Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14574
Laser-cooled 171 Yb+ microwave frequency
standard with √a short-term frequency instability
of 8.5 × 10−13 / τ
N. C. X IN , 1 H. R. Q IN , 2 S. N. M IAO, 1 Y. T. C HEN , 1 Y.
Z HENG , 2 J. Z. H AN , 1 J. W. Z HANG , 1,* AND L. J. WANG 1,2
1 State
Key Laboratory of Precision Measurement Technology and Instruments, Key Laboratory of Photon
Measurement and Control Technology of Ministry of Education, Department of Precision Instrument,
Tsinghua University, Beijing 100084, China
2 Department of Physics, Tsinghua University, Beijing 100084, China
* zhangjw@tsinghua.edu.cn
Abstract: We report on the development of a microwave frequency standard based on a lasercooled 171 Yb + ion trap system. The electronics , lasers, and magnetic shields are integrated
into a single physical package. With over 105 ions are stably trapped, the system offers a high
signal-to-noise ratio Ramsey line-shape. In comparison with previous work,
√ the frequency
instability of a 171 Yb + microwave clock was further improved to 8.5 × 10−13 / τ for averaging
times between 10 and 1000 s. Essential systematic shifts and uncertainties are also estimated.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1.
Introduction
In recent years, numerous atomic clocks have been developed that offer precise and stable time
frequency signals. Such signals are crucial for both basic physics [1,2] and practical applications
[3]. In particular, with the excellent performance, optical clocks are showing extremely strong
promise in the quest for high precision and high stability [4–7]. For instance, optical clocks
based on Al+ exhibit
a systematic uncertainty [8] of 9.4 × 10−19 , and clocks with a stability
√
of 4.8 × 10−17 / τ [6] are also established. In targeting the microwave √
spectral region, Hg+
microwave clock [9] demonstrate a frequency instability of 1.5 × 10−13 / τ[10]. Hg+ [10,11],
Cd+ [12–14], Yb+ [15,16], and Ba+ [17] trap systems have been investigated meticulously during
the past few decades. The investigations show that while ensuring a decent performance, an atomic
microwave frequency standard based on trapped ions has great potential in high transportability
because ions in the ion trap are well isolated from their external environment and the structure of
the system is relative simple [18]. For these reasons, microwave ion clocks are being touted as
promising candidates for the next generation of practical atomic clocks.
Much progress on microwave clocks based on ytterbium ions (171 Yb√+ ) has been achieved. For
the buffer gas cooled Yb+ microwave clock, an instability of 10−11 / τ has been demonstrated
[19], which was limited by collision and pressure related shifts
√ . The instability of the laser-cooled
Yb+ microwave clock was measured to be 2.09 × 10−12 / τ and was limited by second-order
Zeeman effects [20]. In absolute terms, the frequency instability potential of a laser-cooled Yb+
microwave clock is the same as for the microwave clocks based on Hg+ and Cd+ [21–24]. Lately,
researchers at the National Physical Laboratory (NPL) have built a prototype of a laser-cooled
171 Yb + microwave clock that is both compact and portable. The entire system fits into a 6U
19-inch rack unit (51×49×28) cm3 , verifying the feasibility of such miniaturized Ytterbium-based
clocks [25].
In this study, we report the design and prototyping of a laser-cooled 171 Yb + microwave
clock, based on the ground-state hyperfine transition of 171 Yb + at 12.6 GHz. The clock system
is well integrated into a single physical package and operates successfully in the laboratory
#453423
Journal © 2022
https://doi.org/10.1364/OE.453423
Received 10 Jan 2022; revised 18 Mar 2022; accepted 20 Mar 2022; published 19 Apr 2022
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14575
environment. More than 105 ytterbium ions are stably trapped in the system enabling it to offer a
high signal-to-noise ratio (SNR) Ramsey signal of the ground-state transition.
√ The short-term
stability of the 171 Yb + microwave clock has been improved to 8.5 × 10−13 / τ, a current record
level, and its performance in practice demonstrates that the laser-cooled 171 Yb + clock has
promising potential that is as good as other types of ion trap systems. Our research is helpful
for developing a compact microwave clock device that can used in ground-based time-keeping,
navigation station networking, and terrestrial network synchronizing.
2.
System overview
The specifics of the operation of an microwave clock are determined by the electronic energy-level
structure of the trapped ions. The simple energy-level structure of single 171 Yb + is shown in
Fig. 1. To cool the ions and probe their states, a 369 nm laser is used to cycle between the
transition states labeled 2 S1/2 (F = 1) and 2 P1/2 (F = 0). The neutral Yb atoms are ionized using
a combination of two laser beams of wavelengths 369 nm and 399 nm. An extra 935 nm laser
is used to drive ions out of the long-lived state 2 D3/2 (F = 0)[26]. Moreover, to drive ions out
of the state 2 S1/2 (F = 1) during cooling, microwave radiation of 12.6 GHz is also applied. The
ground state hyperfine splitting, (2 S1/2 (F = 1) → 2 S1/2 (F = 0)) of 12.6 GHz, is used as the
clock transition.
Fig. 1. Schematic energy levels of single 171 Yb + ion (not to scale).
In regard to the clock system (Fig. 2), an 171 Yb + cloud is trapped in a well-designed linear
Paul trap. Details of the ion trap are given elsewhere [27], and hence only a brief description is
presented here. The trap is made up of four trisected cylindrical electrodes made of copper. The
radius of each electrode is re = 7.1 mm, and the shortest distance between the electrodes and
the ion trap center is ro = 6.2 mm. The ratio of re/ ro is optimized and set at 1.15, to reduce RF
heating effects and to increase the number of trapped ions [28]. The lengths of the electrode
sections are 20, 40, and 20 mm. Direct current (DC) voltage is applied across adjacent sections,
whereas a radio frequency (RF) voltage, with frequency fixed at 1.144 MHz, is assigned to the
central section. Usually, the amplitude of the RF voltage is set below 200 V, implying that the
trapping q parameter is smaller than 0.11. A DC voltage of 4 V is adopted to provide fine axial
confinement of the ion cloud. Besides the ion pump, a titanium sublimation pump is added to
enhance the vacuum degree because 171 Yb + is found to bond rapidly with some molecules such
as O2 , H2 , and H2 O[29,30] [see Fig. 2]. A background pressure below 10−9 Pa is maintained
during clock operations. For the Helmholtz coil system, three orthogonal pairs of coils are
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14576
arranged compactly around the vacuum chamber and provide a steerable magnetic field that
compensates the residual magnetic field and performs the Zeeman sublevel splitting. An extra
700 µT external magnetic field, generated by a pair of coils, is also placed in the north-south
direction to induce a precession in the dipole moments of the ions and to destabilize the dark states
[31]. To minimize the second-order Zeeman (SOZS) shift, a magnetic shield, composed of three
layers Permalloy, wraps the vacuum chamber and coils. The shielding factors in the north-south,
east-west, and up-down directions are measured to be 1212, 2439, and 235, respectively.
Fig. 2. Detailed sketch of the 171 Yb + ion microwave clock system. System overview:
Upper board : lasers of 369 nm and 935 nm for cooling 171 Yb + ; middle layer: vacuum
chamber, linear Paul trap, magnetic shield, magnetic coils, microwave horn, ion pump , and
titanium sublimation pump ; bottom board: lasers of 369 nm and 399 nm for ionizing 171 Yb.
In regard to the laser system setup [Fig. 3], all wavelengths of the various laser beams,
corresponding to the different state transitions, were found by using saturated absorption
spectroscopy that employed a Yb hollow cathode lamp. For accuracy, a high-resolution
wavemeter (HighFinesse WS8-2) together with a proportional–integral–derivative controller
was used to measure and stabilize the lasers. The 399 nm and 935 nm lasers are respectively
extended-cavity-diode and distributed-feedback (DFB) lasers. Because the 369 nm laser beam
plays a pivotal role in both cooling and probing, to enhance the aseismic capacity and large
mode-hop-free tuning range, this beam is generated by a frequency-tripled laser—the 1108 nm
DFB fiber laser—and two periodically poled crystals. Reviewing the laser system, the 369 nm
laser beam is separated into two beams by a polarization beam splitter. One of the beams is
combined by a dichromatic mirror with the 935 nm laser beam and passed from above into
the vacuum chamber. The other beam is combined with the 399 nm laser beam and directed
in the opposite direction to the 935 nm laser beam. Because the extra strong magnetic field is
recommended to be at a certain oblique angle relative to the polarization of the laser beams, all
beams are linearly polarized by half-wave plates. Our lasers can provide power outputs up to 70
mW, 100 mW, and 80 mW for the 369 nm, 399 nm, and 935 nm beams, respectively, propagating
in free space. The spatial configuration of the ion cloud is captured by an electron-multiplying
charge-coupled device (EMCCD). A photomultiplier tube (PMT) is used to record the number of
photons emitted by the ions. The microwave radiation is generated by a commercial microwave
synthesizer (Agilent E8257D), referenced by the 10 MHz signal obtained from an oven-controlled
crystal oscillator (OCXO).
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14577
Fig. 3. Detailed information of laser system. H: half-wave plate; P: polarization beam
splitter; D: dichromatic mirror; S: optical shutter; B: beam splitter; C: fiber-optic couplers; R:
Reflection mirror; The cabinet with photomultiplier tube, microwave synthesizer, electronmultiplying charge-coupled device (EMCCD) , and hollow cathode lamp (HCL) are not
shown.
3.
Experimental result
To load the ions, the 369 nm laser is detuned by approximately 50 MHz to the low-frequency side
of the corresponding transition. Because the natural Yb solid metal used is a mixture of various
stable isotopes, the 399 nm laser is tuned to the peak-frequency of the 1 S0 → 1 P1 transition
of 171 Yb atom to achieve isotope selection. Confined by both the RF and DC voltages, a long
narrow 171 Yb + cloud is obtained and captured by the CCD camera (Fig. 4).
Fig. 4. EMCCD image of a trapped 171 Yb + ion cloud. The semi-major and semi-minor
axes of the ellipsoid are L1 = 6.07 mm mm and D21 = 0.48 mm. The amplitude of the RF
voltage is approximately 200 V; the DC voltage is set to 4 V. The image has been rotated 90◦
for convenience; the long axis of the ion cloud is actually oriented vertically .
To estimate the number of ions of a cloud, the number densities of ions are estimated by using
low-temperature ion-density theory, from which we have [32]:
n=
2
ε0 VRF
MΩ2 ro4
.
(1)
where ε0 denotes the vacuum permittivity , M is the mass of 171 Yb + . VRF = 200 V, Ω = 2π ×
1.144 MHz , and ro = 6.2 mm. The number density of 171 Yb + ions is estimated to be
approximately nYb+ = 1.634 × 1013 m−3 , the number of trapped ions being NYb+ = 1.1(0.3) × 105 .
To obtain an unambiguous Ramsey signal of the microwave clock transition at 12.6 GHz,
all lasers, microwave radiation, and magnetic fields of the coils must be precisely controlled.
A field-programmable gate-array circuit board is used to generate control signals to within
millisecond order. During measurements, all laser frequencies are first stabilized to within 2
MHz by locking to the wavelength-meter. The ions are then trapped and cooled using the 369
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14578
nm and 935 nm laser beams directed downward into the system. A strong external magnetic
field and high-power microwave radiation is applied to suppress the dark states. To pump the
ions into the ( S1/2 (F = 0)) state via the decay from the off-resonance excited ( 2 P1/2 (F = 1)),
the high-power microwave radiation is turned off. During the interaction with the π/2 pulse, only
the steerable magnetic field is applied to compensate the residual magnetic field and splitting of
the Zeeman sublevels. Finally, the upward-directed low-power 369 nm laser beam is used to
detect the number of ions in the ( 2 S1/2 (F = 1)) states. An interval of approximately 20 ms is set
to wait for closure and establishment of the strong magnetic field, the control of which is based
on a transistor. From the Ramsey fringes (Fig. 5) recorded by the PMT, peak-to-peak values are
calculated along with their amplitudes at half-waist, from which estimates of the SNR, typically
35, for the clock resonance are obtained. The high SNR of the Ramsey signal ensures a decent
stability performance for the designed 171 Yb + microwave frequency standard.
Fig. 5. Typical Ramsey lineshape of the clock transition (12.6 GHz) with a microwave pulse
time of 60 ms, free time of 500 ms, microwave power of -29.8 dBm, and a fluorescence
signal integration time of 150 ms. Each data point is the average of five results. The fitted
curve has a central frequency of 12642812121.47 Hz
3.1.
short-term frequency instability
The short-term frequency instability limitation can be estimated after acquiring the Ramsey
signal. Here, we have mainly considered quantum projection noise, pump noise, and shot noise,
which are typical limitations to the frequency stability of a passive frequency standard [33].
To estimate the quantum projection noise σproj , pump noise σpump and shot noise σshot , the
maximum photon counts of the Ramsey signal Smax , the minimum photon counts of the Ramsey
signal Smin , the photon counts arising from background scattered light Sback , and the number
of ions n need to be confirmed first. Typically, for our system, Smax = 10500, Smin = 2500,
Sback = 1500, n = 100000. Then the noise is calculated from [33]:
√ K
σproj = ηn ,
2
√︁
K
σpump = nη(1 − η) ,
2
√︃
Smax + Smin
σshot =
,
2
(2)
(3)
(4)
Research Article
where K =
(Smax −Sback )
,η
n
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14579
=1−
Smin −Sback
Smax −Sback .
σcal =
√︂
Therefore, the overall calculated noise is:
σproj 2 + σpump 2 + σshot 2 .
(5)
With σcal = 81, the corresponding SNR is around 48.8 , which indicates that the frequency
instability limitation is around 5.9 × 10−13 at 1 s.
We conducted a preliminary closed-loop run to verify the potential of the system. A detailed
description of the measurement setup has been fully discussed [34]. We set the free time at 1 s
and the cooling time at 1.2 s for measurements during the closed-loop run to reduce the linewidth
and increase the theoretical instability limitation. After comparing our 10 MHz output signal with
an active hydrogen maser the Allan deviations of the frequency standard are presented (Fig. 6).
Moreover, the frequency stability of our free-running OCXO, together with the performance
of recent microwave clocks of other research groups, are also presented√for direct comparison
[20,25]. The measured frequency stability of our clock is 8.5 × 10−13 / τ for averaging times
between 10 and 1000 s.
Fig. 6. Allan deviations of the 171 Yb + frequency standard and the free-running OCXO.
Other scholars’ primary and significant work are also plotted, including results from National
Institute of Information and Communications (NICT) and National Physical Laboratory
(NPL) [20,25]. The system was run continuously for 13000s. The frequency reference
during the measurements is generated by a hydrogen maser.
As the number of trapped ions increases to around 105 , the SNR of our system is vastly
enhanced, which explains the reason for the improved performance of our system. However,
our system has not reached the performance limit that we expected from the calculation results.
Fluctuations in the laser frequency, magnetic field, and working temperature are suspected in
increasing noise and degrading system performance [35]. The slight bump in our measured
Allan deviation at around 500 s (Fig. 7) is considered to be related to fluctuations in room
temperature, the fluctuation period of which lasts thousands of seconds. To further enhance the
short-term instability of our system and provide long-term system accuracy and stability, active
feedback must be used to suppress these fluctuations, which is our objective for the next round of
measurements.
3.2.
System shifts and uncertainties
Furthermore, the expected shifts and uncertainties of the system are carefully evaluated . Systematic shifts including Second-order Zeeman shift (SOZS), Second-order Doppler shift(SODS),
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14580
Fig. 7. Fluorescence spectrum of laser cooled Yb+ ions. The scanning range of the laser
frequency is about 1.2 GHz.
Blackbody radiation shift, Quadratic Stark shift, Gravitational redshift, Pressure shift, and Light
shift are listed and calculated, which are mainly shifts affecting the clock performance.
3.2.1.
Second-order Doppler shift
Movement of ions trapped in the trap are quite complicated, which results in the Doppler effect
[36,37]. Since the size of the ion cloud is less than half the wavelength of the 12.6 GHz
microwave signal, the first-order Doppler effect can be approximately ignored [38]. Usually the
total second-order Doppler shift can be calculated as [12]:
2
δfSODS
3kB T
[1 + (NdK )],
=−
2
f0
3
2Mc
(6)
where f0 denotes the ground state hyperfine splitting of 171 Yb + , kB is the Boltzmann constant,
T is the temperature of ion clouds, M is the mass of 171 Yb + , and NdK is usually set as 3 for
microwave clock system [34].
Since the Doppler broadening line-shape of the laser cooling transition can be fitted by a
Voigt curve, from which the ion temperature can be calculated [39]. We scan the 369nm laser
and record the photons signal by the PMT (see Fig. 7), the laser power is maintained below 20
µW/mm2 , to avoid heating and cooling effect during the measurement.
Then the ion temperature is fitted as 14.3(1.2) K, so the SODS is estimated to be −3.5(3)×10−14 .
3.2.2.
Second-order Zeeman shift
In microwave frequency standards, usually the operating magnetic field except the north-south
direction is weaken to near 0 by adjusting the current value of the magnetic field coil, and the
magnetic field in the north-south direction is placed within a few tens of µT. The relationship
between the magnetic field , B, and Second-order Zeeman shift, δfSOZS , can be described as [40]:
δfSOZS = B2 [T2 ] × 3.108 × 1010 [HzT−2 ],
(7)
and small fluctuations, δB, on a background magnetic field, B, will generate the small expansion
[40]:
δfSOZS
= 4.92[T−2 ] × B[T] × δB[T],
(8)
f0
In our experiment, the central frequency of the clock signal is fitted as 12642812121.47 Hz,
which is around 3 Hz deviation from the theoretical value. And the maximum fluctuation of
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14581
the magnetic field is around 180 nT, which is the result of measuring the geomagnetic field and
can be weaken to around 0.18 nT due to the decent magnetic shield. Thus the uncertainty of
Second-order Zeeman shift is estimated to below 9 × 10−15 .
3.2.3.
Blackbody radiation shift
In our experiment, except ions, all components will emit electromagnetic radiation( known as
blackbody radiation), which can be described as [41]:
(︃ )︃ ]︃ 2
(︃ )︃ 4 [︃
T
δfblack
T
1+ϵ
=β
,
(9)
f0
T0
T0
where T0 is the room temperature (300 K), the coefficient β is −9.83 × 10−16 [42] and ϵ is 0.002
[41].
It is reasonable to assume that the temperature variation of our laboratory is within 20K. Thus,
the blackbody radiation shift is estimated to be −1.0(1) × 10−14 .
3.2.4.
Other shifts
The quadratic Stark shift should be also considered for current (AC) electric field produced by
the RF electrodes. Usually, the quadratic Stark shift is expressed by [43]:
)︃ 2
(︃
δfquad
ωs mΩ2
= −4ks
kB T,
(10)
f0
ωm
q2
where Ω is the RF trap frequency, set as 1.144 MHz. The value of the coefficient ks has been
determined to be ks = 2 ± 1 × 10−21 m2 V−2 [44]. q is the elementary electric charge. The ratio
of secular frequencies of the single-ion and ion cloud is set as 1 ± 0.1. The final quadratic Stark
shift is estimated to be −6(3) × 10−16 .
Due to the high degree of vacuum, below 10−9 Pa, the pressure frequency shift is negligible
[13]. The light shift is also supposed to be ignored because all lasers are blocked by optical
shutters during the interrogation process.
A commercial geodetic global positioning system (GPS) is used to determine the altitude of
our lock. With the altitude of 43 ± 1 m, the fractional frequency shift caused by the gravitational
redshift is estimated to be 4.7(1) × 10−15 .
In Table 1, all meticulously estimated systematic uncertainties during the experiment are
listed. It is clear that among all shifts, SODS and SOZS are the dominant errors, though ions are
kept cooling and magnetic field variation are minimized by the decent magnetic shield. Other
shifts are calculated and found negligible. Then better magnetic shield and sympathetic cooling
technique are considered to be adopted [45] to enhance the performance of Yb+ microwave clock.
Table 1. Estimated Systematic Uncertainties
Shift type
Uncertainty (10−15 )
Second-order Doppler shift(SODS)
3
Second-order Zeeman shift (SOZS)
<9
Black body radiation shift
1
Quadratic Stark shift
0.3
Gravitational redshift
0.1
Pressure shift
0
Light shift
0
Total
<9.5
Research Article
4.
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14582
Sympathetic cooling
To further decrease the uncertainty of the SODS frequency shift and the stability limitation arising
from the Dick effect [46] in ion trap frequency standards, the sympathetic cooling technique has
attracted extensive attention of researchers [47,48].
In the sympathetic cooling system, the coolant ions are continuously cooled by lasers, while
the target ion can also be cooled to ultra-low temperature by the Coulomb interactions between
ions. A group in the Tsinghua University has demonstrated large scale ion crystal with the
24 Mg + − 113 Cd + and 40 Ca + − 113 Cd + sympathetic cooling system [27,39]. Temperature of
the 113 Cd + ions has been decreased to around 40 mK [49]. Owing to the sympathetic cooling
technique, the uncertainty of the SODS has been estimated to the level of 10−16 , and the stability
limitation due to the Dick effect has been decreased to the level of 10−14 at 1 s.
Here, we propose to use 114 Cd + ions as the coolant ions to sympathetically cool 171 Yb +
ions. First, the mass ratio of 171 Yb + and 114 Cd + is 1.5, which is quite close to 1, ensuring
a high cooling efficiency [39,50,51]. Secondly, only one laser is needed to achieve cooling of
114 Cd + , which may not greatly increase the complexity of the system. Finally, the cooling
laser of 114 Cd + is insensitive to the polarization due to zero nuclear spin, and the 114 Cd natural
abundance is highest among cadmium isotopes.
By using molecular dynamics (MD) simulation [36], we investigated the feasibility of the
proposed 114 Cd + − 171 Yb + sympathetic cooling system.
As shown in Fig. 8, simulated CCD images of the 114 Cd + − 171 Yb + and 40 Ca + − 113 Cd +
ion crystal are obtained from the simulation outputs. According to the simulation results, the
114 Cd + − 171 Yb + and 40 Ca + − 113 Cd + ion crystals are both stably trapped and can reach
the equilibrium state under the same trapping conditions of our 171 Yb + microwave frequency
standard. After reaching equilibrium state, the temperature of 113 Cd + in the 40 Ca + − 113 Cd +
system is around 5 mK, while the temperature of 171 Yb + in the 114 Cd + − 171 Yb + system
is around 2 mK. The simulated temperature is in the same order of magnitude. It has been
experimentally proved that by using the 40 Ca + − 113 Cd + sympathetic cooling system, a 113 Cd +
ion crystal with 40 mK can be obtained [49]. Then according to the simulation results, we are
confident to obtain a sympathetically cooled 171 Yb + ion crystal with ultra-low temperature in
the 114 Cd + − 171 Yb + system.
Fig. 8. Simulation results of 40 Ca + − 113 Cd + and 114 Cd + − 171 Yb + ion crystal.
Number of each kind of ions is 256. Red dots in 114 Cd + − 171 Yb + and 40 Ca + − 113 Cd +
ion crystal represent 114 Cd + and 40 Ca + respectively. The amplitude of the RF voltage is
250 V; the DC voltage is set to 10 V.
5.
Conclusion
In conclusion, we reported on the development of a highly stable and high performance 171 Yb +
microwave frequency standard. In attaining this microwave frequency standard, cooling, pumping
and detecting of 171 Yb + are achieved. In our experiment, 105 Yb ions are stably trapped. A
high SNR Ramsey fringe line-shape is obtained and the microwave frequency standard shows
Research Article
Vol. 30, No. 9 / 25 Apr 2022 / Optics Express 14583
√
a short-term stability of 8.5 × 10−13 / τ. Still, important systematic shifts and uncertainties
are carefully estimated. To better suppress the SODS frequency shift, a 114 Cd + − 171 Yb +
sympathetic cooling system is also proposed, preliminarily investigated with the help of a
reliable MD simulation. Our work are important for research on ion phase transition, ion spatial
configuration, sympathetic cooling system, and development of frequency standards.
Funding. National Natural Science Foundation of China (12073015); Tsinghua University Initiative Scientific Research
Program; National Key Research and Development Program of China (2016YFA0302101).
Acknowledgments. We would like to thank C. F. Wu, L. M. Guo, T. G. Zhao, and W. X. Shi for reasonable advice
and helpful suggestions.
Disclosures. The authors declare no conflicts of interest.
Data Availability. 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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