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Theoretical study of laser cooling of potassium chloride anion

Wan Ming-Jie Luo Hua-Feng Yuan Di Li Song

Theoretical study of laser cooling of potassium chloride anion

Wan Ming-Jie, Luo Hua-Feng, Yuan Di, Li Song
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  • The potential energy curves and transition dipole moments (TDMs) for three Λ-S states (X2Σ+, A2Π, and B2Σ+) of potassium chloride anion (KCl) are investigated by using multi-reference configuration interaction (MRCI) method. The def2-AQZVPP-JKFI of K atom and AV5Z-DK all-electron basis set of Cl atom are used in all calculations. The Davidson correction, core-valence (CV) correction, and spin-orbit coupling effect (SOC) are also considered. In the complete active self-consistent field (CASSCF) calculations, eight molecular orbitals are selected as active orbitals, which includ K 4s4p and Cl 3s3p shells; K 3p shell is closed orbital, and the remaining shells (K 1s2s3s and Cl 1s2s2p) are frozen orbitals. In the MRCI+Q calculations, K 3p shell is used for the CV correction. There are 15 electrons in the correlation energy calculations. Then, their spectroscopic parameters, Einstein coefficients, Franck-Condon factors, and radiative lifetimes are obtained by solving the radial Schrödinger equation. The spectroscopic properties and transition properties for the Ω states are predicted. Highly diagonally distributed Franck-Condon factor f00 values for the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 transition are 0.8816 and 0.8808, respectively. And the short radiative lifetimes for the (2)1/2 and (1)3/2 excited states are also obtained, i.e. τ[(2)1/2] = 45.7 ns and τ[(1)3/2] = 45.5 ns, which can ensure laser cooling of KCl anion rapidly. The results indicate that the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 quasicycling transitions are suitable to the building of laser cooling projects. For driving the (2)1/2↔(1)1/2 transition, a main pump laser (λ00) and two repumping lasers (λ10 and λ21) are required. Their wavelengths are λ00 = 1065.77 nm, λ10 = 1090.13 nm and λ21 = 1087.76 nm. For driving the (1)3/2↔(1)1/2 transition, the wavelengths are λ00 = 1064.24 nm, λ10 = 1088.54 nm, and λ21 = 1086.17 nm. The cooling wavelengths of KCl- anion for two transitions are both deep in the infrared range. Finally, the Doppler temperature and recoil temperature for two transitions are also calculated, respectively. The Doppler temperatures for (2)1/2↔(1)1/2 and (1)3/2(1)1/2 transitions are 83.57 μK and 83.93 μK, and the recoil temperatures for two transitions are 226 nK and 227 nK, respectively. for two transitions are 226 nK and 227 nK, respectively.
      Corresponding author: Wan Ming-Jie, wanmingjie1983@sina.com
    [1]

    van Veldhoven J, Küpper J, Bethlem H L, Sartakov B, van Roij A J A, Meijer G 2004 Eur. Phys. J. D 31 337

    [2]

    Micheli A, Brennen G K, Zoller P 2006 Nat. Phys. 2 341

    [3]

    Willitsch S, Bell M T, Gingell A D, Procter S R, Softley T P 2008 Phys. Rev. Lett. 100 043203

    [4]

    Shuman E S, Barry J F, de Mille D 2010 Nature 467 820

    [5]

    Hummon M T, Yeo M, Stuhl B K, Collopy A L, Xia Y, Ye J 2013 Phys. Rev. Lett. 110 143001

    [6]

    Zhelyazkova V, Cournol A, Wall T E, Matsushima A, Hudson J J, Hinds E A, Tarbutt M R, Sauer B E 2014 Phys. Rev. A 89 053416

    [7]

    Gao Y, Gao T 2014 Phys. Rev. A 90 052506

    [8]

    张云光, 张华, 窦戈, 徐建刚 2017 物理学报 66 233101

    Zhang Y G, Zhang H, Dou G, Xu J G 2017 Acta Phys. Sin. 66 233101

    [9]

    Cui J, Xu J G, Qi J X, Dou G, Zhang Y G 2018 Chin. Phys. B 27 103101

    [10]

    Yzombard P, Hamamda M, Gerber S, Doser M, Comparat D 2015 Phys. Rev. Lett. 114 213001

    [11]

    Wan M, Huang D, Yu Y, Zhang Y 2017 Phys. Chem. Chem. Phys. 19 27360

    [12]

    万明杰, 李松, 金成国, 罗华锋 2019 物理学报 68 063103

    Wan M J, Li S, Jin C G, Luo H F 2019 Acta Phys. Sin. 68 063103

    [13]

    Zhang Q, Yang C, Wang M, Ma X, Liu W 2017 Spectrochim. Acta, Part A 182 130

    [14]

    Zhang Q, Yang C, Wang M, Ma X, Liu W 2017 Spectrochim. Acta, Part A 185 365

    [15]

    Huber K P, Herzberg G 1979 Constants of Diatomic Molecules (Vol. IV): Molecular Spectra and Molecular Structure (New York: Van Nostrand Reinhold) p358

    [16]

    Ram R S, Dulick M, Guo B, Zhang K Q, Bernath P F 1997 J. Mol. Spectrosc. 183 360

    [17]

    Seth M, Pernpointner M, Bowmaker G A, Schwerdtfeger P 1999 Mol. Phys. 96 1767

    [18]

    Wan M J, Shao J X, Huang D H, Jin C G, Yu Y, Wang F H 2015 Phys. Chem. Chem. Phys. 17 26731

    [19]

    Wan M J, Shao J X, Gao Y F, Huang D H, Yang J S, Cao Q L, Jin C G, Wang F H 2015 J. Chem. Phys. 143 024302

    [20]

    Fu M K, Ma H T, Cao J W, Bian W S 2016 J. Chem. Phys. 144 184302

    [21]

    Wan M J, Yuan D, Jin C G, Wang F H, Yang Y J, Yu Y, Shao J X 2016 J. Chem. Phys. 145 024309

    [22]

    Yuan X, Yin S, Shen Y, Liu Y, Lian Y, Xu H F, Yan B 2018 J. Chem. Phys. 149 094306

    [23]

    Werner H J, Knowles P J, Lindh R, et al. 2010 MOLPRO, version 2010.1, A Package of ab initio Programs, http://www.molpro.net

    [24]

    Knowles P J, Werner H J 1985 J. Chem. Phys. 82 5053

    [25]

    Werner H J, Knowles P J 1988 J. Chem. Phys. 89 5803

    [26]

    Xiao K L, Yang C L, Wang M S, Ma X G, Liu W W 2013 J. Chem. Phys. 139 074305

    [27]

    Weigend F 2008 J. Comput. Chem. 29 167

    [28]

    Woon D E, Dunning Jr T H 1993 J. Chem. Phys. 98 1358

    [29]

    Berning A, Schweizer M, Werner H J, Knowles P J, Palmieri P 2000 Mol. Phys. 98 1823

    [30]

    Le Roy R J Level 8.0: A Computer Program for Solving the Radial Schrödinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663. http://leroy.uwaterloo.ca/programs

    [31]

    Hotop H, Lineberger 1985 J. Phys. Chem. Ref. Data 14 731

    [32]

    Berzinsh U, Gustafsson M, Hanstorp D, Klinkmueller A E, Ljungblad U, Maartensson-Pendrill A M 1995 Phys. Rev. A 51 231

    [33]

    Moore C E 1971 Atomic Energy Levels (Vol. 1) Natl. Stand Ref. Data Ser. Natl. Bur. Stand. No. 35 (Washington, DC: U.S. GPO) p228

    [34]

    Kobayashi J, Aikawa K, Oasa K, Inouye S 2014 Phys. Rev. A 89 021401

    [35]

    Cohen-Tannoudji C N 1998 Rev. Mod. Phys. 70 707

  • 图 1  KCl-阴离子的势能曲线 (a) Λ-S态; (b) Ω态

    Figure 1.  Potential energy curves of KCl anion: (a) Λ-S states; (b) Ω states.

    图 2  KCl-阴离子的偶极矩

    Figure 2.  Dipole moments (DMs) of KCl- anion.

    图 3  KCl阴离子的跃迁偶极矩

    Figure 3.  Transition dipole moments (TDMs) of KCl anion

    图 4  激光冷却KCl阴离子的方案 (a) (2)1/2↔(1)1/2准闭合循环跃迁系统; (b) (1)3/2↔(1)1/2准闭合循环跃迁系统

    Figure 4.  Proposed laser cooling scheme of KCl anion: (a) Using (2)1/2↔(1)1/2 transition; (b) using (1)3/2↔(1)1/2 transition.

    表 1  KCl阴离子Ω电子态的离解极限

    Table 1.  The dissociation relationship for the Ω states of KCl anion.

    原子态Ω态ΔE/cm–1
    计算值实验值[33]
    K(2S1/2) + Cl(1S0)(1)1/200
    K(2P1/2) + Cl(1S0)(2)1/212997.9412985.17
    K(2P3/2) + Cl(1S0)(3)1/2, (1)3/213046.2313042.89
    DownLoad: CSV

    表 2  KCl阴离子的Ω态的光谱常数

    Table 2.  Spectroscopic parameters for the Ω states of KCl anion.

    Ω态对应的Λ-S态Reωe/cm–1Be/cm–1De/eVTe/cm–1
    (1)1/2X2Σ+2.8290212.340.11431.34830
    (2)1/2A2Π2.7839229.640.11801.79769375.30
    (1)3/2A2Π2.7836229.650.11801.80189388.68
    (3)1/2B2Σ+2.7550235.480.12051.386512746.21
    DownLoad: CSV

    表 3  (2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁的FCFs, Aν′ν′′τ

    Table 3.  FCFs, spontaneous emission rates Aν′ν′′ and spontaneous radiative lifetime τ for the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 transitions.

    跃迁ν′′0123
    (2)1/2↔(1)1/2Aν′ν′′/s–11.9384(7)a2.3044(6)1.7867(5)1.1906(4)
    ν′ = 0fν′ν′′0.88160.10900.00880.0006
    τ/ns45.7
    Aν′ν′′/s–12.5793(6)1.4757(7)4.0633(6)5.0295(5)
    ν′ = 1fν′ν′′0.11280.66870.19140.0246
    τ/ns45.5
    Aν′ν′′/s–11.3368(5)4.7122(6)1.0816(7)5.3057(6)
    ν′ = 2fν′ν′′0.00560.20520.48830.2490
    τ/ns45.4
    (1)3/2↔(1)1/2Aν′ν′′/s–11.9451(7)2.3276(6)1.8184(5)1.2242(4)
    ν′ = 0fν′ν′′0.88080.10960.00890.0006
    τ/ns45.5
    Aν′ν′′/s–12.6063(6)1.4777(7)5.1089(5)4.7631(4)
    ν′ = 1fν′ν′′0.11340.66680.19240.0249
    τ/ns45.4
    Aν′ν′′/s–11.3578(5)4.7578(6)1.0803(7)5.3483(6)
    ν′ = 2fν′ν′′0.00570.20630.48570.2499
    τ/ns45.2
    注: a1.9384(7)表示1.9384 × 107.
    DownLoad: CSV

    表 4  (3)1/2↔(1)1/2, (3)1/2↔(2)1/2和(3)1/2↔(1)3/2跃迁的FCF, 总辐射速率A0和辐射寿命

    Table 4.  FCFs, total emission rates A0 and τ for the (3)1/2↔(1)1/2, (3)1/2↔(2)1/2 and (3)1/2↔(1)3/2 transitions.

    跃迁f00A0/s–1τ0/s
    (3)1/2↔(1)1/20.71222.6535(7)3.77(–8)
    (3)1/2↔(2)1/20.94842.3716(5)4.22(–6)
    (3)1/2↔(1)3/20.94902.3435(5)4.27(–6)
    注: a1.9384(7)表示1.9384 × 107.
    DownLoad: CSV
  • [1]

    van Veldhoven J, Küpper J, Bethlem H L, Sartakov B, van Roij A J A, Meijer G 2004 Eur. Phys. J. D 31 337

    [2]

    Micheli A, Brennen G K, Zoller P 2006 Nat. Phys. 2 341

    [3]

    Willitsch S, Bell M T, Gingell A D, Procter S R, Softley T P 2008 Phys. Rev. Lett. 100 043203

    [4]

    Shuman E S, Barry J F, de Mille D 2010 Nature 467 820

    [5]

    Hummon M T, Yeo M, Stuhl B K, Collopy A L, Xia Y, Ye J 2013 Phys. Rev. Lett. 110 143001

    [6]

    Zhelyazkova V, Cournol A, Wall T E, Matsushima A, Hudson J J, Hinds E A, Tarbutt M R, Sauer B E 2014 Phys. Rev. A 89 053416

    [7]

    Gao Y, Gao T 2014 Phys. Rev. A 90 052506

    [8]

    张云光, 张华, 窦戈, 徐建刚 2017 物理学报 66 233101

    Zhang Y G, Zhang H, Dou G, Xu J G 2017 Acta Phys. Sin. 66 233101

    [9]

    Cui J, Xu J G, Qi J X, Dou G, Zhang Y G 2018 Chin. Phys. B 27 103101

    [10]

    Yzombard P, Hamamda M, Gerber S, Doser M, Comparat D 2015 Phys. Rev. Lett. 114 213001

    [11]

    Wan M, Huang D, Yu Y, Zhang Y 2017 Phys. Chem. Chem. Phys. 19 27360

    [12]

    万明杰, 李松, 金成国, 罗华锋 2019 物理学报 68 063103

    Wan M J, Li S, Jin C G, Luo H F 2019 Acta Phys. Sin. 68 063103

    [13]

    Zhang Q, Yang C, Wang M, Ma X, Liu W 2017 Spectrochim. Acta, Part A 182 130

    [14]

    Zhang Q, Yang C, Wang M, Ma X, Liu W 2017 Spectrochim. Acta, Part A 185 365

    [15]

    Huber K P, Herzberg G 1979 Constants of Diatomic Molecules (Vol. IV): Molecular Spectra and Molecular Structure (New York: Van Nostrand Reinhold) p358

    [16]

    Ram R S, Dulick M, Guo B, Zhang K Q, Bernath P F 1997 J. Mol. Spectrosc. 183 360

    [17]

    Seth M, Pernpointner M, Bowmaker G A, Schwerdtfeger P 1999 Mol. Phys. 96 1767

    [18]

    Wan M J, Shao J X, Huang D H, Jin C G, Yu Y, Wang F H 2015 Phys. Chem. Chem. Phys. 17 26731

    [19]

    Wan M J, Shao J X, Gao Y F, Huang D H, Yang J S, Cao Q L, Jin C G, Wang F H 2015 J. Chem. Phys. 143 024302

    [20]

    Fu M K, Ma H T, Cao J W, Bian W S 2016 J. Chem. Phys. 144 184302

    [21]

    Wan M J, Yuan D, Jin C G, Wang F H, Yang Y J, Yu Y, Shao J X 2016 J. Chem. Phys. 145 024309

    [22]

    Yuan X, Yin S, Shen Y, Liu Y, Lian Y, Xu H F, Yan B 2018 J. Chem. Phys. 149 094306

    [23]

    Werner H J, Knowles P J, Lindh R, et al. 2010 MOLPRO, version 2010.1, A Package of ab initio Programs, http://www.molpro.net

    [24]

    Knowles P J, Werner H J 1985 J. Chem. Phys. 82 5053

    [25]

    Werner H J, Knowles P J 1988 J. Chem. Phys. 89 5803

    [26]

    Xiao K L, Yang C L, Wang M S, Ma X G, Liu W W 2013 J. Chem. Phys. 139 074305

    [27]

    Weigend F 2008 J. Comput. Chem. 29 167

    [28]

    Woon D E, Dunning Jr T H 1993 J. Chem. Phys. 98 1358

    [29]

    Berning A, Schweizer M, Werner H J, Knowles P J, Palmieri P 2000 Mol. Phys. 98 1823

    [30]

    Le Roy R J Level 8.0: A Computer Program for Solving the Radial Schrödinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Research Report CP-663. http://leroy.uwaterloo.ca/programs

    [31]

    Hotop H, Lineberger 1985 J. Phys. Chem. Ref. Data 14 731

    [32]

    Berzinsh U, Gustafsson M, Hanstorp D, Klinkmueller A E, Ljungblad U, Maartensson-Pendrill A M 1995 Phys. Rev. A 51 231

    [33]

    Moore C E 1971 Atomic Energy Levels (Vol. 1) Natl. Stand Ref. Data Ser. Natl. Bur. Stand. No. 35 (Washington, DC: U.S. GPO) p228

    [34]

    Kobayashi J, Aikawa K, Oasa K, Inouye S 2014 Phys. Rev. A 89 021401

    [35]

    Cohen-Tannoudji C N 1998 Rev. Mod. Phys. 70 707

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  • Received Date:  04 June 2019
  • Accepted Date:  13 June 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

Theoretical study of laser cooling of potassium chloride anion

    Corresponding author: Wan Ming-Jie, wanmingjie1983@sina.com
  • 1. School of Physics and Electronic Engineering, Yibin University, Yibin 644007, China
  • 2. College of Chemistry & Chemical Engineering, Yibin University, Yibin 644007, China
  • 3. School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China

Abstract: The potential energy curves and transition dipole moments (TDMs) for three Λ-S states (X2Σ+, A2Π, and B2Σ+) of potassium chloride anion (KCl) are investigated by using multi-reference configuration interaction (MRCI) method. The def2-AQZVPP-JKFI of K atom and AV5Z-DK all-electron basis set of Cl atom are used in all calculations. The Davidson correction, core-valence (CV) correction, and spin-orbit coupling effect (SOC) are also considered. In the complete active self-consistent field (CASSCF) calculations, eight molecular orbitals are selected as active orbitals, which includ K 4s4p and Cl 3s3p shells; K 3p shell is closed orbital, and the remaining shells (K 1s2s3s and Cl 1s2s2p) are frozen orbitals. In the MRCI+Q calculations, K 3p shell is used for the CV correction. There are 15 electrons in the correlation energy calculations. Then, their spectroscopic parameters, Einstein coefficients, Franck-Condon factors, and radiative lifetimes are obtained by solving the radial Schrödinger equation. The spectroscopic properties and transition properties for the Ω states are predicted. Highly diagonally distributed Franck-Condon factor f00 values for the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 transition are 0.8816 and 0.8808, respectively. And the short radiative lifetimes for the (2)1/2 and (1)3/2 excited states are also obtained, i.e. τ[(2)1/2] = 45.7 ns and τ[(1)3/2] = 45.5 ns, which can ensure laser cooling of KCl anion rapidly. The results indicate that the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 quasicycling transitions are suitable to the building of laser cooling projects. For driving the (2)1/2↔(1)1/2 transition, a main pump laser (λ00) and two repumping lasers (λ10 and λ21) are required. Their wavelengths are λ00 = 1065.77 nm, λ10 = 1090.13 nm and λ21 = 1087.76 nm. For driving the (1)3/2↔(1)1/2 transition, the wavelengths are λ00 = 1064.24 nm, λ10 = 1088.54 nm, and λ21 = 1086.17 nm. The cooling wavelengths of KCl- anion for two transitions are both deep in the infrared range. Finally, the Doppler temperature and recoil temperature for two transitions are also calculated, respectively. The Doppler temperatures for (2)1/2↔(1)1/2 and (1)3/2(1)1/2 transitions are 83.57 μK and 83.93 μK, and the recoil temperatures for two transitions are 226 nK and 227 nK, respectively. for two transitions are 226 nK and 227 nK, respectively.

    • 超冷分子离子接近静止的特性, 使得其可以用来精确地测量物理基本常数[1], 进行量子计算[2]和研究冷化学[3]. 近年来, 激光冷却双原子分子离子成为了原子分子物理和光学等领域的研究热点. 相对于原子和中性分子而言, 双原子分子阴离子能级结构比较复杂, 其激光冷却更难实现.

      中性分子的激光冷却研究已经比较广泛. 耶鲁大学的Shuman等[4]于2010年首次从实验上成功实现了SrF分子的横向冷却; 随后YO[5]和CaF[6]等分子也在实验上被证实适合激光冷却. 理论上众多分子也被预测为适合激光冷却的潜在分子, 例如MgH[7], OH[8]和CH[9]等自由基. Yzombard等[10]${\rm{C}}_2^ - $阴离子进行了激光冷却研究, 证实了双原子分子阴离子也适合激光冷却. 在我们前期工作中计算了OH[11]和SH[12]阴离子基态和低激发态的光谱性质及跃迁性质, 并给出了激光冷却两种阴离子方案. 激光冷却OH和SH阴离子所需的激光波长分别在绿光和青光范围内. 鲁东大学的Zhang等计算得到了NH[13]和BH[14]阴离子都具有非常大的弗兰克-康登因子以及较短的自发辐射寿命, 并分别预测了其激光冷却路径.

      氯化钾 (KCl) 可以制取钾盐, 也可以用于石油、橡胶和电镀工业, 还是人体需要的正常电解质. Huber和Herzberg[15]归纳了KCl分子基态的平衡核间距(Re)、谐振频率(ωe)、 非谐振频率(ωeχe)和离解能(De)的实验值, 其基态具有很大的势阱, 离解能达到了4.34 eV; 1997年, Ram等[16]从实验上得到了NaCl和KCl分子的分子常数. Seth等[17]用二次组态相互作用方法(QCISD)计算得到了KCl分子的势能曲线, 并拟合得到了基态的光谱常数, 其离解能为4.028 eV, 与实验值[15]的相对误差达到了7.2%, 说明早期针对KCl体系的从头算所选取的计算方法和基组有待改进. 到目前为止, 实验上和理论上对KCl阴离子的光谱性质和跃迁性质没有报道. 氯化物中有很多自由基适合激光冷却, 如BeCl[18], MgCl[19], CaCl[20], AlCl[21]和TlCl[22]等. 本文计算KCl阴离子的势能曲线, 并讨论在自旋-轨道耦合效应(SOC)下激光冷却KCl阴离子的可能性.

    2.   计算细节
    • 采用MOLPRO 2010程序[23]计算了KCl阴离子X2Σ+, A2Σ和B2Σ+态的电子结构. 在核间距2.0—50.0 Å上对KCl阴离子进行了单点能的计算. 首先在限制性Hartree-Fock (HF)方法基础上, 采用完全活性空间自洽场方法(CASSCF)[24]产生多参考波函数, 以CASSCF波函数作为基础构造了组态相互作用(CI)波函数, 并考虑Davidson修正进行MRCI + Q [25]的计算. 与KBe体系的电子结构计算一样[26], 本文对K选取了def2-AQZVPP-JKFI全电子基组[27], 对Cl选取了AV5Z-DK全电子基组[28].

      由于程序的限制, 在KCl阴离子电子结构的计算过程中采用C2V群. C2V群有4个不可约表示A1, B1, B2和A2. 在CASSCF计算中, 选取8个分子轨道作为活性轨道, 包含K (4s4p)和Cl (3s3p)轨道, 9个电子占据(4, 2, 2, 0)活性空间, 可以写为CAS (9, 8), 而K (3p)轨道为双占据的闭壳层轨道, 剩余的K (1s2s2p3s)和Cl (1s2s2p)轨道为冻结轨道; 在MRCI + Q计算中考虑了核-价电子(CV)关联效应, K(3p)轨道参与CV关联计算. 也就是说有15个电子参与了电子关联计算. 最后在MRCI + Q水平下通过Breit-Pauli算符[29]考虑了SOC效应, 得到Ω态的势能曲线.

      通过LEVEL 8.0程序[30]来拟合Λ-S态和Ω态的光谱常数, 并对(2)1/2, (3)1/2和(1)3/2电子态的弗兰克-康登因子 (fν′ν′′), 自发辐射速率 (Aν′ν′′)和自发辐射寿命 (τ)进行了预测.

    3.   结果与讨论
    • K和Cl原子亲合能的实验值分别为0.5015[31]和3.6127 eV[32], 故KCl阴离子最低的两个离解极限分别为K(2Sg) + Cl(1Sg)和K(2Pu) + Cl(1Sg), 来源于K原子4s→4p轨道的跃迁. 本文采用MRCI + Q方法计算了KCl阴离子最低两个离解极限对应的3个电子态的电子结构. X2Σ+态对应于最低离解通道K(2Sg) + Cl(1Sg), A2Π和B2Σ+对应于第二离解通道K(2Pu) + Cl(1Sg).

      考虑SOC效应后, A2Σ态分裂成1/2和3/2态, K(2P)原子态分裂为K(2P3/2)和K(2P1/2). KCl阴离子Ω电子态的离解极限见表1, 可以看出分裂的两条离解极限K(2P1/2) + Cl(1S0)和K(2P3/2) + Cl(1S0)与最低离解极限K(2S1/2) + Cl(1S0)的能量差分别为12997.94和13046.23 cm–1, 与K原子的2P1/22P3/2的实验值的相对误差仅为0.1%和0.03%, 本文计算结果与Moore的实验值[33]符合很好.

      原子态Ω态ΔE/cm–1
      计算值实验值[33]
      K(2S1/2) + Cl(1S0)(1)1/200
      K(2P1/2) + Cl(1S0)(2)1/212997.9412985.17
      K(2P3/2) + Cl(1S0)(3)1/2, (1)3/213046.2313042.89

      Table 1.  The dissociation relationship for the Ω states of KCl anion.

    • 在MRCI + Q水平下计算了KCl阴离子最低的3个Λ-S态(X2Σ+, A2Π和B2Σ+)的势能曲线, 考虑SOC效应后, 原先的3个Λ-S态分裂成4个Ω态, 包含了3个Ω = 1/2和1个Ω = 3/2. Λ-S态和Ω态的势能曲线分别绘于图1(a)图1(b)中. 可以看出4个Ω态都是束缚态, 而且4条曲线没有交叉现象.

      Figure 1.  Potential energy curves of KCl anion: (a) Λ-S states; (b) Ω states.

      4个Ω态的光谱常数列在表2中. (1)1/2对应X2Σ+态, 为KCl阴离子的基态. (2)1/2和(1)3/2对应A2Σ态, (3)1/2对应B2Σ+态. 本文计算得到(1)1/2, (2)1/2, (1)3/2和(3)1/2态的平衡核间距Re分别为2.8290, 2.7839, 2.7836和2.7550 Å. 可以看出激发态和基态的核间距相差不大; 且两个分裂态(2)1/2和(1)3/2的光谱常数基本相同: ΔRe = 0.0003 Å, Δωe = 0.01 cm–1, ΔBe = 0 cm–1以及ΔDe = 0.0042 eV. 同时预测了A2Σ态的分裂常数ASO = 13.38 cm–1, 计算结果表明SOC效应对KCl阴离子的光谱常数影响不大. 由于本文中离解极限能量差的计算值与实验值符合很好, 我们也相信KCl阴离子光谱常数的结果也是可靠的.

      Ω态对应的Λ-S态Reωe/cm–1Be/cm–1De/eVTe/cm–1
      (1)1/2X2Σ+2.8290212.340.11431.34830
      (2)1/2A2Π2.7839229.640.11801.79769375.30
      (1)3/2A2Π2.7836229.650.11801.80189388.68
      (3)1/2B2Σ+2.7550235.480.12051.386512746.21

      Table 2.  Spectroscopic parameters for the Ω states of KCl anion.

    • 偶极矩是离子的固有性质, 反映了离子的成键性质; 同时它在激光与分子离子的相互作用中起着很重要的作用, 较大的偶极矩便于激光冷却实验中外场的操作. 图2描绘了KCl阴离子的4个Ω态的偶极矩. 从图2中可以看出, 当核间距较大时, 4个Ω态的偶极矩都随着核间距的增加呈线性增加, 当核间距R增加至10 Å时, 4个Ω态的偶极矩都超过了20 deb. 这是由于KCl为阴离子体系, 4个电子态的离解极限都为离子对K + Cl. 基态(1)1/2在平衡位置Re处的偶极矩为3.079 deb, 表明该态具有离子键特点. KCl阴离子基态的偶极矩比SrF分子基态的偶极矩略小(SrF: μ = 3.5 deb[4]); 其基态偶极矩要略大于OH阴离子基态偶极矩(OH: μ = 2.859 deb[11]). 可以看出激光冷却KCl阴离子具有足够大的偶极矩.

      Figure 2.  Dipole moments (DMs) of KCl- anion.

      为了预测KCl阴离子的弗兰克-康登因子(FCFs)和自发辐射速率(Aν′ν″), 本文还计算了(2)1/2↔(1)1/2, (1)3/2↔(1)1/2, (3)1/2↔(1)1/2, (3)1/2↔(2)1/2和(3)1/2↔(1)3/2跃迁的跃迁偶极矩, 如图3所示. 由于(3)/1/2和(1)3/2态来源于同一离解极限, 在核间距趋于无穷远时不存在原子态之间的跃迁; 而(2)/1/2和(3)1/2态来源于K(2P1/2) + Cl(1S0)和K(2P3/2) + Cl(1S0), (3)1/2↔(2)1/2在无穷远处的跃迁源自K原子2P两个分裂态的跃迁(2P1/2)↔(2P3/2), K为轻原子, 2P两个分裂态的跃迁可以忽略. 故在图3中可以看出, 在核间距约为10 Å以后, (3)1/2↔(1)3/2和(3)/1/2↔(2)1/2跃迁的跃迁偶极矩趋近于零. 另一方面, 当核间距R = 50 Å时, (2)1/2↔(1)1/2, (1)3/2↔(1)1/2, (3)1/2↔(1)1/2跃迁的跃迁偶极矩都趋于7.85 D, 源自于K原子2S1/22P1/2, 3/2跃迁.

      Figure 3.  Transition dipole moments (TDMs) of KCl anion

    • 在实施激光冷却KCl阴离子的过程中, 可以考虑选择(2)1/2↔(1)1/2和(1)3/2↔(1)1/2直接跃迁以及(3)1/2↔(1)1/2三电子能级跃迁两种方案来构造能级系统. 候选分子离子是否适合激光冷却主要取决于FCFs是否具有高对角化分布. 本文计算得到了(2)1/2↔(1)1/2, (1)3/2↔(1)1/2, (3)1/2↔(1)1/2, (3)1/2↔(2)1/2和(3)1/2↔(1)3/2跃迁的跃迁性质, 表3表4中列出了以上5种跃迁的FCFs (fν′ν″), 自发辐射速率Aν′ν″和自发辐射寿命τ.

      跃迁ν′′0123
      (2)1/2↔(1)1/2Aν′ν′′/s–11.9384(7)a2.3044(6)1.7867(5)1.1906(4)
      ν′ = 0fν′ν′′0.88160.10900.00880.0006
      τ/ns45.7
      Aν′ν′′/s–12.5793(6)1.4757(7)4.0633(6)5.0295(5)
      ν′ = 1fν′ν′′0.11280.66870.19140.0246
      τ/ns45.5
      Aν′ν′′/s–11.3368(5)4.7122(6)1.0816(7)5.3057(6)
      ν′ = 2fν′ν′′0.00560.20520.48830.2490
      τ/ns45.4
      (1)3/2↔(1)1/2Aν′ν′′/s–11.9451(7)2.3276(6)1.8184(5)1.2242(4)
      ν′ = 0fν′ν′′0.88080.10960.00890.0006
      τ/ns45.5
      Aν′ν′′/s–12.6063(6)1.4777(7)5.1089(5)4.7631(4)
      ν′ = 1fν′ν′′0.11340.66680.19240.0249
      τ/ns45.4
      Aν′ν′′/s–11.3578(5)4.7578(6)1.0803(7)5.3483(6)
      ν′ = 2fν′ν′′0.00570.20630.48570.2499
      τ/ns45.2
      注: a1.9384(7)表示1.9384 × 107.

      Table 3.  FCFs, spontaneous emission rates Aν′ν′′ and spontaneous radiative lifetime τ for the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 transitions.

      跃迁f00A0/s–1τ0/s
      (3)1/2↔(1)1/20.71222.6535(7)3.77(–8)
      (3)1/2↔(2)1/20.94842.3716(5)4.22(–6)
      (3)1/2↔(1)3/20.94902.3435(5)4.27(–6)
      注: a1.9384(7)表示1.9384 × 107.

      Table 4.  FCFs, total emission rates A0 and τ for the (3)1/2↔(1)1/2, (3)1/2↔(2)1/2 and (3)1/2↔(1)3/2 transitions.

    • 表3中得到, (2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁都具有高对角分布的f00, 分别为0.8816和0.8808, 比NH+离子12Π↔12Σ+跃迁的FCF值要大(f00 = 0.821)[13], 且对于两种跃迁f00, f01f02之和基本等于1, 保证了两种跃迁的准闭合循环, 满足了激光冷却的第一个条件; 此外(2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁的总辐射速率分别为2.1880 × 107和2.1974 × 107 s–1, 即(2)1/2和(1)3/2激发态的自发辐射寿命为45.7和45.5 ns, 可以保证循环跃迁的快速进行.

      在此基础上构建了(2)1/2↔(1)1/2和(1)3/2↔(1)1/2准闭合循环跃迁系统来对KCl阴离子进行激光冷却. 冷却方案见图4(a)图4(b), 图中的实线表示驱动激光, 虚线表示自发辐射. 驱动(2)1/2↔(1)1/2准闭合循环跃迁需要选用主激光波长λ00 = 1065.77 nm, 增加两束抽运激光来增强激光冷却的效果, 波长分别为λ10 = 1090.13 nm和λ21 = 1087.76 nm; 同样驱动(1)3/2↔(1)1/2准闭合循环跃迁需要选用一束主激光和两束抽运激光, 波长分别为λ00 = 1064.24 nm, λ10 = 1088.54 nm和λ21 = 1086.17 nm. 驱动KCl阴离子的两种跃迁所需的激光波长都在深红外区域. 同时预测了激光驱动(2)1/2↔(1)1/2和(1)3/2↔(1)1/2能级跃迁的散射光子数目, 当加入了两束激光λ00λ10时, 理论上能散射光子数目为Nscat = 1/f03+ ≈ 1500, 再加上λ21可以散射更多数目的光子.

      Figure 4.  Proposed laser cooling scheme of KCl anion: (a) Using (2)1/2↔(1)1/2 transition; (b) using (1)3/2↔(1)1/2 transition.

    • 表4中可以看到, (3)1/2↔(1)1/2跃迁也具有较大的FCF (f00 = 0.7122)和很小的自发辐射寿命 (τ = 0.377 μs), 但在循环跃迁中存在中间态(2)1/2和(1)3/2, 能否适合激光冷却就需要考虑中间态的存在是否影响循环跃迁的闭合性. 本文通过振动分支损失比来分析跃迁的闭合性. 振动分支损失比可以表示为 η1 = γ1/γΣη2 = γ2/γΣ, 在本文中γ1, γ2γΣ分别表示(3)1/2↔(2)1/2, (3)1/2↔(1)3/2和(3)1/2↔(1)1/2跃迁的总自发辐射速率, 通过表4可以得到η1 = 8.94 × 10–3η2 = 8.83 × 10–3. 实验上实现了YO分子的激光冷却[5], YO分子中间态的分支损失比小于4 × 10–4, 本文计算得到的分支损失比约为YO分子的20倍. 由于激光驱动(3)1/2↔(1)1/2能级跃迁的散射光子数目只有约150个, 而中间态的存在还会损失部分散射光子. 所以我们认为不需要考虑构建三电子能级跃迁来激光冷却KCl阴离子.

    • (2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁产生的最大加速度可以由$a = \hbar {k_{\rm{B}}}\varGamma /(4\lambda m)$计算得到[34], 其中$\hbar $表示约化普朗克常数, kB为玻尔兹曼常数, λ为激光驱动跃迁的主激光波长, $\varGamma = 1/(2{\text{π}}\tau )$, 表示线宽, τ为激发态的自发辐射寿命. 计算得到两种跃迁冷却KCl阴离子的最大加速度分别为4379.3和4404.9 m/s2. 由于KCl阴离子比OH自由基的FCFs略小[8], KCl阴离子激发态的自发辐射寿命也比OH自由基的要更短[8], 其操作可以和OH自由基类似. 首先采用缓冲气体冷却技术得到较低温度的离子束源. 分子离子温度达到1 K时可直接激光冷却分子[4]. 本文预测了采用两种跃迁冷却KCl阴离子至1 K所需的最小减速距离分别为0.0368和0.0365 m, 可以看出具有很小的减速距离, 便于实验的操作; 当KCl阴离子冷却至1 K后, 利用(2)1/2↔(1)1/2和(1)3/2↔(1)1/2准闭合循环跃迁产生的自发辐射力对离子束减速.

      为了评估冷却效果, 本文预测了激光冷却KCl阴离子的多普勒温度以及反冲温度. 多普勒温度可以由下面公式计算: ${T_{{\rm{Doppler}}}} = h/(4{k_{\rm{B}}}{\text{π}}\tau )$[35], 其中h为普朗克常数. (2)1/2和(1)3/2激发态的自发辐射寿命分别为45.7和45.5 ns, 可以计算出采用(2)1/2↔(1)1/2和(1)3/2↔(1)1/2循环跃迁进行激光冷却KCl阴离子的多普勒温度分别为83.57和83.93 μK. 而反冲温度可以由下面的公式计算: ${T_{{\rm{recoil}}}} = {h^2}/(m{k_{\rm{B}}}{\lambda ^2})$[35]. 计算得到采用(2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁进行激光冷却KCl阴离子的反冲温度分别可达226和227 nK. 理论预测反冲温度约为多普勒温度的2.7 × 10–3倍.

    4.   结 论
    • 采用MRCI + Q方法计算了KCl阴离子前两个离解极限的3个Λ-S态的电子结构, 在计算过程中考虑了核-价电子关联效应. 在MRCI + Q水平下考虑了SOC效应, 计算了Ω态的势能曲线和跃迁偶极矩. 本文报道了KCl阴离子Ω态的光谱常数和跃迁性质.

      计算得到(2)1/2↔(1)1/2和(1)3/2↔(1)1/2跃迁为直接跃迁, 具有高对角分布的FCF, 分别为0.8816和0.8808; 同时(2)1/2和(1)3/2激发态有很短的自发辐射寿命, 分别为45.7和45.5 ns. 本文给出了(2)1/2↔(1)1/2和(1)3/2↔(1)1/2准闭合能级循环跃迁进行激光冷却KCl阴离子的方案. 分别选取3束激光驱动(2)1/2↔(1)1/2和(2)1/2↔(1)1/2跃迁, 主激光波长分别为1065.77和1064.24 nm. 最后预测了两种方案进行激光冷却KCl阴离子的多普勒温度和反冲温度. 反冲温度约为多普勒温度的2.7 × 10–3倍.

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