Controllable magnetic solitons excitations in an atomic chain of spinor Bose–Einstein condensates confined in an optical lattice

Abstract

We propose an experimental scheme to show that the nonlinear magnetic solitary excitations can be achieved in an atomic spinor Bose–Einstein condensate confined in a blue-detuned optical lattice. Through exact theoretical calculations, we find that the magnetic solitons can be generated by the static magnetic dipole–dipole interaction (MDDI), of which the interaction range can be well controlled. We derive the existence conditions of the magnetic solitons under the nearest-neighboring, the next-nearest-neighboring approximations as well as the long-range consideration. It is shown that the long-range feature of the MDDI plays an important role in determining the existence of magnetic solitons in this system. In addition, to facilitate the experimental observation, we apply an external laser field to drive the lattice, and the existence regions for the magnetic soliton induced by the anisotropic light-induced dipole–dipole interaction are also investigated.

Appendix

• (1) For the NN approximation, the nonlinear equation of motion for ϕ _n can be written as

  $$\begin{aligned} i\frac{\hbox{d}\phi_{n}}{\hbox{d}t} &=\gamma_{B}B_{z}\phi_{n}+\underset{j=n\pm1}{\sum }2SJ_{nj}^{{\rm mag}}\sqrt{1-\left\vert \phi_{j}\right\vert ^{2}}\phi_{n}\\ &\quad+\underset{j=n\pm1}{\sum}4SJ_{nj}\sqrt{1-\left\vert \phi_{n}\right\vert ^{2}}\phi_{j}. \end{aligned}$$

  (24)

  In above equation, the terms $\sqrt{1-\left\vert \phi_{j}\right\vert ^{2}}\approx1-\frac{1}{2}\left\vert \phi_{j}\right\vert ^{2}$ for small distortion from the ground state. Using the slowly varying envelope approximation or long wavelength limit, we have

  $$\frac{\hbox{d}^{2}\phi(y,t)}{\hbox{d}y^{2}}|_{y=y_{n}}\approx\frac{\phi_{n+1}+\phi _{n-1}-2\phi_{n}}{a^{2}}.$$

  (25)

  where a is the lattice constant. Then, we can obtain the continuum limit NLSE for ϕ(y, t), namely

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t} &=4SJ_{01}a^{2}\frac{\hbox{d}^{2}\phi(y,t)}{\hbox{d}y^{2}}\\ &\quad+\left( \gamma_{B}B_{z}+4SJ_{01}^{{\rm mag}}+8SJ_{01}\right) \phi (y,t)\\ &\quad-2S\left( J_{01}^{{\rm mag}}+2J_{01}\right) \left\vert \phi(y,t)\right\vert ^{2}\phi(y,t). \end{aligned}$$

  (26)

• (2) For the NNN approximation, we set $\alpha_{j}=J_{0j}^{{\rm mag}}/J_{0j}, A_{2}=J_{02}/J_{01}, \eta=\gamma_{B}B_{z}+4S\left(J_{01}^{{\rm mag}}+2J_{01}+J_{02}^{{\rm mag}}-2J_{02}\right).$ Additionally, another second-order spatial derivative is introduced

  $$\frac{\hbox{d}^{2}\phi(y,t)}{\hbox{d}y^{2}}|_{y=y_{n}}\approx\frac{\phi_{n+2}+\phi _{n-2}-2\phi_{n}}{(2a)^{2}}.$$

  (27)

  We can easily get

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t} &=4SJ_{01}\left( 1-4A_{2}\right) a^{2}{\frac{\hbox{d}^{2}\phi(y,t)}{\hbox{d}y^{2}}}+\eta\phi(y,t)\\ &\quad-2SJ_{01}\left(\alpha_{1}+A_{2}\alpha_{2}+2-2A_{2}\right) \left\vert \phi(y,t)\right\vert ^{2}\phi(y,t). \end{aligned}$$

  (28)

  In the first case, i.e., the spinor condensate is loaded into a blue-detuned optical lattice and the external laser used to induce the LDDI is absent, we have

  $$J_{01}^{{\rm opt}}=0,J_{01}=\frac{1}{2}J_{01}^{{\rm mag}}\Longrightarrow\alpha_{1} =\alpha_{2}=2.$$

  (29)

  Then, Eq. (28) can be rewritten as

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t} &=2SJ_{01}^{{\rm mag}}\left( 1-4A_{2}\right) a^{2} \frac{\hbox{d}^{2}\phi(y,t)}{\hbox{d}y^{2}}+\eta\phi(y,t)\\ &\quad-4SJ_{01}^{{\rm mag}}\left\vert \phi(y,t)\right\vert ^{2}\phi(y,t). \end{aligned}$$

  (30)

  Second, the external laser is present and strong enough, so that

  $$J_{01}^{{\rm opt}}\gg J_{01}^{{\rm mag}}\Longrightarrow\alpha_{1}=\alpha_{2}\approx0.$$

  (31)

  Then, we can obtain

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t} &=4SJ_{01}\left( 1-4A_{2}\right) \frac{\hbox{d}^{2} \phi(y,t)}{\hbox{d}y^{2}}a^{2}+\eta\phi(y,t)\\ &\quad-4SJ_{01}\left( 1-A_{2}\right) \left\vert \phi(y,t)\right\vert ^{2} \phi(y,t). \end{aligned}$$

  (32)

• (3) For the long-range case, it is a little complicated. First, using $\sqrt{1-\left\vert \phi_{j}\right\vert ^{2}}\approx1-\frac{1}{2}\left\vert \phi_{j}\right\vert ^{2}$ we can rewrite the nonlinear motion equation of ϕ _n as

  $$\begin{aligned} i\frac{\hbox{d}\phi_{n}}{\hbox{d}t} &=\left[ \gamma_{B}B_{z}+\underset{j\neq n}{\sum }2SJ_{nj}^{{\rm mag}}\right] \phi_{n}-\underset{j\neq n}{\sum}SJ_{nj}^{z}\left\vert \phi_{j}\right\vert ^{2}\phi_{n}\\ &\quad-\underset{j\neq n}{\sum}4SJ_{nj}\phi_{j}(-1)^{j-n}+\underset{j\neq n} {\sum}2SJ_{nj}\left\vert \phi_{n}\right\vert ^{2}\phi_{j}(-1)^{j-n}. \end{aligned}$$

  (33)

  Then, we set $\phi_{n}\rightarrow\phi(y,t), J_{nj}^{{\rm mag}}\rightarrow J^{{\rm mag}}(y-y^{^{\prime}}), J_{nj}\rightarrow J(y-y^{^{\prime}}), \gamma _{B}B_{z}+\underset{j\neq n}{\sum}2SJ_{nj}^{{\rm mag}}\rightarrow\omega(y).$ Consider the discreteness of the optical lattice, we can treat the symbolic terms as below

  $$(-1)^{j-n}=\cos\left[(j-n)\pi\right] =\cos\left[\frac{2\pi}{\lambda_{L}}\left( y-y^{^{\prime}}\right) \right].$$

  (34)

  After changing the sum to integration, we get

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t}&=\omega(y)\phi(y,t) \\ &\quad-\frac{2S}{\lambda_{L}}\phi(y,t)\int\limits_{-\infty}^{\infty} J^{{\rm mag}}(y-y^{^{\prime}})\phi^{2}(y^{^{\prime}},t)\hbox{d}y^{^{\prime}}\\ &\quad-\frac{8S}{\lambda_{L}}\int\limits_{-\infty}^{\infty}J(y-y^{^{\prime} })\phi(y^{^{\prime}},t)\cos\left[ \frac{2\pi}{\lambda_{L}}\left( y-y^{^{\prime}}\right) \right] \hbox{d}y^{^{\prime}}\\ &\quad+\frac{4S}{\lambda_{L}}\phi^{2}(y,t)\int\limits_{-\infty}^{\infty }J(y-y^{^{\prime}})\phi(y^{^{\prime}},t)\cos\left[ \frac{2\pi}{\lambda_{L} }\left( y-y^{^{\prime}}\right) \right] \hbox{d}y^{^{\prime}}. \end{aligned}$$

  (35)

  Denoting $y-y^{^{\prime}}=\xi$, the $\phi(y^{^{\prime}},t)$ can be expanded as

  $$\phi(y^{^{\prime}},t)=\phi(y,t) +\frac{\partial\phi( y,t) }{\partial y}\xi+\frac{1}{2!}\frac{\partial^{2}\phi(y,t) }{\partial y^{2}}\xi^{2}+\cdots.$$

  (36)

  Taking above series into Eq. (35) (the integration variable is changed, i.e., $\hbox{d}y^{\prime}\rightarrow-{\rm d}\xi$) and using the assumption ∂ ϕ(y,t)/∂ y≪ 1, finally we get

  $$\begin{aligned} i\frac{\hbox{d}\phi(y,t)}{\hbox{d}t}&=-2\beta_{1}\frac{\partial^{2} \phi\left( y,t\right)}{\partial y^{2}}+\left[\omega(y)-4\beta_{0}\right] \phi(y,t)\\ &\quad+\left(2\beta_{0}-\gamma\right) \left\vert \phi(y,t)\right\vert ^{2} \phi(y,t), \end{aligned}$$

  (37)

  where the coefficients are defined as below

  $$\beta_{n}=\frac{2S}{\lambda_{L}}\int\limits_{-\infty}^{\infty}J(\xi)\xi^{2n} \cos\left(\frac{2\pi}{\lambda_{L}}\xi\right)\hbox{d}\xi, \quad n=0,1,$$

  (38)
  $$\gamma=\frac{2S}{\lambda_{L}}\int\limits_{-\infty}^{\infty}J^{{\rm mag}}(\xi)\hbox{d}\xi.$$

  (39)