Realization of a watt-level 319-nm single-frequency CW ultraviolet laser and its application in single-photon Rydberg excitation of cesium atoms
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摘要: 结合光纤激光器、光纤放大器和非线性光学高效频率转换技术及准位相匹配材料，服务于原子物理领域铯原子单光子跃迁里德堡激发的实际需求，研究并掌握了产生318.6 nm波长连续单频紫外激光的关键技术。采用1 560.5 nm与1 076.9 nm连续激光先通过单次穿过PPLN非线性晶体和频，再经腔增强谐振倍频过程高效地产生了输出功率大于2 W的318.6 nm紫外激光，半小时内，光功率的方均根起伏优于0.87%。采用电子学边带锁频方案，实现了整个紫外激光系统在保持相对于高精细度超稳腔锁定条件下较大范围连续调谐，其连续调谐范围大于4 GHz，残余频率起伏约16 kHz。采用本文研制的高功率窄线宽可调谐318.6 nm紫外激光系统，在铯热原子气室中实现了6S1/2 →nP3/2(n=70~100)的单光子跃迁里德堡激发，并对相关现象作了相关的理论分析与研究。采用纯光学探测方案观察到了318.6 nm紫外激光对磁光阱中铯冷原子系综的单光子跃迁里德堡激发。Abstract: In order to meet the demand for single-photon Rydberg excitation of cesium atoms in the field of atomic physics, we investigated the key technolgies of single-frequency continuous wave(CW) tunable ultraviolet(UV) laser at 318.6 nm. Combining the fiber lasers, fiber amplifiers and the nonlinear crystals, we achieved 318.6 nm UV laser over 2 Watt output with cavity-enhanced second-harmonic generation following the sum-frequency generation of two infrared lasers at 1 560.5 nm and 1 076.9 nm in PPLN crystal. The typical root-mean-square fluctuation of UV laser power was less than 0.87% within 30 minutes. The electronic side-band locking scheme based on a temperature controlled hyper-fine ultra-stable ultra-low-expansion cavity placed in an ultra-high vacuum chamber was used to achieve the continuously tuning of UV laser in a wide range while still keeping it locked. The continuously tunable range was larger than 4 GHz and the residual frequency fluctuation of UV laser was about 16 kHz. We employed this high-power single-frequency continuously tunable UV laser system for the direct 6S1/2→nP3/2(n=70-100) Rydberg excitation of cesium atoms with atomic vapor cells in experiments. After that, relevant theoretical analysis and research have been done. With a magneto-optical trapped cesium atomic ensemble, single-photon Rydberg excitation using the UV laser system was achieved with a pure optical detection scheme.
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图 1 实验方案和技术路线示意图。其中EDFA：掺铒光纤放大器；YDFA：掺镱光纤放大器；PMF：保偏光纤；OI：光隔离器；λ/2：二分之波片；PBS：偏振分光棱镜；λ/4：四分之一波片；DM：双色片；45° HR：45度高反镜；ULE：超稳腔；FG：函数发生器；EOPM：电光相位调制器；LPF：低通滤波器；PD：光电二极管；PS：射频功率分配器；PM：相位调制器；Φ：移相器；HVA&PI：高压放大器以及比例积分差分放大器；APP：整形棱镜对
Figure 1. Schematic diagram of the laser system. Keys to the figure: EDFA, erbium-doped fiber amplifier; YDFA, ytterbium-doped fiber amplifier; PMF, polarization-maintaining optical fiber; OI, optical isolator; λ/2, half-wave plate; PBS, polarization beam splitter cube; λ/4, quarter-wave plate; DM, dichroic mirror; 45° HR, 45° high-reflectivity mirror; FG, Function generator; ULE cavity, ultra-low expansion cavity; EOPM, electro-optic phase modulator; LPF, low-pass filter; PD, photodiode; PS, radio-frequency power splitter; PM:phase modulator; Φ, phase shifter; HVA&PID, high-voltage amplifier and proportional-integration-differential amplifier; APP, anamorphic prism pair
图 2 单次穿过和频PPMgO:LN晶体温度调谐曲线[18]。圆点是实验数据点，实线为使用sinc2函数理论拟合曲线。(a)30 mm×2 mm×1 mm PPMgO:LN晶体(Λ=12.05 μm)，优化的准相位匹配温度为63.0 ℃，温度半高宽为1.5 ℃; (b)40 mm×10 mm×0.5 mm PPMgO:LN晶体(Λ=11.80 μm)，优化的准相位匹配温度为154.0 ℃，温度半高宽为1.2 ℃
Figure 2. The temperature tuning curves of the PPMgO:LN crystals for single-pass sum-frequency generation[18]. Circles are the experimental data, while the solid lines are the theoretically fitted curves using sinc2 function. (a)PPMgO:LN crystal of the dimension 30 mm×2 mm×1 mm(Poling Period:Λ=12.05 μm), and the optimized QPM temperature is 63.0 ℃ with a FWHM of 1.5 ℃; (b)PPMgO:LN crystal of the dimension 40 mm×10 mm×0.5 mm(Poling Period:Λ=11.80 μm), and the optimized QPM temperature is 154.0 ℃ with a FWHM of 1.2 ℃
图 3 637.2 nm和频激光输出功率随两基频光功率的变化[18]。1 076.9 nm激光功率固定为9 W，改变1 560.5 nm激光功率，误差来源于功率计测量误差。(a)30 mm PPMgO:LN晶体实验结果；(b)40 mm PPMgO:LN晶体实验结果。在长度为40 mm的PPMgO:LN晶体中，当1 560.5 nm和1 076.9 nm基频激光功率分别为9 W和14 W时，最大获得了8.75 W的637.2 nm单频红光输出，和频效率高达38%
Figure 3. SFG output power versus power of two fundamental lasers using a 75 mm focusing lens[18]. The 1 076.9 nm laser power was fixed at 9 W and the 1 560.5 nm laser power varied. The error bars come from the measurement error of powermeter. (a)The case of 30-mm-long PPMgO:LN crystal from HC Photonics; (b)the case of 40-mm-long PPMgO:LN crystal. In these crystal, a maximum output power for 637.2 nm single-frequency laser of up to 8.75 W was obtained when the fundamental power of 1 560.5 nm and 1 076.9 nm lasers are tuned to 9 W and 14 W, respectively. The efficiency of SFG is 38%
图 4 和频光束的M2因子测量[18]。方块点(1)和圆点(2)分别代表光束横截面水平和竖直两方向的测量结果，插图为典型的和频光束横截面强度分布。30 mm PPMgO:LN晶体(a)；40 mm PPMgO:LN晶体(b)
Figure 4. M2 factors of the SFG beam[18]. The squares(1) show the measurements of the horizontal direction X, and the circles(2) show the vertical direction Y. Insets show the typical intensity profile of the SFG laser beam. (a)The case of 30-mm-long PPMgO:LN crystal; (b)the case of 40-mm-long PPMgO:LN crystal
图 5 318.6 nm紫外激光功率和倍频效率随输入637.2 nm激光功率的变化情况[19]。方块和圆圈为实验测量值，实线为根据实验参数(T1=2.2%, Lcav=0.67%, Enl =6.5×10-5/W)得到的理论拟合曲线[19]。在4 W的637.2 nm红光注入条件下，可得到2.26 W的318.6 nm紫外激光输出，倍频转化效率约56.5%
Figure 5. 318.6 nm UV laser output and doubling efficiency vary with the incident 637.2 nm laser power[19]. Squares are the experimental data, while the circles are the theoretical results with the parameters T1=2.2%, Lcav=0.67%, and Enl=6.5×10-5/W. The 2.26 W output power for 318.6 nm UV laser is obtained by tuning the power of 637.2 nm laser to 4 W. The efficiency of SHG is 56.5%
图 7 Power stability of the 318.6 nm UV laser output at 1.2 W over 30 min[19]. The typical RMS fluctuation is less than 0.87%
Figure 7. Beam profile and the M2 factors for the 318.6 nm laser output[19]. The left pictures show the UV laser spot profiles (a)before and (b)after shaping. (c)The measured beam quality factors MX2(squares) and MY2(circles)
图 8 紫外激光系统实物图，右上角为种子光光路(光纤激光器和光纤放大器没有显示在图中)。将各光学元件整合到700 mm×1 000 mm的铝板上，并将整个光路系统罩起来，起到了一定的隔振和防尘效果，使得整个激光系统的机械稳定性得到了明显改善
Figure 8. The ultraviolet laser system prototype. Upper right corner for seed light path(fiber lasers and fiber amplifiers are not shown in the figure). Most of optical elements are integrated on the aluminum plate of 700 mm×1 000 mm, and the whole optical path is covered by perspex plates, which play a certain sound insulation and dust prevention effect, and the mechanical stability of the whole laser system has been significantly improved
图 9 锁定1 560.5 nm激光频率，扫描1 076.9 nm激光频率，可得到经位相调制后的637.2 nm激光的腔透射信号(曲线1)，曲线(2)为对应的电子学边带误差信号[20]。其中，调制频率Ω1/2π和Ω2/2π分别为15 MHz和2 MHz，对应射频功率分别为14 dBm和10 dBm
Figure 9. Transmitted signal of the phase-modulated 637.2 nm laser(curve 1) incident on the cavity. It is obtained by sweeping the carrier frequency of the 1 076.9 nm laser while the 1 560.5 nm laser remains locked[20]. The curve 2 represents the corresponding ESB error signal. Here, Ω1/2π and Ω2/2π are equal to 15 MHz and 2 MHz with RF power consumptions of 14 dBm and 10 dBm, respectively
图 10 经电子学边带技术锁定后637.2 nm激光的相对阿伦方差，反映了激光锁定后的频率不稳定性[20]。插图显示锁定30分钟内的频率起伏为~ 8 kHz
Figure 10. Relative Allan standard deviation plots show the relative frequency instability of the 637.2 nm laser using the ESB(squares) locking technique[20]. The inset is a trace of the ESB error signal when the 637.2 nm light is offset-locked. The residual fluctuation is about 8 kHz in 30 min
图 11 1 560.5 nm和1 076.9 nm激光频率分别通过PDH和ESB技术锁定到超稳腔上[20]。在保持倍频腔锁定的条件下，改变EOPM-2所加调制频率，637.2 nm载频可连续调谐1.95 GHz(a)，同时318.6 nm紫外激光调谐至少4 GHz(b)。两激光的调谐范围通过两自由光谱区分别为~487 MHz和~500 MHz的光学腔测得。
Figure 11. The 1 560.5 and 1076.9 nm lasers are locked to the ULE cavity using the PDH and ESB methods[20], respectively. By changing the modulation frequency of the EOPM-2, (a)the carrier frequency of the 637.2 nm red light is continuously tuned over 1.95 GHz; (b)Simultaneously, the 318.6 nm UV laser is tuned over 4 GHz under the condition of the doubling cavity also remains locked. The tuning ranges of the two lasers are monitored by an optical cavity with a FSR of ~487 and 500 MHz, respectively.
图 12 (a) Cs原子单步里德堡激发相关能级图[21]。852.3 nm探测光共振于6S1/2(F=4)→ 6P3/2(F′=5)，318.6 nm耦合光在6S1/2(F=4)→nP3/2跃迁线附近扫描；(b)实验装置图。318.6 nm耦合光与852.3 nm探测光同向穿过长度为10 cm的Cs原子气室
Figure 12. (a)Relevant hyperfine levels for Cs atomic single-photon Rydberg excitation[21]. The 852.3 nm probe laser is resonant on the transition 6S1/2(F=4)→6P3/2(F′=5), and the 318.6 nm coupling laser is scanned over the Rydberg transition 6S1/2(F=4)→nP3/2. (b)Schematic of the experimental setup. The 318.6 nm coupling laser co-propagating with the 852.3 nm probe laser in a 10-cm-long Cs vapor cell
图 13 (a) 852.3 nm探测光锁定到6S1/2(F=4)→6P3/2(F′=5)循环线上时，Cs原子6S1/2(F=4)→71P3/2里德堡激发光谱。耦合光和探测光的Rabi频率分别为~0.30 MHz和~8.53 MHz。(b)852.3 nm探测光加70 MHz射频频率调制时，里德堡光谱边带标定结果，考虑多普勒因子λp/λc≈2.675，观察到的光谱超精细分裂间隔为~671 MHz。红色曲线(2)为多峰Lorentz拟合[21]
Figure 13. (a)The excitation spectra of 6S1/2(F=4)→71P3/2 Rydberg transition in a Cs vapor cell when the 852.3 nm probe laser is locked to Cs 6S1/2(F=4)→6P3/2(F′=5) cycling transition. The Rabi frequencies of coupling and probe beams are ~0.30 and ~8.53 MHz, respectively. (b)Sideband calibration result with a frequency modulation of 70 MHz for 852.3 nm probe laser, considering the Doppler factor of λp/λc≈2.675, the observed hyperfine interval becomes 671 MHz. Red curve(2) is a multi-peak Lorentzian fitting[21]
图 16 在铯冷原子磁光阱中，利用俘获损耗光谱观察到的Cs 6S1/2(F=4)→71P3/2单光子跃迁的Rydberg发光谱。两个峰是由于冷却光未关断导致的Autler-Townes分裂[23]
Figure 16. In the cesium atom magneto-optical, Rydberg spectroscopy of Cs 6S1/2(F=4)→71P3/2 single-photon transition observed by capture loss spectrum. Two peaks appeared in spectrum are due to the Autler-Townes splitting that caused by the cooling laser beams[23]
图 17 不同主量子数n(71、84、90)的铯原子里德堡态nP3/2激发信号随紫外激光Rabi频率的变化情况
Figure 17. Rydberg state nP3/2 excitation signals with different principal quantum numbers n(71, 84, 90) vary with Rabi frequency of ultraviolet laser
图 18 不同主量子数下，磁光阱荧光强度随紫外激光失谐量的变化关系[23]。图(a)、(b)、(c)和(d)分别对应于84P3/2、90P3/2、95P3/2和100P3/2里德堡态激发信号。方块为实验数据，实线为多峰Voigt函数拟合
Figure 18. Fluorescence intensities of magneto-optical trap change with the detuning of UV laser(n=84, 90, 95 and 100 Rydberg states for figure(a), (b), (c) and (d), respectively[23]). The squares are experimental data, and line is fitting results by the multi-peak Voigt function
图 19 利用高激发态(主量子数n=71, 84, 90, 95和100)里德堡原子测量背景直流电场的结果[23]
Figure 19. Background DC electric field sensed by highly-excited cesium nP3/2(n=71, 84, 90, 95 and 100) Rydberg states[23]. The error bars are obtained by fitting the Stark map of each Rydberg state. The line represents the average value for experimental data(diamond cube)
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[1] GALLAGHER T F. Rydberg Atoms[M]. Cambridge:Cambridge University Press, 1994. [2] PONOMARENKO D V, SHESTAKOV A F. Polarizabilities of ns and np Rydberg states of the Li atom[J]. Chemical Physics Letters, 1993, 210(1-3):269-273. doi: 10.1016/0009-2614(93)89132-2 [3] BALUKTSIAN T, HUBER B, LÖW R, et al.. Evidence for strong van der Waals type Rydberg-Rydberg interaction in a thermal vapor[J]. Physical Review Letters, 2013, 110(12):123001. doi: 10.1103/PhysRevLett.110.123001 [4] SAFFMAN M, WALKER T G, MØLMER K. Quantum information with Rydberg atoms[J]. Reviews of Modern Physics, 2010, 82(3):2313-2363. doi: 10.1103/RevModPhys.82.2313 [5] MÜLLER M, LESANOVSKY I, WEIMER H, et al.. Mesoscopic Rydberg gate based on electromagnetically induced transparency[J]. Physical Review Letters, 2009, 102(17):170502. doi: 10.1103/PhysRevLett.102.170502 [6] SEDLACEK J A, SCHWETTMANN A, KVBLER H, et al.. Microwave electrometry with Rydberg atoms in a vapour cell using bright atomic resonances[J]. Nature Physics, 2012, 8(11):819-824. doi: 10.1038/nphys2423 [7] HANKIN A M, JAU Y Y, PARAZZOLI L P, et al.. Two-atom Rydberg blockade using direct 6S to nP excitation[J]. Physical Review A, 2014, 89(3):033416. doi: 10.1103/PhysRevA.89.033416 [8] TONG D, FAROOQI S M, STANOJEVIC J, et al.. Local blockade of Rydberg excitation in an ultracold gas[J]. Physical Review Letters, 2004, 93(6):063001. doi: 10.1103/PhysRevLett.93.063001 [9] ZHANG H, WANG L M, ZHANG L J, et al.. Stark-induced L-mixing interferences in ultracold cesium Rydberg atoms[J]. Physical Review A, 2013, 87(3):033405. doi: 10.1103/PhysRevA.87.033405 [10] 王丽, 张好, 张临杰.超冷里德堡原子的俘获损耗光谱[J].量子光学学报, 2018, 24(2):178-183. http://d.old.wanfangdata.com.cn/Periodical/lzgxxb201802009WANG L, ZHANG H, ZHANG L J. Trap loss spectroscopy of ultracold cesium Rydberg atoms[J]. Journal of Quantum Optics, 2018, 24(2):178-183.(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/lzgxxb201802009 [11] 韩小萱, 焦月春, 赵建明.ns6s里德堡分子的势能曲线[J].量子光学学报, 2017, 23(1):46-51. http://d.old.wanfangdata.com.cn/Periodical/lzgxxb201701006HAN X X, JIAO Y CH, ZHAO J M. Potential of long-range ns6s Rydberg molecule[J]. Journal of Quantum Optics, 2017, 23(1):46-51.(in Chinese) http://d.old.wanfangdata.com.cn/Periodical/lzgxxb201701006 [12] BARREDO D, RAVETS S, LABUHN H, et al.. Demonstration of a strong Rydberg blockade in three-atom systems with anisotropic interactions[J]. Physical Review Letters, 2014, 112(18):183002. doi: 10.1103/PhysRevLett.112.183002 [13] JOHNSON J E, ROLSTON S L. Interactions between Rydberg-dressed atoms[J]. Physical Review A, 2010, 82(3):033412. doi: 10.1103/PhysRevA.82.033412 [14] WILSON A C, OSPELKAUS C, VANDEVENDER A P, et al.. A 750-mW, continuous-wave, solid-state laser source at 313 nm for cooling and manipulating trapped 9Be+ ions[J]. Applied Physics B, 2011, 105(4):741-748. doi: 10.1007/s00340-011-4771-1 [15] LO H Y, ALONSO J, KIENZLER D, et al.. All-solid-state continuous-wave laser systems for ionization, cooling and quantum state manipulation of beryllium ions[J]. Applied Physics B, 2014, 114(1-2):17-25. doi: 10.1007/s00340-013-5605-0 [16] BRIDGE E M, KEEGAN N C, BOUNDS A D, et al.. Tunable cw UV laser with < 35 kHz absolute frequency instability for precision spectroscopy of Sr Rydberg states[J]. Optics Express, 2016, 24(3):2281-2292. doi: 10.1364/OE.24.002281 [17] RENGELINK R J, NOTERMANS R P M J W, VASSEN W. A simple 2 W continuous-wave laser system for trapping ultracold metastable helium atoms at the 319.8 nm magic wavelength[J]. Applied Physics B, 2016, 122(5):122. doi: 10.1007/s00340-016-6395-y [18] WANG J Y, BAI J D, HE J, et al.. Realization and characterization of single-frequency tunable 637.2 nm high-power laser[J]. Optics Communications, 2016, 370:150-155. doi: 10.1016/j.optcom.2016.02.067 [19] WANG J Y, BAI J D, HE J, et al.. Development and characterization of a 2.2 W narrow-linewidth 318.6 nm ultraviolet laser[J]. Journal of the Optical Society of America B, 2016, 33(10):2020-2025. doi: 10.1364/JOSAB.33.002020 [20] BAI J D, WANG J Y, HE J, et al.. Electronic sideband locking of a broadly tunable 318.6 nm ultraviolet laser to an ultra-stable optical cavity[J]. Journal of Optics, 2017, 19(4):045501. doi: 10.1088/2040-8986/aa5a8c [21] WANG J Y, BAI J D, HE J, et al.. Single-photon cesium Rydberg excitation spectroscopy using 318.6-nm UV laser and room-temperature vapor cell[J]. Optics Express, 2017, 25(19):22510-22518. doi: 10.1364/OE.25.022510 [22] BAI J D, WANG J Y, LIU S, et al.. Autler-Townes doublet in single-photon Rydberg spectra of cesium atomic vapor with a 319 nm UV laser[J]. Applied Physics B, 2019, 125(3):33. doi: 10.1007/s00340-019-7151-x [23] BAI J D, LIU S, WANG J Y, et al.. Single-photon Rydberg excitation and trap-loss spectroscopy of cold cesium atoms in a magneto-optical trap by using of a 319-nm ultra-violet laser system[J]. arXiv: 1811.05092v2, 2018. [24] 白建东, 王杰英, 王军民.基于光纤延时声光频移自差拍法快速测量激光线宽[J].激光与光电子学进展, 2016, 53(6):061407. http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jgygdzxjz201606027BAI J D, WANG J Y, WANG J M. Rapid measurement of laser linewidth based on fiber-delayed AOM-shifted self-heterodyne scheme[J]. Laser & Optoelectronics Progress, 2016, 53(6):061407.(in Chinese) http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jgygdzxjz201606027 [25] THOUMANY P, HÄNSCH T, STANIA G, et al.. Optical spectroscopy of rubidium Rydberg atoms with a 297 nm frequency-doubled dye laser[J]. Optics Letters, 2009, 34(11):1621-1623. doi: 10.1364/OL.34.001621 [26] AUTLER S H, TOWNES C H. Stark effect in rapidly varying fields[J]. Physical Review, 1955, 100(2):703-722. doi: 10.1103/PhysRev.100.703 [27] SCULLY M O, ZUBAIRY M S. Quantum Optics[M] Cambridge:Cambridge University Press, 1997. [28] FLEISCHHAUER M, IMAMOGLU A, MARANGOS J P. Electromagnetically induced transparency:Optics in coherent media[J]. Reviews of Modern Physics, 2005, 77(2):633-673. doi: 10.1103/RevModPhys.77.633 [29] HAU L V, HARRIS S E, DUTTON Z, et al.. Light speed reduction to 17 meters per second in an ultracold atomic gas[J]. Nature, 1999, 397(6720):594-598. doi: 10.1038/17561 [30] BRAJE D A, BALIĆ V, GODA S, et al.. Frequency mixing using electromagnetically induced transparency in cold atoms[J]. Physical Review Letters, 2004, 93(18):183601. doi: 10.1103/PhysRevLett.93.183601 [31] AHMED E H, INGRAM S, KIROVA T, et al.. Quantum control of the spin-orbit interaction using the Autler-Townes effect[J]. Physical Review Letters, 2011, 107(16):163601. doi: 10.1103/PhysRevLett.107.163601 [32] RAITZSCH U, HEIDEMANN R, WEIMER H, et al.. Investigation of dephasing rates in an interacting Rydberg gas[J]. New Journal of Physics, 2009, 11(5):055014. doi: 10.1088/1367-2630/11/5/055014 [33] MITSUNAGA M, IMOTO N. Observation of an electromagnetically induced grating in cold sodium atoms[J]. Physical Review A, 1999, 59(6):4773-4776. doi: 10.1103/PhysRevA.59.4773 [34] LIANG Q B, YANG B D, YANG J F, et al.. Autler-Townes doublet in the absorption spectra for the transition between excited states of cold cesium atoms[J]. Chinese Physics B, 2010, 19(11):113207. doi: 10.1088/1674-1056/19/11/113207 [35] MENON S, AGARWAL G S. Gain components in the Autler-Townes doublet from quantum interferences in decay channels[J]. Physical Review A, 1999, 61(1):013807. doi: 10.1103/PhysRevA.61.013807 [36] SINGER K, REETZ-LAMOUR M, AMTHOR T, et al.. Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms[J]. Physical Review Letters, 2004, 93(16):163001. doi: 10.1103/PhysRevLett.93.163001 [37] BOISSEAU C, SIMBOTIN I, CÔTÉ R. Macrodimers:ultralong range Rydberg molecules[J]. Physical Review Letters, 2002, 88(13):133004. doi: 10.1103/PhysRevLett.88.133004 [38] MARINESCU M. Dispersion coefficients for the nP-nP asymptote of homonuclear alkali-metal dimers[J]. Physical Review A, 1997, 56(6):4764-4773. doi: 10.1103/PhysRevA.56.4764 -
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