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Journals >Acta Physica Sinica >Volume 68 >Issue 4 >Page 043703-1 > Article

Acta Physica Sinica
Vol. 68, Issue 4, 043703-1 (2019)

Magnetic excitation of ultra-cold atoms trapped in optical lattice

Xing-Dong Zhao¹, Ying-Ying Zhang¹, and Wu-Ming Liu^2,*

Author Affiliations

¹College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China

²Laboratory of Condensed Matter Theory and Materials Computation, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

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DOI: 10.7498/aps.68.20190153 Cite this Article

Xing-Dong Zhao, Ying-Ying Zhang, Wu-Ming Liu. Magnetic excitation of ultra-cold atoms trapped in optical lattice[J]. Acta Physica Sinica, 2019, 68(4): 043703-1 Copy Citation Text

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Abstract

Spinor condensates trapped in optical lattices have become potential candidates for multi-bit quantum computation due to their long coherence and controllability. But first, we need to understand the generation and regulation of spin and magnetism in the system. This paper reviews the origin and manipulation of the magnetism of atomic spin chains in optical lattices. The theoretical study of the whole process is described in this paper, including laser cooling, the spinor Bose-Einstein condensate preparations, the optical lattice, and the atomic spin chain. Then, the generation and manipulation of magnetic excitations are discussed, including the preparation of magnetic solitons. Finally, we discuss how to apply atomic spin chains to quantum simulation. The theoretical study of magnetic excitations in optical lattices will play a guiding role when the optical lattice is used in cold atomic physics, condensed matter physics and quantum information.

Keywords

magnetic soliton optical lattice quantum simulation spinor Bose-Einstein condensates

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\begin{aligned} \hat{H} = & \int d r {\hat{Ψ}}^{†} (r) [- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{e x t}] \hat{Ψ} (r) \\ + \frac{1}{2} \int d r d r^{'} {\hat{Ψ}}^{†} (r) {\hat{Ψ}}^{†} (r^{'}) V (r - r^{'}) \hat{Ψ} (r^{'}) \hat{Ψ} (r), \end{aligned}

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(1)

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${\rm{i}}\hbar \frac{\partial }{{\partial t}}\Phi ({{r}},t) = \left( { - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + {V_{{\rm{ext}}}} + g{{\left| {\Phi ({{r}},t)} \right|}^2}} \right)\Phi ({{r}},t),$

(2)

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\begin{aligned} \hat{H} = & \int d r {\hat{Ψ}}^{†} (r) [- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{e x t}] \hat{Ψ} (r) \\ + Ω_{α, β, μ, ν} \int d r d r^{'} {\hat{Ψ}}_{α}^{†} (r) {\hat{Ψ}}_{β}^{†} (r^{'}) \\ \times V (r - r^{'}) {\hat{Ψ}}_{μ} (r^{'}) {\hat{Ψ}}_{ν} (r), \end{aligned}

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(3)

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\begin{aligned} \hat{H} = & \sum_{α} \int d r {\hat{Ψ}}_{α}^{†} (r) [- \frac{ℏ^{2}}{2 m} \nabla^{2} + V_{e x t}] {\hat{Ψ}}_{α} (r) \\ + \frac{c_{0}}{2} \sum_{α, β} \int d r {\hat{Ψ}}_{α}^{†} (r) {\hat{Ψ}}_{β}^{†} (r) {\hat{Ψ}}_{β} (r) {\hat{Ψ}}_{α} (r) \\ + \frac{c_{2}}{2} \sum_{α, β, α^{'}, β^{'}} \int d r {\hat{Ψ}}_{α}^{†} (r) {\hat{Ψ}}_{β}^{†} (r) \\ \times F_{α α^{'}} F_{β β^{'}} {\hat{Ψ}}_{β^{'}} (r) {\hat{Ψ}}_{α^{'}} (r), \end{aligned}

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(4)

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$

\begin{aligned} E = & \int d^{3} r \frac{ℏ^{2}}{2 m} {{[\nabla \sqrt{n}]}^{2} + {[\nabla ζ (r)]}^{2} n} \\ - \int d^{3} r {[μ - U (r)] n - \frac{n^{2}}{2} [c_{0} + c_{2} {⟨ F ⟩}^{2}]} \end{aligned}

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(5)

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${K_{\rm{s}}} = c{\left\langle {{F}} \right\rangle ^2} - p\left\langle {{F_z}} \right\rangle + q\left\langle {{F_z}^2} \right\rangle $

(6)

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\begin{aligned} \hat{H} = & - J \sum_{i, j} \sum_{α} {\hat{a}}_{α i}^{+} {\hat{a}}_{α j}^{+} + \sum_{i} \sum_{α} ε_{i} {\hat{a}}_{α i}^{+} {\hat{a}}_{α i}^{+} \\ + \frac{U_{0}}{2} \sum_{i} \sum_{α, β} {\hat{a}}_{α i}^{+} {\hat{a}}_{β i}^{+} {\hat{a}}_{β i}^{} {\hat{a}}_{α i} \\ + \frac{U_{2}}{2} \sum_{i} \sum_{α, β, α^{'}, β^{'}} {\hat{a}}_{α i}^{+} {\hat{a}}_{β i}^{+} F_{α α^{'}} F_{β β^{'}} {\hat{a}}_{β^{'} i}^{} {\hat{a}}_{α^{'} i}, \end{aligned}

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(7)

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\begin{aligned} V_{d d}^{i j} = & \frac{μ_{0}}{4 π} \int d r \int d r^{'} {| ϕ (r - r_{i}) |}^{2} {| ϕ (r^{'} - r_{j}) |}^{2} \\ \times [\frac{μ_{i} \cdot μ_{j}}{{| r - r^{'} |}^{3}} - \frac{3 [μ_{i} (r - r^{'}) \cdot μ_{j} (r - r^{'})]}{{| r - r^{'} |}^{5}}], \end{aligned}

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(8)

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\begin{aligned} H = & \sum_{i} [λ_{a}^{'} {\hat{S}}_{i}^{2} - γ_{B} \sum_{j \neq i} λ_{i j} {\hat{S}}_{i} \cdot {\hat{S}}_{j} \\ - 3 γ_{B} \sum_{j \neq i} {\hat{S}}_{i} \cdot Λ_{i j} \cdot {\hat{S}}_{j} - γ_{B} {\hat{S}}_{i} \cdot B] \end{aligned}

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(9)

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\begin{aligned} H_{1 d} = & λ_{a}^{'} {\hat{S}}^{2} - γ_{B} \hat{S} \cdot [(B_{z} + 2 \sum_{j \neq i} λ_{i j}^{} S_{j}^{z}) \hat{z} \\ + (B_{x} - 2 \sum_{j \neq i} λ_{i j}^{} S_{j}^{x}) \hat{x} - \sum_{j \neq i} λ_{i j}^{} S_{j}^{y} \hat{y}], \end{aligned}

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(10)

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${H_{2d}} = \sum\limits_{ij} {\left[ {\frac{{{\lambda _a}}}{2}\hat {{S}}_{ij}^2 + \frac{{{\gamma _B}{\mu _0}}}{{4{\text{π}}}}\sum\limits_{kl \ne ij} {\hat {{S}}_{{\rm{ij}}}^{\rm{T}}{\Lambda _{ij,kl}}{{\hat {{S}}}_{kl}}} } \right]} .$

(11)

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$

\begin{aligned} Q_{m m^{'} n^{'} n} = & 3 \frac{γ}{Δ} U_{0} \exp (- \frac{r_{⊥}^{2} + r_{⊥}^{^{'} 2}}{W_{L}^{2}}) \cos (k_{L} y) \\ \times \cos (k_{L} y^{'}) \times [\frac{4}{9} δ_{m^{'} n^{'}} δ_{m n} e_{0} \\ \times W (r - r^{'}) \cdot e_{0} - d_{m^{'} n^{'}} \\ \times W (r - r^{'}) \cdot d_{m n}], \end{aligned}

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(12)

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\begin{aligned} H = & \sum_{i} [λ_{a}^{'} {\hat{S}}_{i}^{2} - γ_{B} {\hat{S}}_{i} \cdot B - \sum_{j \neq i} J_{i j}^{z} {\hat{S}}_{i}^{z} {\hat{S}}_{j}^{z} \\ - \sum_{j \neq i} J_{i j} ({\hat{S}}_{i}^{-} {\hat{S}}_{i}^{+} + {\hat{S}}_{i}^{+} {\hat{S}}_{j}^{-})], \end{aligned}

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(13)

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\begin{aligned} J_{i j}^{z} = & \frac{μ_{0} γ_{B}^{2}}{16 π ℏ^{2}} \int d r \int d r^{'} \frac{{| r^{'} |}^{2} - 3 {y^{'}}^{2}}{{| r^{'} |}^{5}} {| ϕ_{i} (r) |}^{2} \\ \times {| ϕ_{j} (r - r^{'}) |}^{2}, \end{aligned}

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(14)

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\begin{aligned} J_{i j} = & \frac{U}{144 ℏ^{2} k_{L}^{3}} \int d r \int d r^{'} f_{c} (r^{'}) \\ \times \exp (\frac{r_{⊥}^{2} + {| r_{⊥} - {r^{'}}_{⊥} |}^{2}}{W_{L}^{2}}) \cos (k_{L} y) \\ \times \cos [k_{L} (y - y^{'})] \times e_{+ 1} \cdot W (r^{'}) \cdot e_{- 1} \\ \times {| ϕ_{i} (r) |}^{2} {| ϕ_{j} (r - r^{'}) |}^{2} + \frac{1}{2} J_{i j}^{z} . \end{aligned}

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(15)

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${\rm{i}}\frac{\partial }{{\partial t}}\hat S_q^{( - )} = ({\omega _0} + \Delta {\omega _q})\hat S_q^{( - )} - \sum\limits_{j \ne n} {{\chi _{qj}}\hat S_q^{( - )}} ,$

(16)

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${\rm{i}}\frac{{\partial S(y,t)}}{{\partial t}} = \left[ { - \frac{{{\beta _1}}}{2}\frac{{{\partial ^2}}}{{\partial {y^2}}} - {\beta _0} + \omega (y)} \right]S(y,t).$

(17)

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$H = \sum\limits_k {{\omega _k}\widehat a_k^\dagger } {\widehat a_k} + \sum\limits_{k,k',q} {{V_{k,k',q}}\widehat a_{k - q}^\dagger \widehat a_{k' + q}^\dagger } {\widehat a_{k'}}{\widehat a_k}.$

(18)

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\begin{aligned} N_{k} (t) = & \sinh^{2} (γ_{k}) + \sinh^{2} (λ_{k}) + 2 \sinh^{2} (γ_{k}) \sinh^{2} (λ_{k}) \\ - \frac{1}{2} \sinh (2 γ_{k}) \sinh (2 λ_{k}) | \sin (w_{k} t) | . \end{aligned}

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(19)

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\begin{aligned} \frac{\partial}{\partial t} ε (r, t) \hat{E} (r, t) = \nabla \times \hat{H} (r, t), \\ \frac{\partial}{\partial t} \hat{H} (r, t) = - \nabla \times ε (r, t) \hat{E} (r, t) . \end{aligned}

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(20)

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Xing-Dong Zhao, Ying-Ying Zhang, Wu-Ming Liu. Magnetic excitation of ultra-cold atoms trapped in optical lattice[J]. Acta Physica Sinica, 2019, 68(4): 043703-1

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