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

Journals >Acta Physica Sinica >Volume 69 >Issue 1 >Page 010303-1 > Article

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,*|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

show less

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.

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)

View in Article

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

(2)

View in Article

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

(3)

View in Article

$

\begin{aligned} {\hat{A}}_{F M} (r, r^{'}) = & \sum_{m_{1}, m_{2} = - f}^{f} ⟨ F, M | f, m_{1}; f, m_{2} ⟩ \\ {\hat{ψ}}_{m_{1}}^{} (r) ψ_{m_{2}}^{} (r^{'}) . \end{aligned}

$

(4)

View in Article

$

\begin{aligned} {\hat{V}}^{F} = & \frac{1}{2} \int d r \int d r^{'} v^{(F)} (r - r^{'}) \\ \times \sum_{M = - F}^{F} {\hat{A}}_{F M}^{†} (r, r^{'}) {\hat{A}}_{F M} (r, r^{'}), \end{aligned}

$

(5)

View in Article

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

(6)

View in Article

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

(7)

View in Article

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

(8)

View in Article

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

(9)

View in Article

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

(10)

View in Article

$ \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)

View in Article

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

(12)

View in Article

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

(13)

View in Article

$

\begin{aligned} {\hat{V}}_{2} & = \frac{g_{2}}{2} \int d^{3} r \sum_{M = - 2}^{2} {\hat{A}}_{2 M}^{†} (r) {\hat{A}}_{2 M} (r) \\ = \frac{g_{2}}{2} \int d^{3} [: {\hat{n}}^{2} (r) : - {\hat{A}}_{00}^{†} (r) {\hat{A}}_{00} (r)], \end{aligned}

$

(14)

View in Article

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

(15)

View in Article

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

(16)

View in Article

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

(17)

View in Article

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

(18)

View in Article

$

\begin{aligned} \hat{V} = & \frac{1}{2} \int d^{3} [c_{0} : {\hat{n}}^{2} (r) : + c_{1} : {\hat{F}}^{2} (r) : \\ + c_{2} {\hat{A}}_{00}^{†} (r) {\hat{A}}_{00} (r)], \end{aligned}

$

(19)

View in Article

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

(20)

View in Article

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

(21)

View in Article

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

(22)

View in Article

$

\begin{aligned} \hat{V} = & \frac{1}{2} \int d^{3} [c_{0} : {\hat{n}}^{2} (r) : + c_{1} : {\hat{F}}^{2} (r) : \\ + c_{2} {\hat{A}}_{00}^{†} (r) {\hat{A}}_{00} (r) \\ + c_{3} \sum_{M = - 2}^{2} {\hat{A}}_{2 M}^{†} (r) {\hat{A}}_{2 M} (r)], \end{aligned}

$

(23)

View in Article

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

(24)

View in Article

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

(25)

View in Article

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

(26)

View in Article

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

(27)

View in Article

$\left\{ {

\begin{aligned} S_{x} = \frac{2 | ψ_{1} | | ψ_{2} |}{{| Ψ |}^{2}} \cos (θ_{1} - θ_{2}) \\ S_{y} = - \frac{2 | ψ_{1} | | ψ_{2} |}{{| Ψ |}^{2}} \sin (θ_{1} - θ_{2}) \\ S_{z} = \frac{{| ψ_{1} |}^{2} - {| ψ_{2} |}^{2}}{{| Ψ |}^{2}} \end{aligned}

} \right.,$

(28)

View in Article

$\left\{ {

\begin{aligned} S_{x} = \sin θ \cos φ \\ S_{y} = \sin θ \sin φ \\ S_{z} = \cos θ \end{aligned}

} \right.,$

(29)

View in Article

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

(30)

View in Article

$

\begin{aligned} Q (S_{x}, S_{y}, S_{z}) & = \int \frac{1}{8 π} ε^{i j} S \cdot \partial_{i} S \times \partial_{j} S d r \\ = \frac{1}{4 π} \int | \begin{array}{c} S_{x} & S_{y} & S_{z} \\ \frac{\partial S_{x}}{\partial x} & \frac{\partial S_{y}}{\partial x} & \frac{\partial S_{z}}{\partial x} \\ \frac{\partial S_{x}}{\partial y} & \frac{\partial S_{y}}{\partial y} & \frac{\partial S_{z}}{\partial y} \end{array} | d r \\ = - \frac{1}{4 π} \int | \begin{array}{c} S_{x} & S_{z} & S_{y} \\ \frac{\partial S_{x}}{\partial x} & \frac{\partial S_{z}}{\partial x} & \frac{\partial S_{y}}{\partial x} \\ \frac{\partial S_{x}}{\partial y} & \frac{\partial S_{z}}{\partial y} & \frac{\partial S_{y}}{\partial y} \end{array} | d r \\ = - Q (S_{x}, S_{z}, S_{y}) . \end{aligned}

$

(31)

View in Article

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

(32)

View in Article

$\left( {

\begin{matrix} ψ_{1} \\ ψ_{0} \\ ψ_{- 1} \end{matrix}

} \right) = \sqrt n \left( {

\begin{matrix} c o s^{2} \frac{β}{2} \\ \sqrt{2} e^{i ϕ} \sin \frac{β}{2} \cos \frac{β}{2} \\ e^{2 i ϕ} s i n^{2} \frac{β}{2} \end{matrix}

} \right),$

(33)

View in Article

$\left( {

\begin{matrix} \begin{matrix} ψ_{1} \\ ψ_{0} \end{matrix} \\ ψ_{- 1} \end{matrix}

} \right) = \sqrt n \left( {

\begin{matrix} e^{- i ϕ} c o s^{2} \frac{β}{2} \\ \frac{1}{\sqrt{2}} \sin β \\ e^{i ϕ} s i n^{2} \frac{β}{2} \end{matrix}

} \right).$

(34)

View in Article

$

\begin{aligned} (\begin{array}{c} ψ_{1} \\ ψ_{0} \\ ψ_{- 1} \end{array}) & = \sqrt{n} e^{i θ} U (α, β, γ) (\begin{array}{c} 1 \\ 0 \\ 0 \end{array}) \\ = \sqrt{n} e^{i (θ - γ)} (\begin{array}{c} e^{- i α} c o s^{2} \frac{β}{2} \\ \frac{1}{\sqrt{2}} \sin β \\ e^{i α} s i n^{2} \frac{β}{2} \end{array}) . \end{aligned}

$

(35)

View in Article

$ \left( {

\begin{matrix} ψ_{1} \\ ψ_{0} \\ ψ_{- 1} \end{matrix}

} \right) = \left(

\begin{matrix} {(\cos \frac{f (r) n}{2} - i \cos θ \sin \frac{f (r) n}{2})}^{2} \\ - \sqrt{2} i (\cos \frac{f (r) n}{2} - i \cos θ \sin \frac{f (r) n}{2}) \sin \frac{f (r) n}{2} \sin θ e^{i ϕ} \\ - s i n^{2} \frac{f (r) n}{2} s i n^{2} θ e^{2 i ϕ} \end{matrix}

\right), $

(36)

View in Article

Get PDF(in Chinese)

Figures&Tables (15)

References (125)

Copy Citation Text

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

Download Citation

Save the article for my favorites

Paper Information

Recommended Topics

Nuclear physics

Particles and fields

Particle and nuclear astrophysics and cosmology

Set citation alerts for the article

Please enter your email address