Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11834003, 11874151)
Received Date:18 March 2021
Accepted Date:12 April 2021
Available Online:07 June 2021
Published Online:20 August 2021
Abstract:Laser cooling and trapping of neutral molecules has made substantial progress in the past few years. On one hand, molecules have more complex energy level structures than atoms, thus bringing great challenges to direct laser cooling and trapping; on the other hand, cold molecules show great advantages in cold molecular collisions and cold chemistry, as well as the applications in many-body interactions and fundamental physics such as searching for fundamental symmetry violations. In recent years, polar diatomic molecules such as SrF, YO, and CaF have been demonstrated experimentally in direct laser cooling techniques and magneto-optic traps (MOTs), all of which require a comprehensive understanding of their molecular internal level structures. Other suitable candidates have also been proposed, such as YbF, MgF, BaF, HgF or even SrOH and YbOH, some of which are already found to play important roles in searching for variations of fundamental constants and the measurement of the electron’s Electric Dipole Moment (eEDM). As early as 2004, the CaH molecule was selected as a good candidate for laser cooling and magneto-optical trapping. In this article, we first theoretically investigate the Franck?Condon factors of CaH in the ${{\rm{A}}}^{2}\Pi _{1/2}\leftarrow {{\rm{X}}}^{2}\Sigma _{1/2}$ transition by the Morse potential method, the closed-form approximation method and the Rydberg-Klein-Rees method separately, and prove that Franck?Condon factor matrix between $ {\mathrm{X}}^{2}\Sigma _{1/2} $ state and $ {\mathrm{A}}^{2}\Pi _{1/2} $state is highly diagonalized, and indicate that sum of f00, f01 and f02 for each molecule is greater than 0.9999 and almost 1 × 104 photons can be scattered to slow the molecules with merely three lasers. The molecular hyperfine structures of $ {X}^{2}\Sigma _{1/2} $, as well as the transitions and associated hyperfine branching ratios in the ${{\rm{A}}}^{2}\Pi _{1/2}\left(J=1/2, \mathrm{ }+\right)\leftarrow {{\rm{X}}}^{2}\Sigma _{1/2}\left(N=1, \mathrm{ }-\right)$ transition of CaH, are examined via the effective Hamiltonian approach. According to these results, in order to fully cover the hyperfine manifold originating from $ |X, \mathrm{ }N=1, -\rangle $, we propose the sideband modulation scheme that at least two electro-optic modulators (EOMs) should be required for CaH when detuning within 3Γ of the respective hyperfine transition. In the end, we analyze the Zeeman structures and magnetic g factors with and without J mixing of the $ |X, \mathrm{ }N=1, -\rangle $ state to undercover more information about the magneto-optical trapping. Our work here not only demonstrates the feasibility of laser cooling and trapping of CaH, but also illuminates the studies related to spectral analysis in astrophysics, ultracold molecular collisions and fundamental physics such as exploring the fundamental symmetry violations. Keywords:cold molecules/ laser cooling/ CaH molecule/ Franck-Condon factors
表2用三种方法(闭合近似法、莫尔斯势法和RKR反演法)计算的CaH分子的部分F-C因子 Table2.Calculated Franck-Condon factors of CaH by the closed-form approximation method, the Morse potential method and the RKR inversion method.
表3CaH分子$ {\mathrm{X}}^{2}\Sigma _{1/2} $态和$ {\mathrm{A}}^{2}\Pi _{1/2} $间跃迁波长的计算值和实验值, 括号内的数值代表最后位的不确定度(标准偏差) Table3.Comparison between the calculated and experimental results of the transition wavelengths between $ {\mathrm{X}}^{2}\Sigma _{1/2} $ and $ {\mathrm{A}}^{2}\Pi _{1/2} $ states of CaH. Numbers in parentheses indicate the uncertainty (standard deviation) in the last figures.
根据上面的讨论, 我们确定了CaH在${\mathrm{A}}^{2}\Pi _{1/2}\to $$ {\mathrm{X}}^{2}\Sigma _{1/2}$跃迁上具有高度对角化的F-C因子. 因此, 我们选择$ {\mathrm{A}}^{2}\Pi _{1/2} $态与$ {\mathrm{X}}^{2}\Sigma _{1/2} $态为构建激光冷却准封闭跃迁循环的上下两个电子态. 图2是本文设计的电子振动态激光冷却方案, 其中的蓝线表示泵浦激光, 绿色虚线表示从$ {\mathrm{A}}^{2}\Pi _{1/2} $态自发辐射回$ {\mathrm{X}}^{2}\Sigma _{1/2} $态各振动能级的衰变率, 即F-C因子. 由于f00 + f01 + f02 > 0.9999, 在这个准封闭跃迁循环当中可以散射几乎1 × 104个光子. 这里, $ {\lambda }_{\upsilon {\upsilon }'} $表示从$ {\mathrm{X}}^{2}\Sigma _{1/2} $态到$ {\mathrm{A}}^{2}\Pi _{1/2} $态不同振动能级的跃迁波长, 其中主跃迁$ {\lambda }_{00} $ = 692.996 nm. 图 2 CaH分子准封闭跃迁循环的激光冷却方案. 其中的蓝线表示泵浦激光, $ {\lambda }_{\upsilon {\upsilon }'} $表示泵浦激光的跃迁波长, 绿色虚线表示从$ {\mathrm{A}}^{2}\Pi _{1/2} $态自发辐射的衰变率, 即F-C因子$ {f}_{{\upsilon }'\upsilon } $ Figure2. Proposed scheme to create a quasi-closed cycling transition for laser cooling of CaH. Blue solid lines indicate the laser-driven transitions at the wavelengths $ {\lambda }_{\upsilon {\upsilon }'} $, while green dotted lines indicate the spontaneous decays from the $ {\mathrm{A}}^{2}\Pi _{1/2} $ state along with the corresponding F-C factors $ {f}_{{\upsilon }'\upsilon } $.
表4CaH分子$ {\mathrm{X}}^{2}\Sigma _{1/2} $态的转动常数和超精细结构常数 Table4.Rotational and hyperfine structure parameters for the $ {\mathrm{X}}^{2}\Sigma _{1/2} $state of CaH.
洪特情况(b)体系下的基矢可以写作$|\eta, N, $$ S, J, I, F, {M}_{F}\rangle$, $ N $表示除了电子自旋外的总角动量, $ S $表示总的电子自旋, $ J $表示包含电子自旋的总角动量, $ I $表示核自旋, $ F $表示包含核自旋后的总角动量, $ {M}_{F} $表示$ F $在z轴方向的投影, $ \eta $表示其余的量子数. $ {H}_{\mathrm{e}\mathrm{f}\mathrm{f}} $的每一项矩阵元在该表象下可以表示为
表6CaH分子$ |X, \upsilon =0, \mathrm{ }N=1, \mathrm{ }-\rangle $态理想的组分和考虑J混合的组分 Table6.Nominal labels and actual labels due to J mixing for the $ |X, N=1, -\rangle $state of CaH molecules.
为了构建准封闭的跃迁循环, 需要同时泵浦$ |X, \upsilon =0, \mathrm{ }N=1, \mathrm{ }-\rangle $态下的四个子能级, 四束泵浦光的相对强度对应于表7中$|A, \upsilon =0, \mathrm{ }R=0, J= $$ 1/2\rangle$态自发辐射到$ |J=1/2, F=0\rangle $态、$|J=1/2, F= $$ 1\rangle$态、$ |J=3/2, F=1\rangle $态和$ |J=3/2, F=2\rangle $态的相对概率. 得到了相应能级的超精细结构和跃迁分支比后, 便可以设计边带调制方案使主冷却光同时覆盖$ |X, \upsilon =0, \mathrm{ }N=1, \mathrm{ }-\rangle $态的四个子能级. 对于CaH分子, 衰减率Γ = 1/τ = 2π × 4.794 MHz, 而饱和光强IS = πhcΓ/(3λ3) = 1.88 mW/cm2. 之前被研究比较多的CaF分子具有更小的激发态寿命(19.2 ns)和更短的波长(606 nm)[32]. 因此相较于CaF的4.87 mW/cm2的饱和光强, 本文研究的CaH的饱和光强更小, 其用于磁光囚禁所需要的激光器功率要求就更小. 此外, CaF所需的波段(λ00~606 nm)还没有容易获得的二极管激光器, 仍需要在泵浦激光作用下产生, 因而代价较高, 而CaH所需的波长(λ00~693 nm)对应的二极管激光器性价比高, 且已经实现商用化. 边带调制方案如图4所示, 黑实线表示基频光, 蓝线对应四个子能级的中心频率. 该方案需要用到两个电光调制器(EOM), 图中的红色实线和黑色虚线分别对应了两个EOM产生的边带, 调制频率为: fmod1 = 994.25 MHz和fmod2 = 941.25 MHz, 失谐量都在3Γ内. 图 4 能同时覆盖CaH分子$ |X, \mathrm{ }N=1, -\rangle $态下四个子能级的边带调制方案. 中间的黑色实线表示基频光; 蓝色实线表示超精细能级的中心频率; 黑色虚线与红色实线分别表示两个EOM的边带, 调制频率分别为fmod1 = 994.25 MHz 和 fmod2 = 941.25 MHz. 每个边带的失谐量均控制在3Γ内 Figure4. Proposed sideband modulation scheme to simultaneously cover all hyperfine transitions originating from the $ |X, \mathrm{ }N=1, -\rangle $ state of CaH. The black solid line in the middle indicates the fundamental laser frequency, while the blue solid line corresponds to the central frequency of the hyperfine transitions. The black dash line and the red solid line represent the sidebands of two EOMs respectively with the modulating frequencies fmod1 = 994.25 MHz and fmod2 = 941.25 MHz. All the hyperfine levels are well addressed for detuning within 3Γ of the respective hyperfine transition.
图 1 CaH分子高度对角化的F-C因子矩阵
图 2 CaH分子准封闭跃迁循环的激光冷却方案. 其中的蓝线表示泵浦激光, 
图 3 CaH转动能级和超精细能级的能级分裂以及准封闭的跃迁循环. 根据选择定则, 由
图 4 能同时覆盖CaH分子
图 5 CaH在
