Hidden vortices of quantum droplets in quasi-two dimensional space

Chen Yi-Xi; Cai Xiao-Yan; Liu Bin; Jiang Xun-Da; Li Yong-Yao

doi:10.7498/aps.71.20220709

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:

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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References

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Cited By

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$\text{δ} g = 0.5$

Figure 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 $\text{δ} g = 0.5$.

DownLoad: Full-Size Img PowerPoint

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

Figure 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 ${S_{1, 2}} = \pm 1$.

DownLoad: Full-Size Img PowerPoint

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

Figure 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 $m = 0, 1, 2, 3$.

DownLoad: Full-Size Img PowerPoint

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

Figure 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 $t = 6400$.

DownLoad: Full-Size Img PowerPoint

map

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[9]	贾瑞煜, 方乒乒, 高超, 林机 2021 70 180303 Google Scholar Jia R Y, Fang P P, Gao C, Lin J 2021 Acta Phys. Sin. 70 180303 Google Scholar
[10]	陈海军, 李向富 2013 62 070302 Google Scholar Chen H J, Li X F 2013 Acta Phys. Sin. 62 070302 Google Scholar
[11]	Li Y E, Xue J K 2016 Chin. Phys. Lett. 33 100502 Google Scholar
[12]	周嘉仪, 李画眉 2014 浙江师范大学学报 38 303 Zhou J Y, Li H M 2014 J. Zhejiang Normal Univ. 38 303
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[14]	Wen L, Sun Q, Wang H Q, Ji A C, Liu W M 2012 Phys. Rev. A 86 043602 Google Scholar
[15]	Wang L X, Dai C Q, Wen L, Liu T, Jiang H F, Saito H, Zhang Sh G, Zhang X F 2018 Phys. Rev. A 97 063607 Google Scholar
[16]	Peng P, Li G Q, Zhao L C, Yang W L, Yang Z Y 2019 Phys. Lett. A 383 2883 Google Scholar
[17]	Segev M, Valley G C, Crosignani B, DiPorto P, Yariv A 1994 Phys. Rev. Lett. 73 3211 Google Scholar
[18]	Huang J S, Jiang X D, Chen H Y, Fan Z W, Pang W, Li Y Y 2015 Front. Phys. 10 100507
[19]	Maucher F, Henkel N, Saffman M, Królikowski W, Skupin S, Pohl T 2011 Phys. Rev. Lett. 106 170401 Google Scholar
[20]	Wang Y Y, Chen L, Dai C Q, Zheng J, Fan Y 2017 Nonlinear Dyn. 90 1269 Google Scholar
[21]	Dai C Q, Chen R P, Wang Y Y, Fan Y 2017 Nonlinear Dyn. 87 1675 Google Scholar
[22]	Chen Y X, Zheng L H, Xu F Q 2018 Nonlinear Dyn. 93 2379 Google Scholar
[23]	Li J T, Zhu Y, Han J Z, Qin W, Dai C Q, Wang S H 2018 Nonlinear Dyn. 91 757 Google Scholar
[24]	Chen X W, Deng Z G, Xu X X, Li S L, Fan Z W, Chen Z P, Liu B, Li Y Y 2020 Nonlinear Dyn. 101 569 Google Scholar
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[26]	Liu B, Zhong R X, Chen Z P, Qin X Z, Zhong H H, Li Y Y, Malomed B A 2020 New J. Phys. 22 043004 Google Scholar
[27]	Li Y Y, Luo Z H, Liu Y, Chen Z P, Huang C Q, Fu S H, Tan H S, Malomed B A 2017 New J. Phys. 19 113043 Google Scholar
[28]	Hu X H, Zhang X F, Zhao D, Luo H G, Liu W M 2009 Phys. Rev. A 79 023619 Google Scholar
[29]	李吉, 刘斌, 白晶, 王寰宇, 何天琛 2020 69 140301 Google Scholar Li J, Liu B, Bai J, Wang H Y, He T C 2020 Acta Phys. Sin. 69 140301 Google Scholar
[30]	陈海军, 任元, 王华 2022 71 056701 Google Scholar Chen H J, Ren Y, Wang H 2022 Acta Phys. Sin. 71 056701 Google Scholar
[31]	Schmitt M, Wenzel M, Böttcher F, Ferrier-Barbut I, Pfau T 2016 Nature 539 259 Google Scholar
[32]	Cabrera C R, Tanzi L, Sanz J, Naylor B, Thomas P, Cheiney P, Tarruell L 2018 Science 359 301 Google Scholar
[33]	Lee T D, Huang K, Yang C N 1957 Phys. Rev. 106 1135 Google Scholar
[34]	Tylutki M, Astrakharchik G E, Malomed B A, Petrov D S 2020 Phys. Rev. A 101 051601 Google Scholar
[35]	Boudjemâa A 2018 Phys. Rev. A 98 033612 Google Scholar
[36]	Hu H, Liu X J 2020 Phys. Rev. A 102 043302 Google Scholar
[37]	Pylak M, Gampel F, Płodzień M, Gajda M 2022 Phys. Rev. Res. 4 013168 Google Scholar
[38]	Parisi L, Giorgini S 2020 Phys. Rev. A 102 023318 Google Scholar
[39]	Bisset R N, Peña Ardila L A, Santos L 2021 Phys. Rev. Lett. 126 025301 Google Scholar
[40]	Zin P, Pylak M, Gajda M 2021 Phys. Rev. A 103 013312 Google Scholar
[41]	Ferrier-Barbut I, Wenzel M, Böttcher F, Langen T, Isoard M, Stringari S, Pfau T 2018 Phys. Rev. Lett. 120 160402 Google Scholar
[42]	Kartashov Y V, Malomed B A, Torner L 2020 Phys. Rev. Res. 2 033522 Google Scholar
[43]	Otajonov S R, Tsoy E N, Abdullaev F K 2020 Phys. Rev. E 102 062217
[44]	Luo Z H, Pang W, Liu B, Li Y Y, Malomed B A 2021 Front. Phys. 16 32201 Google Scholar
[45]	Liu B, Zhang H F, Zhong R X, Zhang X L, Qin X Z, Huang C Q, Li Y Y, Malomed B A 2019 Phys. Rev. A 99 053602 Google Scholar
[46]	Zhao F Y, Yan Z T, Cai X Y, Li C L, Chen G L, He H X, Liu B, Li Y Y 2021 Chaos Solitons Fractals 152 111313 Google Scholar
[47]	Huang H, Wang H C, Chen M, Lim C S, Wong K C 2022 Chaos Solitons Fractals 158 112079 Google Scholar
[48]	Guo M Y, Pfau T 2021 Front. Phys. 16 32202 Google Scholar
[49]	Malomed B A 2021 Front. Phys. 16 22504 Google Scholar
[50]	Cui X L 2018 Phys. Rev. A 98 023630 Google Scholar
[51]	Wang Y Q, Guo L F, Yi S, Shi T 2020 Phys. Rev. Res. 2 043074 Google Scholar
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Vol. 71, Issue 20 - 2022-10-20

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