• Acta Physica Sinica
  • Vol. 69, Issue 1, 010303-1 (2020)
Li Wang1, Jing-Si Liu2, Ji Li3, Xiao-Lin Zhou4..., Xiang-Rong Chen1, Chao-Fei Liu5,* and Wu-Ming Liu6,7,*|Show fewer author(s)
Author Affiliations
  • 1College of Physics, Sichuan University, Chengdu 610065, China
  • 2Beijing Jingshan School Chaoyang Branch School, Beijing 100012, China
  • 3College of Physics, Taiyuan Normal University, Jinzhong, 030619, China
  • 4School of Physics and Electronic engineering, Sichuan Normal University, Chengdu 610101, China
  • 5School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
  • 6Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 7School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
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    DOI: 10.7498/aps.69.20191648 Cite this Article
    Li Wang, Jing-Si Liu, Ji Li, Xiao-Lin Zhou, Xiang-Rong Chen, Chao-Fei Liu, Wu-Ming Liu. The research progress of topological properties in spinor Bose-Einstein condensates[J]. Acta Physica Sinica, 2020, 69(1): 010303-1 Copy Citation Text show less

    Abstract

    Most of the atoms that realize Bose-Einstein condensation have internal spin degree of freedom. In the optical potential trap, the internal spin of the atom is thawed, and the atom can be condensed into each hyperfine quantum state to form the spinor Bose-Einstein condensate. Flexible spin degrees of freedom become dynamic variables related to the system, which can make the system appear novel topological quantum states, such as spin domain wall, vortex, magnetic monopole, skymion, and so on. In this paper, the experimental and theoretical study of spinor Bose-Einstein condensation, the types of topological defects in spinor Bose-Einstein condensate, and the research progress of topological defects in spinor two-component and three-component Bose-Einstein condensate are reviewed.
    $\left[ {{{\hat \psi }_m}\left( {{r}} \right),{\hat \psi}_n^{\dagger} \left( {{{r}}'} \right)} \right] = {\delta _{nm}}\delta \left( {{{r}} - {{r}}'} \right),$ (1)

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    $\left[ {{{\hat \psi }_n}\left( {{r}} \right),{{\hat \psi }_m}\left( {{{r}}'} \right)} \right] = 0,$ (2)

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    $\left[ {{\hat \psi}_m^{\dagger} \left( {{r}} \right),{\hat \psi}_n^{\dagger} \left( {{r}} \right)} \right] = 0.$ (3)

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    $ A^FM(r,r)=m1,m2=ffF,M|f,m1;f,m2ψ^m1(r)ψm2(r).$ (4)

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    $ V^F=12drdrv(F)(rr)×M=FFA^FM(r,r)A^FM(r,r),$ (5)

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    $\sum _{{F}} \sum_{{{M}} = - F}^F \hat A_{{\rm{FM}}}^{\dagger} ( {{r}},{{r'}} ){\hat A_{{\rm{FM}}}} \left({{{r}},{{r'}}} \right)\! =\! :\hat n ( {{r}} )\hat n ( {{{r'}}} ):,$ (6)

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    ${\rm{\hat n}}\left( r \right) \equiv \mathop \sum \limits_{m = - f}^f {\hat \psi}_m^{\dagger} \left( {{r}} \right){{\hat \psi}_m}\left( {{r}} \right),$ (7)

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    ${v^{\left( F \right)}} = {g_{\rm{F}}}\delta \left( {{r}} \right),$ (8)

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    ${g_{\rm{F}}} = \frac{{4{\text{π}}{\hbar ^2}}}{m}{a_{\rm{F}}},$ (9)

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    ${\hat V^F} = \frac{{{g_{\rm{F}}}}}{2}\int {\rm{d}}{{r}}\mathop \sum \limits_{M = - F}^F \hat A_{{\rm{FM}}}^{\dagger} \left( {{r}} \right){\hat A_{{\rm{FM}}}}\left( {{r}} \right),$ (10)

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    $ \langle 0,0|f,{m_1};f,{m_2} \rangle = {\delta _{{m_1} + {m_2},0}}\frac{{\left( { - 1} \right){f^{ - {m_1}}}}}{{\sqrt {2f + 1} }},$ (11)

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    ${\hat A_{00}}\left( {{{r}},{{r'}}} \right) \!=\! \frac{1}{{\sqrt {2f \!+\! 1} }}\!\!\mathop \sum \limits_{ - f}^f \!{\left( { - 1} \right)^{f - m}}{{\hat \psi}_m}\left( {{r}} \right){{\hat \psi}_{ - m}}\left( {{{r}}'} \right).$ (12)

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    ${\hat V_0} = \frac{{{g_0}}}{2}\int{{\rm{d}}^3}{{r}}\hat A_{00}^{\dagger} \left( {{r}} \right){\hat A_{00}}\left( {{r}} \right),$ (13)

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    $V^2=g22d3rM=22A^2M(r)A^2M(r)=g22d3[:n^2(r):A^00(r)A^00(r)],$ (14)

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    $\hat V = \mathop \int \limits {{\rm{d}}^3}\left[ {\frac{{{g_2}}}{2}:{{\hat n}^2}\left( {{r}} \right): + \frac{{{g_0} - {g_2}}}{2}\hat A_{00}^{\dagger} \left( {{r}} \right){{\hat A}_{00}}\left( {{r}} \right)} \right],$ (15)

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    $\hat V = \frac{1}{2}\int {{\rm{d}}^3}\left[ {{c_0}:{{\hat n}^2}\left( {{r}} \right): + {c_1}:{{{{\hat F}}}^2}\left( {{r}} \right):} \right],$ (16)

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    ${c_0} = \frac{{{g_0} + 2{g_2}}}{3},$ (17)

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    ${c_1} = \frac{{{g_2} - {g_0}}}{3},$ (18)

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    $V^=12d3[c0:n^2(r):+c1:F^2(r):+c2A^00(r)A^00(r)],$ (19)

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    ${c_0} = ({{4{g_2} + 3{g_4}}})/{7},$ (20)

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    ${c_1} = ({{{g_4} - {g_2}}})/{7},$ (21)

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    ${c_2} = ({{7{g_0} - 10{g_2} + 3{g_4}}})/{7},$ (22)

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    $V^=12d3[c0:n^2(r):+c1:F^2(r):+c2A^00(r)A^00(r)+c3M=22A^2M(r)A^2M(r)],$ (23)

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    ${c_0} = ({{9{g_4} + 2{g_6}}})/{{11}},$ (24)

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    ${c_1} = ({{{g_6} - {g_4}}})/{{11}},$ (25)

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    ${c_2} = ({{11{g_0} - 21{g_4} + 10{g_6}}})/{{11}},$ (26)

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    ${c_3} = (11g_0 - 18g_4 + 7g_6)/11.$ (27)

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    $\left\{ {Sx=2|ψ1||ψ2||Ψ|2cos(θ1θ2)Sy=2|ψ1||ψ2||Ψ|2sin(θ1θ2)Sz=|ψ1|2|ψ2|2|Ψ|2} \right.,$ (28)

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    $\left\{ {Sx=sinθcosφSy=sinθsinφSz=cosθ} \right.,$ (29)

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    $q\left( r \right) = \frac{1}{{8{\text{π}} }}{\varepsilon ^{ij}}{\bf{S}} \cdot {\partial _i}{{S}} \times {\partial _j}{{S}},$ (30)

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    $Q(Sx,Sy,Sz)=18πεijSiS×jSdr=14π|SxSySzSxxSyxSzxSxySyySzy|dr=14π|SxSzSySxxSzxSyxSxySzySyy|dr=Q(Sx,Sz,Sy).$ (31)

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    $\mathop \oint \nolimits_C {{{v}}_{\rm{s}}} \cdot {\rm{d}}{{l}} = {n_{\rm{w}}}\kappa,$ (32)

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    $\left( {ψ1ψ0ψ1} \right) = \sqrt n \left( {cos2β22eiϕsinβ2cosβ2e2iϕsin2β2} \right),$ (33)

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    $\left( {ψ1ψ0ψ1} \right) = \sqrt n \left( {eiϕcos2β212sinβeiϕsin2β2} \right).$ (34)

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    $(ψ1ψ0ψ1)=neiθU(α,β,γ)(100)=nei(θγ)(eiαcos2β212sinβeiαsin2β2).$ (35)

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    $ \left( {ψ1ψ0ψ1} \right) = \left( (cosf(r)n2icosθsinf(r)n2)22i(cosf(r)n2icosθsinf(r)n2)sinf(r)n2sinθeiϕsin2f(r)n2sin2θe2iϕ \right), $ (36)

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    Li Wang, Jing-Si Liu, Ji Li, Xiao-Lin Zhou, Xiang-Rong Chen, Chao-Fei Liu, Wu-Ming Liu. The research progress of topological properties in spinor Bose-Einstein condensates[J]. Acta Physica Sinica, 2020, 69(1): 010303-1
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