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Acta Physica Sinica
Vol. 69, Issue 1, 010303-1 (2020)

The research progress of topological properties in spinor Bose-Einstein condensates

Li Wang¹, Jing-Si Liu², Ji Li³, Xiao-Lin Zhou⁴, Xiang-Rong Chen¹, Chao-Fei Liu^5、*, and Wu-Ming Liu^6、7、*

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

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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.

Keywords

magnetic monopole skymion spin domain wall spinor Bose - Einstein condensation vortex

$\left[ {{{\hat \psi }_m}\left( {{r}} \right),{\hat \psi}_n^{\dagger} \left( {{{r}}'} \right)} \right] = {\delta _{nm}}\delta \left( {{{r}} - {{r}}'} \right),$

(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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$ \begin{split}{\hat A_{{\rm{FM}}}}\left( {{{r}},{{r'}}} \right) = & \mathop \sum \limits_{{m_1},{m_2} = - f}^f \langle F,M|f,{m_1};f,{m_2} \rangle \\ & {\hat \psi}_{{m_1}}^{}\left( {{r}} \right)\psi _{{m_2}}^{}\left( {{{r}}'} \right).\end{split}$

(4)

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$ \begin{split} {\hat V^F} =\, & \frac{1}{2} \mathop \int {\rm{d}}{{r}}\mathop \int \limits {\rm{d}}{{r}}'{v^{\left( F \right)}}\left( {{{r}} - {{r}}'} \right)\\ & \times\mathop \sum \limits_{M = - F}^F \hat A_{{\rm{FM}}}^{\dagger} \left( {r,r'} \right){\hat A_{{\rm{FM}}}}\left( {{{r}},{{r}}'} \right), \end{split}$

(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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$\begin{split} {\hat V_2} \, & = \frac{{{g_2}}}{2}\int {{\rm{d}}^3}{{r}}\mathop \sum \limits_{M = - 2}^2 \hat A_{2M}^{\dagger} \left( {{r}} \right){\hat A_{2M}}\left( {{r}} \right) \\ & = \frac{{{g_2}}}{2}\int {{\rm{d}}^3}\left[ {:{{\hat n}^2}\left( {{r}} \right): - \hat A_{00}^{\dagger} \left( {{r}} \right){{\hat A}_{00}}\left( {{r}} \right)} \right],\end{split}$

(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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$\begin{split} {\hat V} =\, & \frac{1}{2}\int {{\rm{d}}^3}\left[{c_0}:{{\hat n}^2}\left( {{r}} \right): + {c_1}:{{{{\hat F}}}^2}\left( {{r}} \right):\right. \\ & +\left. {c_2}\hat A_{00}^{\dagger} \left( {{r}} \right){{\hat A}_{00}}\left( {{r}} \right)\right], \end{split}$

(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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$\begin{split} \hat V = \, &\frac{1}{2} \int{{\rm{d}}^3}\bigg[{c_0}:{{\hat n}^2}\left( {{r}} \right): + {c_1}:{{{{\hat F}}}^2}\left( {{r}} \right): \\& + {c_2}\hat A_{00}^{\dagger} \left( {{r}} \right){{\hat A}_{00}}\left( {{r}} \right) \\& + {c_3}\mathop \sum \limits_{M = - 2}^2 \hat A_{2M}^{\dagger} \left( {{r}} \right){{\hat A}_{2M}}\left( {{r}} \right) \bigg], \end{split}$

(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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$\left\{ {\begin{aligned} & {{S_{{x}}} = \frac{{2\left| {{\psi _1}} \right|\left| {{\psi _2}} \right|}}{{{{\left| \varPsi \right|}^2}}}\cos \left( {{\theta _1} - {\theta _2}} \right)}\\ & {{S_{{y}}} = - \frac{{2\left| {{\psi _1}} \right|\left| {{\psi _2}} \right|}}{{{{\left| \varPsi \right|}^2}}}\sin \left( {{\theta _1} - {{\rm{\theta }}_2}} \right)}\\ & {{S_{{z}}} = \frac{{{{\left| {{\psi _1}} \right|}^2} - {{\left| {{\psi _2}} \right|}^2}}}{{{{\left| \varPsi \right|}^2}}}} \end{aligned}} \right.,$

(28)

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$\left\{ {\begin{aligned} & {{S_{{x}}} = \sin \theta \cos \varphi }\\ & {{S_{{y}}} = \sin \theta \sin \varphi }\\ & {{S_{{z}}} = \cos \theta } \end{aligned}} \right.,$

(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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$\begin{split} {\cal Q}\left( {{S_{{x}}},{S_{{y}}},{S_{{z}}}} \right) & = \int \dfrac{1}{{8{\text{π}}}}{\varepsilon ^{ij}}{{S}} \cdot {\partial _i}{{S}} \times {\partial _j}{{S}}{\rm{d}}{{r}}\\ & = \dfrac{1}{{4{\text{π}}}}\int {\left| {\begin{array}{*{20}{c}} {{S_{{x}}}}&{{S_{{y}}}}&{{S_z}}\\ {\dfrac{{\partial {{{S}}_{{x}}}}}{{\partial x}}}&{\dfrac{{\partial {S_{{y}}}}}{{\partial x}}}&{\dfrac{{\partial {S_{\rm{z}}}}}{{\partial x}}}\\ {\dfrac{{\partial {S_{{x}}}}}{{\partial y}}}&{\dfrac{{\partial {S_{{y}}}}}{{\partial y}}}&{\dfrac{{\partial {S_{{z}}}}}{{\partial y}}} \end{array}} \right|{\rm{d}}{{r}}}\\ &= - \dfrac{1}{{4{\text{π}}}}\int {\left| {\begin{array}{*{20}{c}} {{S_{{x}}}}&{{S_{{z}}}}&{{S_y}}\\ {\dfrac{{\partial {{{S}}_{{x}}}}}{{\partial x}}}&{\dfrac{{\partial {S_{{z}}}}}{{\partial x}}}&{\dfrac{{\partial {S_{{y}}}}}{{\partial x}}}\\ {\dfrac{{\partial {S_{{x}}}}}{{\partial y}}}&{\dfrac{{\partial {S_{{z}}}}}{{\partial y}}}&{\dfrac{{\partial {S_{{y}}}}}{{\partial y}}} \end{array}} \right|{\rm{d}}{{r}}} \\ & = - {\cal Q}\left( {{S_{{x}}},{S_z},{S_{{y}}}} \right).\\[-14pt] \end{split}$

(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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$\left( {\begin{array}{*{20}{c}} {{\psi _1}}\\ {{\psi _0}}\\ {{\psi _{ - 1}}} \end{array}} \right) = \sqrt n \left( {\begin{array}{*{20}{c}} {{\rm{co}}{{\rm{s}}^2}\dfrac{\beta }{2}}\\ {\sqrt 2 {{\rm{e}}^{{\rm{i}}\phi }}\sin \dfrac{\beta }{2}\cos \dfrac{\beta }{2}}\\ {{{\rm{e}}^{2{\rm{i}}\phi }}{\rm{si}}{{\rm{n}}^2}\dfrac{\beta }{2}} \end{array}} \right),$

(33)

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$\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{\psi _1}}\\ {{\psi _0}} \end{array}}\\ {{\psi _{ - 1}}} \end{array}} \right) = \sqrt n \left( {\begin{array}{*{20}{c}} {{{\rm{e}}^{ - {\rm{i}}\phi }}{\rm{co}}{{\rm{s}}^2}\dfrac{\beta }{2}}\\ {\dfrac{1}{{\sqrt 2 }}\sin \beta }\\ {{{\rm{e}}^{{\rm{i}}\phi }}{\rm{si}}{{\rm{n}}^2}\dfrac{\beta }{2}} \end{array}} \right).$

(34)

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$\begin{split} \left( {\begin{array}{*{20}{c}} {{\psi _1}}\\ {{\psi _0}}\\ {{\psi _{ - 1}}} \end{array}} \right) & = \sqrt n {{\rm{e}}^{{\rm{i}}\theta }}U\left( {\alpha,\beta,\gamma } \right)\left( {\begin{array}{*{20}{c}} 1\\ 0\\ 0 \end{array}} \right) \\ & = \sqrt n {{\rm{e}}^{{\rm{i}}\left( {\theta - \gamma } \right)}}\left( {\begin{array}{*{20}{c}} {{{\rm{e}}^{ - {\rm{i}}\alpha }}{\rm{co}}{{\rm{s}}^2}\dfrac{\beta }{2}}\\ {\dfrac{1}{{\sqrt 2 }}\sin \beta }\\ {{{\rm{e}}^{{\rm{i}}\alpha }}{\rm{si}}{{\rm{n}}^2}\dfrac{\beta }{2}} \end{array}} \right).\end{split}$

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$ \left( {\begin{array}{*{20}{c}} {{\psi _1}}\\ {{\psi _0}}\\ {{\psi _{ - 1}}} \end{array}} \right) = \left( \begin{array}{c} {\left( {\cos \dfrac{{f\left( r \right)n}}{2} - {\rm{i}}\cos \theta \sin \dfrac{{f\left( r \right)n}}{2}} \right)^2} \\ - \sqrt 2 {\rm{i}}\left( {\cos \dfrac{{f\left( r \right)n}}{2} - {\rm{i}}\cos \theta \sin \dfrac{{f\left( r \right)n}}{2}} \right)\sin \dfrac{{f\left( r \right)n}}{2}\sin \theta {{\rm{e}}^{{\rm{i}}\phi }}\\ - {\rm{si}}{{\rm{n}}^2}\dfrac{{f\left( r \right)n}}{2}{\rm{si}}{{\rm{n}}^2}\theta {{\rm{e}}^{2{\rm{i}}\phi }} \end{array} \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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