler-free hyperfine transitions of the iodine lines. The frequency stability was evaluated, and the absolute frequency was measured using an optical frequency comb. The measured hyperfine structures were fitted to a four-term Hamiltonian with a standard deviation of approximately 1 kHz. The main hyperfine constant eQq was extracted from the fit with an uncertainty at the kHz level. Consequently, the measured hyperfine transitions of the three iodine lines provide useful frequency references for the laser cooling of Yb atoms using the intercombination transition at 556 nm. Furthermore, the hyperfine constants derived for the three iodine lines were used to improve the empirical formulae for the hyperfine structure of the molecular iodine.
Fig. 1. Frequency atlas of the 127I2 absorption lines near the 1S0 – 3P1 intercombination transition of Yb at 556 nm. The transition frequencies of six stable isotopes of Yb [38,39] are also indicated. The relative intensity of iodine lines was taken from Ref. [37]. The intensity of the Yb intercombination transition is not on the same scale with the iodine lines.
2. Experiment
2.1 Experimental setup
Figure 2 shows a schematic of the experimental setup for Doppler-free spectroscopy and the absolute frequency measurement of the iodine lines at 556 nm. ECDL with a wavelength of 1112 nm (MOGLabs: CEL-002) was used as the light source. The ECDL output was sent to a semiconductor optical amplifier (SOA) (Innolume: SOA-1130-20-YY-35dB) for power amplification. The output of the SOA was then injected into a periodically poled lithium niobate waveguide (PPLN-WG) for SHG at 556 nm. A maximum output power of 60 mW at 556 nm was obtained when the fundamental power from SOA was 200 mW. The light from the PPLN-WG was separated into fundamental and SHG laser beams using a dichroic mirror (DM). The SHG laser beam was sent to an iodine spectrometer for Doppler-free spectroscopy and laser frequency stabilization. In the present experiment, the optical power used for iodine spectroscopy was 5 mW. Therefore, a remaining power of 55 mW at 556 nm can be used for laser cooling. Doppler-free spectroscopy of molecular iodine is based on saturation spectroscopy using the modulation transfer technique [40,41]. The reflected SHG light was first split into pump and probe beams using a half-wave plate (HWP2) and a polarizing beam splitter. The pump light was frequency-shifted by 80 MHz using an acousto-optic modulator (AOM) and then phase-modulated at 350 kHz using an electro-optic modulator (EOM). Consequently, sidebands with a frequency spacing of 350 kHz are generated in the pump beam. The pump and probe beams were then overlapped in a 45-cm iodine cell. The sidebands of the pump light are transferred to the probe light via a four-wave mixing process when saturation absorption occurs [40,41]. The probe light was then reflected by a Glan-Thompson polarizer and detected using a photodetector (PD2). The detected signal was demodulated using a double-balanced mixer and signal from a local oscillator, of which the phase was offset from the signal applied to the EOM. The demodulated signal contained Doppler-free spectroscopic information of molecular iodine and was used as an error signal for servo control of the laser frequency via the piezoelectric transducer of the ECDL. The fundamental laser beam passing through the DM was combined with the light from an optical frequency comb through an optical fiber coupler for the beat frequency measurement. We used two frequency combs for laser frequency measurement:1) an optical frequency standard (OFS)-based comb for main usage and 2) a global positioning system (GPS)-based comb for the verification of the comb mode numbers and the measured frequency values. The OFS-based comb was an erbium-doped fiber comb operating at a repetition rate of 50 MHz, using an iodine-stabilized Nd:YAG laser as a frequency Ref. [42]. The GPS-based comb was also an erbium-doped fiber comb but operated at a repetition rate of 107 MHz using a GPS disciplined oscillator (GPS-DO) (Freqtime: FT-001S) as a frequency reference. The beat signal observed by the photodetector (PD1) was measured using a dead-time free frequency counter (Pendulum: CNT-91). The time base of the frequency counter was connected to the GPS-DO.
Fig. 2. Diagram of the experimental setup. ECDL: external cavity diode laser, HWP: half-wave plate, SOA: semiconductor optical amplifier, PPLN-WG: periodically poled lithium niobate waveguide, TEC: thermoelectric cooler, DM: dichroic mirror, PBS: polarizing beam splitter, AOM: acousto-optic modulator, EOM: electro-optic modulator, GTP: Glan-Thompson polarizer, PD: photodetector, DBM: double-balanced mixer, LO: local oscillator, LPF: low-pass filter, GPS-DO: global positioning system-disciplined oscillator, OFS: optical frequency standard, Nd:YAG /I2 : iodine-stabilized Nd:YAG laser.
2.2 Saturation spectroscopy of molecular iodine
Figures 3(a)–3(c) show the saturation spectroscopic signals of the R(53)24-1, P(49)24-1, and R(95)25-1 lines observed using the modulation transfer technique [40,41]. Each signal in the figures has a dispersion-like shape (although the shape is quite narrow and may not be clearly discernible in the figure) and represents one hyperfine transition in the absorption line. The scanning of the laser frequency was accomplished by tuning the temperature of the laser continuously. The cold-finger temperature of the iodine cell was set to −0.2 °C, corresponding to the iodine pressure of 4.0 Pa. The optical powers of the pump and probe beams were 2.0 mW and 0.3 mW, respectively. When the rotational quantum number of the ground state J” is odd (53, 49, and 95 for the observed lines), the rovibrational energy level is split into 21 sublevels, resulting in 21 hyperfine transitions (a1 to a21). In the three lines that were observed, all 21 hyperfine transitions were resolved from one another. In the P(49)24-1 line, a weak signal was detected between the a18 and a19 hyperfine transitions and was assigned to be the first hyperfine transition (b1) of the R(158)25-0 line. More hyperfine transitions of the R(158)25-0 line were observed beyond the frequency range in Fig. 3(b). The signal-to-noise ratio (S/N) for the a1 transition of the P(49)24-1 line was 676, which was recorded using a low-pass filter with a bandwidth of 30 Hz.
Fig. 3. Observed hyperfine structures of the R(53)24-1, P(49)24-1, and R(95)25-1 lines of 127I2 near 556 nm. The low-pass filter for recording had a bandwidth of 30 Hz. Vertical scales are the same for all graphs.
2.3 Frequency stabilization
The laser frequency was stabilized and evaluated using the a1 hyperfine transition of the P(49)24-1 line. The red solid curve with solid circles in Fig. 4 shows the Allan standard deviation calculated from the measured beat frequency between the frequency-stabilized laser and OFS-based comb. The Allan standard deviation was calculated using two series of data with counter gate times of 0.01 and 1 s, respectively. The Allan standard deviation was 2.0 × 10−12 at an average of 0.01 s, decreasing to 1.4 × 10−13 after an average of 15 s, increasing to 5.0 × 10−13 at 750 s, and decreasing again for τ > 750 s. For comparison, the frequency instability of the iodine-stabilized Nd:YAG laser (the frequency reference of the OFS-based comb) is shown in Fig. 4 as a black dashed line. As the iodine-stabilized Nd:YAG laser is more stable than the developed iodine-stabilized laser, the observed Allan standard deviation (red solid curve with solid circles) indicates the instability of the developed laser. The degradation of the laser stability after τ = 15 s is attributed to power fluctuations in the SOA. The solid blue curve with solid circles in Fig. 4 shows the free-running frequency instability of the developed laser. The frequency stability of the locked laser is improved such that it is more than 60 times higher than that of the free-running laser at 0.01 s and more than three orders of magnitude higher at 1 s. We also performed laser frequency measurements using a GPS-based frequency comb. The gray solid curve with solid circles in Fig. 4 shows the Allan standard deviation calculated from the measured beat frequency between the frequency-stabilized laser and GPS-based comb. In this case, the observed Allan standard deviation indicates the instability of the GPS comb. The GPS-based comb cannot be used to evaluate the frequency stability of the developed laser. However, the absolute frequency of the developed laser can be measured with an uncertainty of < 1 × 10−12 (< 0.5 kHz at 556 nm) at an average time longer than 4400 s using the GPS-based comb. The laser frequency measured using the GPS-based comb can be used to confirm the measurement results using the OFS-based comb within the measurement uncertainties.
Fig. 4. Allan standard deviations calculated from the beat frequency of the iodine-stabilized 1112 nm ECDL and the optical frequency comb (red solid curve with solid circles). The instabilities of the OFS-based comb (black dashed line) and the GPS-based comb (gray solid curve with solid circles) are shown for comparison. The blue solid curve with solid circles is the measured free-running frequency instability of the 1112 nm ECDL.
2.4 Uncertainty evaluation and frequency measurement
Figure 5 shows the results of the frequency measurement of the laser locked to the a1 hyperfine transition of the P(49)24-1 line. Eight frequency measurements were performed over the course of a week. Each measurement was calculated from 600 beat frequency data points measured with a counter gate time of 1 s. The uncertainty bar in this figure is given by the Allan standard deviation at the longest averaging time. The averaged frequency of the eight measurements was 539 385 032 239.1 kHz. The standard deviation of the eight measurements was 1.4 kHz, indicating the repeatability of the developed laser.
Fig. 5. Measured absolute frequency of the 1112 nm ECDL locked on the a1 component of the P(49)24-1 line. The blue line represents the average frequency of 8 measurements.
To evaluate the systematic uncertainty of the measured frequency, we investigated the effects of the frequency shift of iodine-stabilized laser (Table 1). Figure 6(a) shows the pressure shift of the laser stabilized to the a1 hyperfine transition of the P(49)24-1 line. The slope of the measured pressure shift is −1.8 kHz/Pa. The uncertainty of the iodine pressure in the cell was determined by the uncertainty of the temperature of the solid-state iodine crystal in the cold finger, which was estimated to be less than 0.5 K. This corresponds to an uncertainty of < 0.2 Pa of the pressure and results in a frequency uncertainty of < 0.4 kHz. Figure 6(b) shows the frequency shift of the laser due to the optical power of the pump beam. The slope of the measured power shift was −2.6 kHz/mW. As the uncertainty in the determination of the pump power was less than 10%, the frequency uncertainty was estimated to be less than 0.5 kHz. Figure 6(c) shows the frequency shift owing to the phase between the modulation and demodulation signals. The phase 0° in Fig. 6(c) is defined as the phase where the modulation transfer signal is at its maximum. The measured slope of the frequency shift due to the phase adjustment was 150 Hz/degree. Because the uncertainty of the phase adjustment was set to < 10°, the corresponding frequency uncertainty was < 1.5 kHz. When the laser is locked, the offset voltage of the servo system must be adjusted to coincide with the baseline of the modulation transfer signal. Figure 6(d) shows the frequency shift owing to the servo electronic offset. The measured slope of this effect was −19.7 kHz/mV. In this experiment, the offset voltage was carefully adjusted to an uncertainty level of < 50 µV, resulting in a frequency uncertainty of < 1.0 kHz. The frequency shift owing to the misalignment of the pump and probe beams was also investigated. With multiple optimizations of the alignment to maximize the modulation transfer signal, the frequency uncertainty due to misalignment was estimated to be 3.7 kHz. Contamination of iodine cells causes a frequency shift. The uncertainty of the cell impurity (5 kHz) is given by the CIPM recommended frequency of 474 THz-I2 (wavelength = 633 nm) [43]. The uncertainty of the frequency reference used in the OFS-based comb was the repeatability (1.5 kHz) of the iodine-stabilized Nd:YAG laser. The standard deviation (1.4 kHz) calculated from the repeatability measurement was used as the statistical uncertainty. By combining these uncertainties, the total uncertainty of the absolute frequency measurement in this study was estimated to be 7 kHz (relatively 1.3 × 10−11).
Fig. 6. Frequency shift of the laser stabilized to the a1 component of the P(49)24-1 transition. (a) Pressure shift, (b) power shift, (c) demodulation phase shift, and (d) input offset voltage shift.
Table 1. Frequency shift
The measured absolute frequencies of the a1 hyperfine transitions of the R(53)24-1, P(49)24-1, and R(95)25-1 lines at 556 nm are presented in Table 2. The frequency of the a1 hyperfine transition of the P(49)24-1 line was derived from the measurement, as shown in Fig. 5. The frequencies of the a1 hyperfine transition of the R(53)24-1 and R(95)25-1 lines were measured as the average of the beat frequency data for 600 s, where each beat frequency value was measured by setting the gate time of the frequency counter to 1 s. The frequencies were directly measured under the experimental conditions described in Section 2.2. The measured frequencies were confirmed by using a GPS-based frequency comb. The frequency of the 1S0 – 3P1 intercombination transition of 174Yb [38,39] is also listed in Table 2 for comparison.
Table 2. Measured absolute frequency of the transitions
2.4 Measurement of hyperfine structures
To measure the hyperfine splittings, the iodine-stabilized laser was locked in succession to all 21 hyperfine transitions of the R(53)24-1, P(49)24-1, and R(95)25-1 lines. The frequency of each hyperfine transition was measured for 600 s by setting the gate time of the frequency counter to 1 s. The values in the columns with the header “Obs.” In Table 3 are the measured hyperfine splittings of the three absorption lines obtained by taking the frequency difference between each hyperfine transition and a1 transition. Because the frequency shift is similar for each hyperfine transition, the measurement uncertainty of the interval of the hyperfine splittings is not affected by systematic uncertainties, such as cell impurity, optical power, and iodine pressure shifts. The measurement uncertainty of the hyperfine splittings is limited by the repeatability of the developed iodine-stabilized laser.
Table 3. Observed and calculated hyperfine splittings of the three transitionsa
3. Calculation of hyperfine splittings and coupling constants
The measured hyperfine splittings were used to determine the hyperfine constants of the iodine lines. The hyperfine interactions of molecular iodine can be described by the following four-term Hamiltonian, HHFS [20]:
The theoretical fit enables us to only obtain the differences in the hyperfine constants between the upper and lower levels [20].
where eQq’, C’, d’, δ’, and eQq”, C”, d”, δ” are the upper- and lower-state hyperfine constants, respectively. Conversely, the hyperfine splitting of the iodine lines can be precisely reproduced using the hyperfine constants ΔeQq, ΔC, Δd, and Δδ. Table 4 lists the fitted hyperfine constants for the three observed lines. The uncertainties of the main hyperfine constant ΔeQq for the lines were at the 2 kHz level, which are comparable to those obtained previously [27]. Highly accurate ΔC, Δd, and Δδ values for the three observed lines were also obtained from the theoretical fit.Table 4. Fitted hyperfine constantsa
We also listed the hyperfine constants of other iodine lines with v$^{\prime\prime}$ = 1 [24,25,33–35]. The uncertainty of the main hyperfine constant ΔeQq for these lines is mostly at the tens or hundreds of kilohertz level.
4. Discussion and conclusion
The instability of the developed iodine-stabilized laser was < 2.0 × 10−12 (corresponding to 1.1 kHz at 556 nm) when the average time was longer than 0.01 s. This is more than two orders of magnitude smaller than the linewidth of the 1S0 – 3P1 Yb intercombination transition. We also measured the instantaneous linewidth of the iodine-stabilized laser using a spectrum analyzer. Figure 7 shows the observed beat signal between the fundamental beam of the frequency-stabilized laser and OFS-based comb. The resolution bandwidth and sweep time of the spectrum analyzer were 10 kHz and 120 ms, respectively. The observed linewidth of approximately 30 kHz (corresponding to 60 kHz at 556 nm) was mainly attributed to the linewidth of the developed iodine-stabilized laser, because the linewidth of the OFS-based comb is at the kilohertz level. Therefore, the linewidth of the developed iodine-stabilized laser was approximately threefold narrower compared to that of the 1S0 – 3P1 intercombination transition of Yb.
Fig. 7. Beat signal between the fundamental beam of the frequency-stabilized laser and the OFS-based comb, at a resolution of 10 kHz. The observed linewidth was approximately 30 kHz (corresponding to 60 kHz at 556 nm).
The observed 63 hyperfine transitions of the three iodine lines examined in the present study can be used as frequency references for research on Yb atoms. For example, the frequency of the a1 hyperfine transition of the P(49)24-1 line is approximately 560 MHz higher than that of the 1S0 – 3P1 intercombination transition of the 171Yb atoms used in optical lattice clock experiments [45]. In addition, the frequency of the a21 hyperfine transition of the P(49)24-1 line is approximately 660 MHz lower than that of the 1S0 – 3P1 intercombination transition of 174Yb atoms, of which the natural abundance is the highest (32%) among all isotopes. Frequency separations of 560 MHz and 660 MHz can be easily bridged using acousto-optic modulators. Because the absolute frequency of the stabilized laser is known to have an uncertainty of 7 kHz (a factor of 26 smaller than the linewidth of the Yb transition), it is possible to set the laser frequency to the desired value for cold atom experiments.
The highly accurate hyperfine constants obtained in the present study can be used to improve our knowledge of molecular iodine. In our previous study [27], we derived a formula to describe the vibrational dependence of the hyperfine constant of the tensor spin-spin interaction Δd:
for iodine lines with a lower vibrational quantum number v$^{\prime\prime}$ = 1. Here, v’ is the vibrational quantum number of the upper state. No clear vibrational dependence was observed for the hyperfine constant of the scalar spin-spin interaction, Δδ [27]. We note that rotational dependences are much smaller than vibrational dependence. Therefore, for Δd and Δδ, we used the average values of the R(53)24-1 and P(49)24-1 lines. In Fig. 8(a), Δd is plotted as a function of v’ using the data obtained in the present study and Ref. [27]. Δd can be fitted using the following formula:The fitting results are shown in Fig. 8(a) as a solid blue line. The uncertainties of the coefficients in Eq. (7) were reduced by an order of magnitude by adding the data generated in this study to the fit.
Figure 8(b) shows Δδ as a function of v’ using the data produced in the present study and Ref. [27]. Δδ can be fitted using the following formula:
The fitting results are shown in Fig. 8(b) as a solid blue line. For the first time, the dependence of Δδ on the vibrational quantum number v’ was clearly observed for iodine lines with a lower vibrational quantum number v$^{\prime\prime}$ = 1.
Optical frequency references based on the hyperfine transitions of molecular iodine at 556 nm can also be used for length measurements [2]. For example, gauge block measurements require several light sources emitting at different wavelengths [46]. Furthermore, a more compact iodine-stabilized laser system operating at 556 nm would be useful in various applications. We previously developed a compact iodine-stabilized laser at 531 nm using a coin-sized laser module [47]. All optical parts of the laser system were arranged on a 20 cm × 30 cm breadboard, which was much smaller than that in the present experimental setup. The same laser module is already available at 561 nm [48], and a similar coin-sized module at 556 nm could, in principle, be constructed. This module would be suitable for a wide range of applications.
In conclusion, we studied the hyperfine structure of three absorption lines of molecular iodine, the R(53)24-1, P(49)24-1, and R(95)25-1 lines near the 556 nm transition of Yb. All the hyperfine transitions of these three lines were observed by precise spectroscopy using the modulation transfer technique. The absolute frequency of each transition was measured using an optical frequency comb. The hyperfine constants of each iodine line were calculated by fitting the obtained hyperfine structure to a four-term Hamiltonian equation. The 63 hyperfine transitions of the three iodine lines studied in this work are close to the 1S0 – 3P1 intercombination transition and can be used as frequency references for research on Yb atoms.
Funding
Japan Society for the Promotion of Science (KAKENHI 18H03886).
Acknowledgments
The authors thank T. Kobayashi for helpful discussions on the calculation of hyperfine splitting and coupling constants.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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