准二维空间中的隐秘涡旋量子液滴

陈逸熙; 蔡晓妍; 刘彬; 江迅达; 黎永耀

doi:10.7498/aps.71.20220709

摘要

本文探究了弱囚禁条件下玻色-玻色混合凝聚体中的准二维隐秘涡旋量子液滴及其动力学特性. 前期研究表明, 在完全自由的三维空间中, 隐秘涡旋量子液滴难以稳定; 在二维系统中, 当囚禁尺度是窄囚禁条件时, 系统仅支持拓扑荷${S_{1,2}} = \pm 1$的隐秘涡旋量子液滴; 因此当横向囚禁尺度较弱时, 准二维空间中李-黄-杨修正项仍然采用三维空间中的表达式来描述, 此时隐秘涡旋量子液滴能否保持稳定是一个重要的科学问题. 本文采用虚时间方法获得了拓扑荷${S_{1,2}}$达到$ \pm 4$的隐秘涡旋量子液滴; 进一步论证了隐秘涡旋量子液滴的有效面积${A_{{\text{eff}}}}$和化学势$\mu $与总粒子数$N$之间的依赖关系; 并采用线性稳定性分析结合实时传输方法获得了总粒子数临界值${N_{{\text{th}}}}$分别与拓扑荷${S_1}$和非线性系数${\text{δ}}g$之间的依赖关系. 在动力学部分, 本文研究了由两个不同拓扑荷的隐秘涡旋量子液滴构造的复合涡旋模式, 即嵌套涡旋量子液滴. 结合量子液滴的密度分布具有“平顶型”的特点, 采用Thomas-Fermi近似对数值结果进行了有效验证.

关键词:

Abstract

In this work, we study the quasi-two-dimensional hidden vortices of quantum droplets (QDs) trapped by a thicker transverse confinement and investigate their dynamical properties. Previous studies demonstrated that the hidden vortices of QDs in a three-dimensional free space are unstable and stable two-dimensional hidden vortices of QDs only with ${S_{1,2}} = \pm 1$ can be supported by a thin transverse confinement. Under the conditions of thicker transverse confinement, the Lee-Huang-Yang correction term in quasi-two-dimensional space is still described in the form of the three-dimensional space. Hence, under this condition, the stability and characteristics of the hidden vortices of QDs are worth studying. By using the imaginary time method, the hidden vortices of QDs with topological charge ${S_{1,2}}$ up to $ \pm 4$ are obtained for the first time. Furthermore, the dependence of the effective area${A_{{\text{eff}}}}$and the chemical potential$\mu $on the total norm$N$of the hidden vortices of QDs are demonstrated. Besides, by using the linear stability analysis combined with the direct simulations, we obtain the dependence of the threshold norm${N_{{\text{th}}}}$ on the topological charge ${S_1}$ and the nonlinear coefficient ${\text{δ}}g$. Finally, we study the composite vortex pattern constructed by two hidden vortices of QDs, namely nested vortex QDs. Based on the fact that the hidden vortices of QDs generally have flat-top density profiles, the Thomas-Fermi approximation can be used to verify the numerical results effectively. The results of this paper can be extended in some directions, and provide a theoretical basis for the experimental realization of the hidden vortices of QDs.

Keywords:

作者及机构信息

1.
佛山科学技术学院物理与光电工程学院, 佛山　528225

2.
佛山科学技术学院, 物理与光电工程学院粤港澳智能微纳光电技术联合实验室, 佛山　528225

通信作者: 刘彬, binliu@fosu.edu.cn

基金项目: 国家自然科学基金(批准号: 11905032, 11874112)、广东省自然科学基金面上项目(批准号: 2021A1515010214)、广东省普通高校重点科研项目(批准号: 2019KZDXM001)、广东省基础与应用基础研究基金(批准号: 2021A1515111015)和粤港澳智能微纳光电技术联合实验室(批准号: 2020B1212030010)资助的课题.

Authors and contacts

1.
School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528225, China

2.
Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528225, China

Corresponding author: Liu Bin, binliu@fosu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11905032, 11874112), the Natural Science Foundation of Guangdong Province, China (Grant No. 2021A1515010214), the Key Research Projects of General Colleges in Guangdong Province, China (Grant No. 2019KZDXM001), the Guang Dong Basic and Applied Basic Research Foundation, China (Grant No. 2021A1515111015), and Research Fund of Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, China (Grant No. 2020B1212030010).

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施引文献

图 1 隐秘涡旋量子液滴的典型例子 (a1), (a2)总粒子数$N = 2500$时, 拓扑荷为${S_{1, 2}}{\text{ = }} \pm {\text{3}}$的两分量密度分布图; (b1), (b2) 分别对应其相位分布; (c1), (c2)总粒子数$N = 30000$时, 拓扑荷为${S_{1, 2}}{\text{ = }} \pm {\text{4}}$的两分量的密度分布图; (d1), (d2) 分别对应其相位分布. 系统参数为$g = 10$和$\text{δ} g = 0.5$

Fig. 1. Typical examples of stable hidden vortices of QDs: (a1), (a2) Density patterns of the two components with $(N, {S_{1, 2}}) = (2500, \pm {\text{3}})$; (b1), (b2) the corresponding phase diagrams of the hidden vortices of QDs are in panels (a1) and (a2), respectively; (c1), (c2) density patterns of the two components with $(N, {S_{1, 2}}) = (30000, \pm 4)$; (d1), (d2) the corresponding phase diagrams of the hidden vortices of QDs are in panels (c1) and (c2), respectively. The other parameters are $g = 10$ and $\text{δ} g = 0.5$.

下载: 全尺寸图片幻灯片

图 2 (a) 隐秘涡旋量子液滴的有效面积${A_{{\text{eff}}}}$与总粒子数$N$之间的依赖关系, ${N_{{\text{th}}}}$表示稳定与不稳定区域的临界值, 红色实线表示稳定区域, 黑色点虚线表示不稳定区域; (b) 蓝色点表示固定${\text{δ}}g = 0.5$时, 临界值${N_{{\text{th}}}}$与拓扑荷$ S_1$之间的依赖关系; 黑色“★”表示临界值下的隐秘涡旋量子液滴的内半径大小${R_{{\text{in}}}}$; (c) 固定${S_{1, 2}} = \pm 1$时, 不同$\delta g$值下的临界值${N_{{\text{th}}}}$; (d) 固定${S_{1, 2}} = \pm 1$时, 化学势$\mu $与总粒子数$N$之间的依赖关系

Fig. 2. (a) The dependence of the effective area${A_{{\text{eff}}}}$on the norm$N$of the hidden vortices of QDs, ${N_{{\text{th}}}}$is the threshold norm, red solid curve shows the stable region and the black dash curve shows the unstable region; (b) the blue dot represents the dependence of the threshold${N_{{\text{th}}}}$on the topological charge$ S_1$when${\text{δ}}g = 0.5$, the black star represents the inner radius of the hidden vortices of QDs at the threshold norm; (c) the threshold norm${N_{{\text{th}}}}$for hidden vortices of QDs with ${S_{1, 2}} = \pm 1$ as a function of ${\text{δ}}g$; (d) the dependence of the chemical potential$\mu $on the total norm$N$with ${S_{1, 2}} = \pm 1$.

下载: 全尺寸图片幻灯片

图 3 (a1)—(a3)$ N = 300 $时, ${\varPhi _1}$分量传输到$t = 0$, $t = 7300$和$t = 10000$时密度分布情况, 显然此时传输不稳定; (a4)涡旋拓扑荷数的扰动值$m = 0, 1, 2, 3$时不稳定增长率$\lambda $的实部和虚部的关系图; (b1)—(b3)$ N = 1500 $时, ${\varPhi _1}$分量传输到$t = 0$, $t = 5200$和$t = 10000$时密度分布情况, 显然此时传输稳定; (b4)涡旋拓扑荷数的扰动值$m = 0, 1, 2, 3$时不稳定增长率$\lambda $的实部和虚部的关系图

Fig. 3. (a1)–(a3) The density pattern of the ${\varPhi _1}$ component with $N = 300$ and $t = 0, 7300, 10000$, which is obviously unstable; (a4) perturbation eigenvalues for the corresponding hidden vortices of QDs with $N = 300$ and ${S_{1, 2}} = \pm 1$ for different azimuthal index $m = 0, 1, 2, 3$; (b1)–(b3) the density pattern of the ${\varPhi _1}$ component with $ N = 1500 $ and $t = 0, 5200, 10000$; (b4) perturbation eigenvalues for the corresponding hidden vortices of QDs with $N = 1500$ and ${S_{1, 2}} = \pm 1$ for different azimuthal index $m = 0, 1, 2, 3$.

下载: 全尺寸图片幻灯片

图 4 嵌套涡旋的典型例子 (a1)—(a3)$N = {\text{800}}, {S_{1, {\text{2}}}} = \pm 1$和$N = {\text{30000}}, {S_{1, 2}} = \pm 4$嵌套形成的嵌套涡旋量子液滴; (a4) 液滴传输到$t = 10000$时的相位分布; (b1)—(b3) $N = {\text{800}}, {S_{1, {\text{2}}}} = \pm 1$和$N = {\text{26000}}, {S_{1, 2}} = \pm 4$嵌套形成的嵌套涡旋量子液滴; (b4) 液滴传输到$t = 4500$时的相位分布; (c1)—(c3)$N = {\text{800}}, {S_{1, {\text{2}}}} = \pm 1$和$N = {\text{30000}}, {S_{1, 2}} = \mp 4$嵌套形成的嵌套涡旋量子液滴; (c4) 液滴传输到$t = 10000$时的相位分布; (d1)—(d3) $N = {\text{800}}, {S_{1, {\text{2}}}} = \pm 1$和$N = {\text{26000}}, {S_{1, 2}} = \mp 4$嵌套形成的嵌套涡旋量子液滴; (d4) 液滴传输到$t = 6400$时的相位分布

Fig. 4. Typical examples of the nested vortex QDs: (a1)—(a3) The hidden vortices of QDs with$(N, {S_{1, 2}}) = (800, \pm 1)$nests inside $(N, {S_{1, 2}}) = (30000, \pm 4)$, which has a larger inner radius; (a4)output pattern of the phase structure for the nested hidden vortices of QDs at $t = 10000$; (b1)–(b3) the hidden vortices of QDs with$(N, {S_{1, 2}}) = (800, \pm 1)$nests inside $(N, {S_{1, 2}}) = (26000, \pm 4)$; (b4) output pattern of the phase structure for the nested hidden vortices of QDs at $t = 4500$; (c1)–(c3) the hidden vortices of QDs with$(N, {S_{1, 2}}) = (800, \pm 1)$nests inside $(N, {S_{1, 2}}) = (30000, \mp 4)$; (c4) output pattern of the phase structure for the nested hidden vortices of QDs at $t = 10000$; (d1)–(d3) the hidden vortices of QDs with$(N, {S_{1, 2}}) = (800, \pm 1)$nests inside $(N, {S_{1, 2}}) = (26000, \mp 4)$; (d4) output pattern of the phase structure for the nested hidden vortices of QDs at $t = 6400$.

下载: 全尺寸图片幻灯片

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