Topological defects and inhomogeneous spin patterns induced by the quadratic Zeeman effect in spin-1 Bose-Einstein condensates

We obtain analytically two kinds of inhomogeneous spin domain configurations endowed with time-dependent periodic domain walls in spin-1 Bose-Einstein condensate, which result from the positive and negative quadratic Zeeman effects, respectively. It is shown analytically that the topological defects are given by the unmagnetized atoms, at which the order parameter is of polar phase, but besides them, the order parameter is of axisymmetry-broken phase. The quadratic Zeeman effect can be tuned to induce the dynamical phase transition of the spin domain, and its sign can affect the topological structure of the spin pattern. These features arise from the axisymmetry breaking due to the pointwise-different population exchange between the sublevels determined uniquely by the quadratic Zeeman effects. The related experimental observations for the spin-1 ${}^{87}\mathrm{Rb}$ and ${}^{23}\mathrm{Na}$ condensates are also discussed.

Figure 1

Initial states of ${\Phi }_{\pm 1},{\Phi }_0$ given by (5) for ${}^{87}\mathrm{Rb}$: (a) $q=2$, the periodic case; (b) $q=\frac{430}{217} \left(\lambda =1\right)$, the soliton case. The other parameters are $p=1,c_0=216,c_1=-1,k_1=1,x\in [-5,5]$. Such initial states should be prepared in the presence of positive quadratic Zeeman effect. Take $n=2.8\times {10}^{20}{\text{m}}^{-3}$, then $p=1$ corresponds to $\overline{p}=10.02$ Hz; $q=2$ corresponds to $\overline{q}=20.05$ Hz; and $q=\frac{430}{217}$ corresponds to $\overline{q}=19.86$ Hz.

Figure 2

Evolution of the polar angle determined by solution (4): (a) $q=2$, the periodic case; (b) $q=\frac{430}{217}(\lambda =1)$, the soliton case. The other parameters are $x\in [-10,10],t\in [0,3],p=1,c_0=216,c_1=-1,k_1=1$.

Figure 3

The spatial distribution of $F_x,F_y,F_z$ and topological defects: (a) the periodic case $\left(q=2\right)$, the common zeros of the three curves are the topological defects; (b) the soliton case $\left(q=\frac{430}{217}\right)$, the only common zero of the three curves indicates the topological defect. The other parameters are $p=1,c_1=-1,c_0=216,k_1=1,t=1,x\in [-8,8]$.

Figure 4

Inhomogeneous spin structure and topological defects induced by positive quadratic Zeeman effect $q$. (a) The space-time evolution of the spin density vector $F$, the point $O$ corresponds to topological defects. (b) The field plot of the transverse magnetization vector $F_{\perp }$, the green and blue lines between the yellow and pink regions correspond to the evolution of the topological defects. In (a) and (b), $t\in [0,10].$ (c) The spatial spin configuration at $t=1$, and (d) the spatial spin configuration at $t=0.$ In (c) and (d), the periodic domain walls are the topological defects. (e) The spatial spin configuration of local magnetism (the soliton case) at $t=1$, where the unique domain wall is the topological defect. (f) The spatial distribution of $|F_{\perp }|$ and $|F_z|$ of local magnetism (the soliton case) at $t=1$, where the only common zero of the two curves indicates the topological defect. In (a)–(d), $q=2$; in (e) and (f), $q=430/217 \left(\lambda =1\right)$. The other parameters are $x\in [-5,5],k_1=1,c_1=-1,c_0=216,p=1.$

Figure 5

Inhomogeneity of the order parameter $\left({\Phi }_{-1},{\Phi }_0,{\Phi }_1\right)$ in the spin space determined by the positive quadratic Zeeman effect. As a comparison for the spin state, (a) and (b) are for the same position $\left(x=4\right)$ and quadratic Zeeman effect $\left(q=3\right)$ but different time: $t=2$ in (a), $t=4.5$ in (b); (a) and (c) are for the same position $\left(x=4\right)$ and time $\left(t=2\right)$ but different quadratic Zeeman effect: $q=3$ in (a), $q=9$ in (c). The other parameters are taken as $p=1,k_1=1,c_1=-1,c_0=216$. Notice that the three figures have different morphologies near the centers, so they cannot be identified in the spin space via a rotation.

Figure 6

Initial states of ${\Phi }_{\pm 1},{\Phi }_0$ determined by solution (10): (a) $q=-3$, the periodic case; (b) $q=-3.875(\lambda =1)$, the soliton case. The other parameters are $t=0,x\in [-5,5],p=1,c_1=1,c_0=32,A_{-1}=-0.5,k_1=1$.

Figure 7

(a) The spatial distribution of $F_x,F_y,F_z$ for $q=-3.87$ (the periodic case), the common zeros of the three curves show the topological defects. (b) The spatial distribution of $F_x,F_y,F_z$ for $q=-3.875(\lambda =1)$ (the soliton case), the only common zero of the three curves indicates the topological defect. (c) The spatial distribution of $|F_{\perp }|$ and $|F_z|$ for $q=-3.875(\lambda =1)$, the only common zero of the two curves is the topological defect. The other parameters are $p=1,c_1=1,c_0=32,A_{-1}=-0.5,k_1=1,t=1,x\in [-5,5]$.

Figure 8

Evolution of the polar angle determined by solution (10): (a) $q=-3$, the periodic case; (b) $q=-\frac{31}{8}(\lambda =1)$, the soliton case. The other parameters are $x\in [-3,3],t\in [0,3],p=1,c_1=1,c_0=32.k_1=1,A_{-1}=-0.5$.

Figure 9

Inhomogeneous spin structure and the topological defects determined by solution (10) for different $t$ and $q$: (a) the spatial spin configuration at $t=1$ for $q=-3$, the periodic case; (b) the spatial spin configuration at $t=0$ for $q=-3$; and (c) the spatial spin configuration at $t=1$ for $q=-\frac{31}{8}(\lambda =1)$, the soliton case. The other parameters are: $x\in [-3,3],p=1,c_1=1,c_0=32,k_1=1,A_{-1}=-0.5$. The domain walls are formed by the topological defects.

Figure 10

Inhomogeneous spin structure induced by negative quadratic Zeeman effect: (a) the space-time evolution of the spin density vector $F$, the color indicates the intensity of $F_z$; (b) the influence of $q$ on $F,F_{\perp }$ and $F_z$. In (a), $p=1,q=-3,x\in [-5,5],t\in [0,15]$; in (b), $x=0,t=1,q\in [-15,0]$. The other parameters are $A_{-1}=-0.5,p=1,k_1=1,c_1=1,c_0=32$.

Figure 11

Pointwise evolution of the spin density vector $F$ in the linear and quadratic Zeeman effects $p$ and $q$ at the same time $\left(t=1\right)$ but different position, the color indicates the intensity of $F_z$: (a, b) for $q>0$, where $x=2$ in (a), $x=3.3$ in (b), $q\in [0,10]$; (c, d) for $q<0$, where $x=1.7$ in (c), $x=2.7$ in (d), $q\in [-3.87,0],A_{-1}=-0.5$. In both cases $p\in [0,2\pi ],k_1=1$.