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Feshbach resonance is a fundamental phenomenon in cold atomic physics, where interatomic interactions can be precisely changed into a scattering resonance by varying an external magnetic field. This effect plays a crucial role in ultracold atomic experiments, enabling the control of interaction strength, the formation of molecular bound states, and the realization of strongly correlated quantum systems. With the rapid development of cold atom experiments, numerous Feshbach resonances corresponding to different partial waves, such as s-wave, p-wave, and even higher partial wave ones, have been experimentally identified. While s-wave resonances have been widely utilized due to their isotropic nature and strong coupling, and higher partial-wave resonances (including p-wave and d-wave resonances), provide unique opportunities for exploring anisotropic interactions and novel quantum phases. In this study, by using the multichannel quantum defect theory (MQDT), we predict that two d-wave Feshbach resonances are existent in 7Li at 1039.24 G and 1055.64 G, repectively. Physical properties of the two resonances are presented, such as the resonance width and closed channel dimer energy. In addition, we optimize the computational parameters by using the Nelder-Mead algorithm and investigate the possible resonance splitting induced by dipole-dipole interactions in higher partial waves. The presence of these d-wave resonances at high magnetic fields provides a new platform for investigating the interplay between higher-order partial wave interactions and quantum many-body effects. Our results provide opportunities for investigating the effects of higher partial wave Feshbach resonances in high magnetic fields. Our theoretical predictions thus serve as a useful reference for future experimental studies of higher-order resonance phenomena in lithium and other atomic species.
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Keywords:
- Feshbach resonance /
- MQDT calculation
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图 1 7Li原子内态能量随外磁场的变化, 其中$ E_6\equiv $$ E_{\text{VdW}}/4 $, van der Waals能量为$ E_{\text{VdW}}\equiv \dfrac{\hbar^2}{2\mu R^2_{\text{VdW}}} $, van der Waals长度为$ R_{\text{VdW}} \equiv\dfrac{1}{2}\left( \dfrac{2\mu C_6}{\hbar^2}\right)^{\frac{1}{4}} $
Figure 1. The internal state energies of 7Li atoms v.s. the external magnetic field B, where $ E_6\equiv E_{\text{VdW}}/4 $, and $ E_{\text{VdW}}\equiv \dfrac{\hbar^2}{2\mu R^2_{\text{VdW}}} $ is the van de Waals energy and $ R_{\text{VdW}} \equiv $$ \dfrac{1}{2}\left( \dfrac{2\mu C_6}{\hbar^2}\right)^{\frac{1}{4}} $ is the van de Waals length.
图 2 7Li原子d波广义散射长度$ a_2 $在两个共振磁场1039.24 G和1055.64 G附件随磁场的变化处. 图中黑色虚线标注出了共振的位置
Figure 2. The d-wave generalized scattering length $ a_2 $ for 7Li atoms v.s. the external magnetic field B in the vicinity of two resonances at 1039.24 G and 1055.64 G. The black dashed lines represent the resonance points.
图 3 闭通道中低能束缚态能量$ E_{\mathrm{b}} $在两个共振点附件随磁场的变化. 蓝色点为计算的数据点, 绿色线是线性拟合的数据, 红色虚线标记共振点的位置. 其中$ E_6\equiv E_{\text{VdW}}/4 $, van der Waals能量为$ E_{\text{VdW}}\equiv \dfrac{\hbar^2}{2\mu R^2_{\text{VdW}}} $, van der Waals长度为$ R_{\text{VdW}} \equiv\dfrac{1}{2}\left( \dfrac{2\mu C_6}{\hbar^2}\right)^{\frac{1}{4}} $
Figure 3. The energy $ E_{\mathrm{b}} $ of the low energy bound state in the closed channels v.s. the external magnetic field B in the vicinity of the two resonances. Here $ E_6\equiv E_{\text{VdW}}/4 $, and $ E_{\text{VdW}}\equiv \dfrac{\hbar^2}{2\mu R^2_{\text{VdW}}} $ is the van de Waals energy and $ R_{\text{VdW}} \equiv $$ \dfrac{1}{2}\left( \dfrac{2\mu C_6}{\hbar^2}\right)^{\frac{1}{4}} $ is the van de Waals length.
图 4 磁偶极矩相互作用导致的共振点劈裂. 两子图中$ a_2 $发散的磁场从小到大分别对应于$ m = 0 $、$ m = 1 $、不考虑劈裂、$ m = 2 $的共振点位置
Figure 4. The resonance splittings due to the magnetic dipole-dipole interaction. In the plots the divergences of $ a_2 $ correspond to the resonances of $ m = 0 $, $ m = 1 $, no splitting, and $ m = 2 $ from small B to big B.
图 B1 三个参数$ a_{\mathrm{s}} $、$ a_{\mathrm{t}} $以及$ C_6 $根据迭代次数的变化
Figure B1. The three parameters $ a_{\mathrm{s}} $、$ a_{\mathrm{t}} $ and $ C_6 $ vs the number of iterations for optimization.
图 B2 输出函数$ f(B) $的迭代情况
Figure B2. The iteration behavior of the output function $ f(B) $.
表 1 7Li的d-波Feshbach共振
Table 1. D-wave Feshbach resonances in 7Li
B0 (G) $ \text{a}_{2, {\mathrm{bg}}} $/$ \bar a_{2} $ Δ (G) $ \delta\mu/\mu_0 $ sres ζ 1039.24 1.529 $ 6.397 \times 10^{-4} $ 3.924 $ 2.19 \times 10^{-4} $ $ 6.49 \times 10^{-11} $ 1055.64 1.529 0.0615 3.953 0.021 $ 6.29 \times 10^{-9} $ 
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表 3 不同原子的Feshbach共振
Table 3. Feshbach resonances in other alkali atoms.
元素 入射通道 l $ B_{0} $ (G) $ a_{l, {\mathrm{bg}}}/\bar a_l $ Δ (G) 7Li ab p 977.12 1.1 121.1 d 72.9 2.63 148.3 7Li bc p 514.3 –1.235 –61.07 1083 –1.3 –15.85 23Na aa d 578.7 –1.179 –0.04575 656.2 –8.462 –0.1149 23Na ab s 977.1 1.464 $ 3.689\times 10^{-3} $ p 880.1 1.753 0.1491 d 694.3 –1.147 –0.05086 23Na bc s 1017 –1.152 –0.0523
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