Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 61475198, 11705108)
Received Date:21 October 2020
Accepted Date:01 December 2020
Available Online:19 March 2021
Published Online:05 April 2021
Abstract:Based on the three-dimensional spinor Gross-Pitaevskii (GP) equation, the dynamic behavior of the Bose-Einstein condensate under the action of a time-dependent periodic external magnetic field is studied. The results show that the Bose-Einstein condensate with spin-1 in a ferromagnetic state will undergo topological deformation under the action of an external magnetic field periodically varying with time. When the two zero points of the magnetic field enter into the condensate, the density pattern of the spin-up state forms small convexities protruding upward and downward on the z-axis, respectively. As the two zero points of the magnetic field gradually coincide in the condensate, the upward and downward protruding convexities are elongated. Finally, the spin-up state in the shape of a line is distributed on the z-axis, which is consistent with the scenario of the isolated Dirac string predicted by theoretical analysis. As far as we know, magnetic monopole can be divided into positive monopole and negative monopole. The positive magnetic monopole means that all magnetic induction lines are emitted from the center of the circle. And only the Dirac string points to the center of the circle. The negative monopole is that all the magnetic induction lines point from the outside to the center of the circle, and only the Dirac string emits from the center of the circle. Magnetic monopole is a topological defect in vector field, which accords with both quantum mechanics and gauge invariance of electromagnetic field. Single magnetic monopole has been studied a lot in theory, and its analogues have been observed in experiment. But multiple monopoles and the interaction between them are still rarely studied. In this paper, multiple monopoles are produced based on the fact that the periodic magnetic field has multiple zeros. We use a new periodic magnetic field to generate a positive and negative magnetic monopole. Due to the strong external magnetic field, the vorticity in the condensate is consistent with the magnetic field of the monopole. Finally, by calculating the superfluid vorticity of the condensate, the characteristic diagram of the magnetic monopole is obtained. The results show that the condensate forms a pair of positive and negative magnetic monopoles at the two zero points of the magnetic field, corresponding to the two small convexities protruding upward and downward on the z-axis of the spin-up state, respectively. As the two zero points of the magnetic field coincide, the two Dirac strings in the positive and negative magnetic monopole gradually approach to each other, and after about 5 ms, they are completely connected, finally forming an isolated Dirac string. This result provides a new idea for further studying the isolated Dirac strings. Keywords:magnetic monopole/ positive and negative monopoles/ Bose-Einstein condensate/ isolated Dirac string
${{\mathit{\boldsymbol{B}}}}({{\mathit{\boldsymbol{r}}}},t) = (bx, - by,a\cos (kz) + c - vt),$
其中$b = 3.7\;{{\rm{G}} / {{\rm{cm}}}}$是四极磁场梯度的强度系数, $c = 10\;{\rm{mG}}$为偏置场在$t = 0$时刻的磁场强度以及$v = 0.25\;{{\rm{G}} / {\rm{s}}}$为偏置磁场的减少速率. 等效磁场中的系数$a = 3.6\;{\rm{mG}}$与激光强度有关而系数$k = $$ 0.2\;{{{{\text{μ}}{\rm{ m}}}}^{{{ - 1}}}}$与激光的频率相关. 图1给出了不同时刻的磁场在xoz平面的分布图. 从中可以看出, 在$t = $$ 0$时, 凝聚体内的磁场全部指向z轴; 在$t = 50\;{\rm{ms}}$ 时, 由于等效磁场的周期性, 凝聚体内将会出现两个零点, 分别位于$z = 4\;{{\text{μ}} {\rm{m}}}$和$z = - 4\;{\text{μ}} {\rm{m}}$; 在$t = $$ 55\;{\rm{ms}}$ 时, 两个零点在原点处合并变成一个零点. 图 1 凝聚体处不同时刻的磁场在xoz平面的分布图. 箭头的方向和长度分别表示磁场的方向和大小. 绿色图案为凝聚体所在位置, 凝聚体的半径${R_{{\rm{TF}}}}$可以通过托马斯-费米近似得到: ${R_{{\rm{TF}}}} = 5 N{c_0}/(4 m{\omega ^2}) = 6\;{\rm{{\text{μ}} m}}$, 其中各图分别对应时刻 (a) t = 0; (b) t = 50 ms; (c) t = 55 ms Figure1. Distributions of the magnetic field at xoz plane for different timesaround the condensation. Direction and length of arrows indicate the direction and size of the magnetic field. The green pattern indicates the condensation, radius of which is determined by Thomas-Fermi approximation, i.e., ${R_{{\rm{TF}}}} = 5 N{c_0}/(4 m{\omega ^2}) = 6\;{\rm{{\text{μ}} m}}$. (a) t = 0; (b) t = 50 ms; (c) t = 55 ms.
在数值模拟过程中, 首先通过GP方程(1)的虚时演化获取$t = 0$时刻凝聚体的基态, 并将其作为初态. 此时, 由于凝聚体处的磁场都指向z轴正方向, 所以为了使能量最低, 凝聚体中粒子的自旋将会与磁场同向, 即自旋矢量为${{\mathit{\boldsymbol{S}}}} = {(0, 0, 1)^{{{\rm{T}}}}}$, 对应的旋量为${{\mathit{\boldsymbol{\xi}}}} = {(1, 0, 0)^{\rm{T}}}$. 如图2(a)所示, 此时所有粒子都布居在${\psi _1}$态上. 图 2 不同时刻凝聚体各个组分密度在xoz平面的分布图, 其中各图分别对应时刻 (a) t = 0; (b) t = 50 ms; (c) t = 55 ms; (d) t = 60 ms Figure2. Distributions of density of the condensation at xoz-plane for different times: (a) t = 0; (b) t = 50 ms; (c) t = 55 ms; (d) t = 60 ms.
由于超流涡度对应着磁单极, 这样可以通过凝聚体演化过程中涡度场的变化来探究对应的磁单极的行为. 我们数值计算出不同时刻的涡度, 如图3所示. 注意由于初始时刻$t = 0$所有粒子都布居在${{\psi} _1}$态上, 涡度始终为0, 故在此不做讨论. 图 3 第一行是不同时刻涡度场在xoz平面的分布图. 第二行是归一化的涡旋场, 只保留了涡旋场的方向 (a), (b) t = 50 ms; (c), (d) t = 55 ms; (e), (f) t = 60 ms Figure3. First row is the distributions of vorticity at xoz-plane for different times. Second row is the corresponding normalized field. (a), (b) t = 50 ms; (c), (d) t = 55 ms; (e), (f) t = 60 ms.