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The Casimir effect has received extensive attention theoretically and experimentally in recent years. It arises from the macroscopic manifestation of quantum vacuum fluctuations, and this Casimir interaction force can be an effective means of driving and controlling components in micro-electro-mechanical system (MEMS) and nano-electromechanical system (NEMS). Due to the new possibilities provided by photonic topological insulator for designing and using photonic devices, in this work, the Casimir force between the multilayer structures of non-reciprocal photonic topological insulators with broken time-reversal symmetry is investigated, and the influences of the dielectric tensor of the photonic topological insulator, the spatial structural parameters of the multilayer system, and the rotational degree of freedom on the Casimir force are examined. It is found that there exists Casimir repulsive force in such a multilayer system, and the Casimir stable equilibrium and restoring force can be further realized and controlled. Continuous variation between anti-mirror-symmetric configuration and mirror-symmetric configuration is examined. Both the Casimir attraction and repulsion can be generally enhanced through structural optimization by increasing layer number and individual layer thickness. Furthermore, we focus on the detailed analysis of how the optical axis angle difference within the photonic topological insulator layers can be used to adjust the Casimir force. The overall relative rotation of the multilayer system may adjust the magnitude and the direction of the Casimir force, and some inflection points can be found from the influence curve of the optical axis angle difference between internal layers of the multilayer on the Casimir force, allowing the rotational degrees of freedom in the multilayer system to be used for fine-adjusting the Casimir interaction. This work introduces the enhanced degrees of freedom for probing and manipulating the interaction between small objects in micro/nano systems, thereby suppressing adverse Casimir forces and effectively using them.
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Keywords:
- photonic topological insulator /
- Casimir effect /
- repulsive Casimir force /
- multilayered system
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Google Scholar
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Google Scholar
[26] Grushin A G, Cortijo A 2011 Phys. Rev. Lett. 106 020403
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Google Scholar
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Google Scholar
[30] Nefedov I S, Valagiannopoulos C A, Melnikov L A 2013 J. Opt. 15 114003
Google Scholar
[31] Zeng R, Chen L, Nie W, Bi M, Yang Y, Zhu S 2016 Phys. Lett. A 380 2861
Google Scholar
[32] Chiadini F, Fiumara V, Lakhtakia A, Scaglione A 2019 Appl. Opt. 58 1724
Google Scholar
[33] Zeng R, Gao T, Ni P, Fang S, Li H, Yang S, Zeng Z 2024 J. Opt. 26 075602
Google Scholar
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Google Scholar
[35] Silveirinha M G 2015 Phys. Rev. B 92 125153
Google Scholar
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Google Scholar
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Google Scholar
[38] Bittencourt J A 2004 Fundamentals of Plasma Physics (Springer: New York
[39] Silveirinha M G, Terças H, Antezza M 2023 Phys. Rev. B 108 235154
Google Scholar
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图 1 光子拓扑绝缘体多层结构间Casimir效应示意图, 其中左右多层结构间距L, 左右侧结构相应的参考系分别对应为x-y-z和x'-y'-z坐标系, 红色箭头表示光子拓扑绝缘体的x(x')方向光轴 (a) 两侧结构的该光轴反平行; (b) 两侧结构该光轴的夹角为$\theta $; (c) 系统内部各层相互间也存在光轴夹角
Figure 1. Sketch of the Casimir effect between multilayered structures made of photonic topological insulators, where L is the separation between two structures, x-y-z and x'-y'-z are the coordinates in the left and right structures, respectively. The red arrows indicate the x(x') optical axes of photonic topological insulators: (a) Anti-parallel on the two sides, (b) an angle $\theta $ between the two sides; (c) exist angles between layers within the system.
图 2 反镜像对称的两个N层光子拓扑绝缘体结构(N = 1, 2, 3)间的Casimir作用力相对幅值随结构间距L的变化
Figure 2. Relative Casimir force between two N-layer photonic topological insulator structures (N = 1, 2, 3) with inversion-mirror symmetry as a function of the separation L.
图 3 反镜像对称N层光子拓扑绝缘体结构间 (a) Casimir平衡回复力; (b) Casimir力随${\varepsilon _n}$的变化曲线
Figure 3. Within the anti-mirror symmetric N-layer photonic topological insulator structure: (a) The restoring Casimir force; (b) the Casimir force as a function of ${\varepsilon _n}$.
图 4 反镜像对称N层光子拓扑绝缘体结构(N = 1, 2, 3)间的Casimir作用力相对幅值随单元层厚度的变化, 其中结构间距L = 1.0λ, 其他系统参数同图2中的取值
Figure 4. Relative Casimir force between two N-layer photonic topological insulator structures (N = 1, 2, 3) with inversion-mirror symmetry as a function of the layer thickness, where L = 1.0λ and other parameters are the same as in Fig. 2.
图 5 N层光子拓扑绝缘体结构(N = 1, 2, 3)间的Casimir作用力相对幅值 (a)在镜像对称情况下随间距L的变化; (b)随光轴夹角$\theta $的变化
Figure 5. (a) Relative amplitude of Casimir force between N-layer photonic topological insulator structures (N = 1, 2, 3): (a) Variation with the separation L with mirror symmetry; (b) variation with the $\theta $.
图 6 锑化铟三层系统间Casimir作用力相对幅值在不同 内部层间光轴夹角下随整体旋转角度$\theta $的变化, 其中相 关参数为${\omega _{\text{L}}} = 3.62 \times {10^{13}} {\text{ rad/s}}$, ${\omega _{\text{T}}} = 3.39 \times {10^{13}} {\text{ rad/s}}$, $\varGamma = 5.65 \times {10^{11}} {\text{ rad/s}}$, $\gamma = 3.39 \times {10^{12}} {\text{ rad/s}}$, $N = 1.07 \times $$ {10^{17}}\;{\text{c}}{{\text{m}}^{{{ - 3}}}}$, B = 10 T, 其他参数同图4中的取值
Figure 6. Relative Casimir force between the three-layer InSb systems as a function of the overall rotation angle $\theta $ under different optical axis angles between internal layers, where ${\omega _{\text{L}}} = 3.62 \times {10^{13}}\; {\text{rad/s}}$, ${\omega _{\text{T}}} = 3.39 \times {10^{13}} {\text{ rad/s}}$, $\varGamma = 5.65 \times $$ {10^{11}} {\text{ rad/s}}$, $\gamma = 3.39 \times {10^{12}} {\text{ rad/s}}$, $N = 1.07 \times {10^{17}}\;{\text{c}}{{\text{m}}^{{{ - 3}}}}$, B = 10 T, and other parameters are the same as in Fig. 4.
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[1] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[2] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[3] Xia Y, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor Y S, Cava R J, Hasan M Z 2009 Nat. Phys. 5 398
Google Scholar
[4] Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438
Google Scholar
[5] Luo W, Qi X L 2013 Phys. Rev. B 87 085431
Google Scholar
[6] Wang P, Ge J, Li J, Liu Y, Xu Y, Wang J 2021 Innovation 2 100098
Google Scholar
[7] Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904
Google Scholar
[8] Wang Z, Chong Y, Joannopoulos J D, Soljačić M 2009 Nature 461 772
Google Scholar
[9] 王子尧, 陈福家, 郗翔, 高振, 杨怡豪 2024 物理学报 73 064201
Google Scholar
Wang Z Y, Chen F J, Xi X, Gao Z, Yang Y H 2024 Acta Phys. Sin 73 064201
Google Scholar
[10] Wang Y, Lu Y H, Gao J, Chang Y J, Ren R J, Jiao Z Q, Zhang Z Y, Jin X M 2022 Chip 1 100003
Google Scholar
[11] Yang Y H, Yamagami Y, Yu X B, Pitchappa P, Webber J, Zhang B L, Fujita M, Nagatsuma T, Singh R 2020 Nat. Photonics 14 446
Google Scholar
[12] Webber J, Yamagami Y, Ducournau G, Szriftgiser P, Iyoda K, Fujita M 2021 J. Lightwave Technol. 39 7609
Google Scholar
[13] Tschernig K, Jimenez-Galán Á, Christodoulides D N, Ivanov M, Busch K, Bandres M A, Perez-Leija A 2021 Nat. Commun. 12 1974
Google Scholar
[14] Chen Y, He X T, Cheng Y J, Qiu H Y, Feng L T, Zhang M, Dai D X, Guo G C, Dong J W, Ren X F 2021 Phys. Rev. Lett. 126 230503
Google Scholar
[15] Dai T X, Ao Y T, Bao J M, Mao J, Chi Y L, Fu Z R, You Y L, Chen X J, Zhai C H, Tang B, Yang Y, Li Z H, Yuan L Q, Gao F, Lin X, Thompson M G, O’Brien J L, Li Y, Hu X Y, Gong Q H, Wang J W 2022 Nat. Photonics 16 248
Google Scholar
[16] Tang G J, He X T, Shi F L, Liu J W, Chen X D, Dong J W 2022 Laser Photonics Rev. 16 2100300
Google Scholar
[17] Lustig E, Maczewsky L J, Beck J, Biesenthal T, Heinrich M, Yang Z, Plotnik Y, Szameit A, Segev M 2022 Nature 609 931
Google Scholar
[18] Teo H T, Xue H R, Zhang B L 2022 Phys. Rev. A 105 053510
Google Scholar
[19] Devi K M, Jana S, Chowdhury D R 2021 Opt. Mater. Express 11 2445
Google Scholar
[20] Casimir H B G 1948 Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 51 793
[21] Palasantzas G, Sedighi M, Svetovoy V B 2020 Appl. Phys. Lett. 117 120501
Google Scholar
[22] Vasilyev O A, Marino E, Kluft B B, Schall P, Kondrat S 2021 Nanoscale 13 6475
Google Scholar
[23] 周帅, 柳开鹏, 戴士为, 葛力新 2025 物理学报 74 014202
Google Scholar
Zhou S, Liu K P, Dai S W, Ge L X 2025 Acta Phys. Sin 74 014202
Google Scholar
[24] Zeng R, Wang C, Zeng X D, Li H Z, Yang S N, Li Q L, Yang Y P 2020 Opt. Express 28 7425
Google Scholar
[25] Küçüköz B, Kotov O V, Canales A, Polyakov A Y, Agrawal A V, Antosiewicz T J, Shegai T O 2024 Sci. Adv. 10 eadn1825
Google Scholar
[26] Grushin A G, Cortijo A 2011 Phys. Rev. Lett. 106 020403
Google Scholar
[27] Fuchs S, Lindel F, Krems R V, Hanson G W, Antezza M, Buhmann S Y 2017 Phys. Rev. A 96 062505
Google Scholar
[28] Lindel F, Hanson G W, Antezza M, Buhmann S Y 2018 Phys. Rev. B 98 144101
Google Scholar
[29] Masyukov M S, Grebenchukov A N 2021 Phys. Rev. B 104 165308
Google Scholar
[30] Nefedov I S, Valagiannopoulos C A, Melnikov L A 2013 J. Opt. 15 114003
Google Scholar
[31] Zeng R, Chen L, Nie W, Bi M, Yang Y, Zhu S 2016 Phys. Lett. A 380 2861
Google Scholar
[32] Chiadini F, Fiumara V, Lakhtakia A, Scaglione A 2019 Appl. Opt. 58 1724
Google Scholar
[33] Zeng R, Gao T, Ni P, Fang S, Li H, Yang S, Zeng Z 2024 J. Opt. 26 075602
Google Scholar
[34] Kenneth O, Klich I 2006 Phys. Rev. Lett. 97 160401
Google Scholar
[35] Silveirinha M G 2015 Phys. Rev. B 92 125153
Google Scholar
[36] Xu J, He P P, Feng D L, Luo Y M, Fan S Q, Yong K L, Tsakmakidis K L 2023 Opt. Express 31 42388
Google Scholar
[37] Holmes A M, Sabbaghi M, Hanson G W 2021 Phys. Rev. B 104 214433
Google Scholar
[38] Bittencourt J A 2004 Fundamentals of Plasma Physics (Springer: New York
[39] Silveirinha M G, Terças H, Antezza M 2023 Phys. Rev. B 108 235154
Google Scholar
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