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In this work, a one-dimensional lattice theory scheme is proposed based on superconducting microwave cavity, which includes two different types of microwave cavity unit cells. The coupling between the unit cells is controlled by flux qubits to simulate and study their topological insulator characteristics. Specifically, by mapping the counter-rotating wave terms into the p-wave superconducting pairing term, a one-dimensional superconducting microwave cavity lattice scheme with a p-wave superconducting pairing term is obtained. It is found that the p-wave superconducting pairing term can modulate the topological quantum state of the system, allowing the topological quantum information transmission channels with four edge states to be created. In addition, when the p-wave superconducting pairing term interacts with the nearest-neighbor, the energy band undergoes fluctuations, thus inducing new energy bands to be generated, but the degeneracy of the edge states remains stable, which can realize the multiple topological quantum state transmission paths. However, when its regulation exceeds the threshold, the energy gap of the system will close, causing the edge states to annihilate in a new energy band. Furthermore, with defects considered to exist in the system, when the strength of the defect is small, the edge state produces small fluctuations, but it can be clearly distinguished, showing its robustness. When the strength of the defect exceeds the threshold, the edge state and energy band will cause irregular fluctuations, allowing the edge state to integrate into an energy band. Our research results have important theoretical value and practical significance, and can be applied to quantum optics and quantum information processing in the future.
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Keywords:
- quantum optics /
- superconducting quantum circuits /
- topological insulators /
- quantum states
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[4] Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys. 88 035005
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[5] Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606
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[6] Li L, Xu Z, Chen S 2014 Phys. Rev. B 89 085111
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[7] Li L, Chen S 2015 Phys. Rev. B 92 085118
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[8] Mei F, Zhu S L, Zhang Z M, Oh C H, Goldman N 2012 Phys. Rev. A 85 013638
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[9] Wray L A, Xu V, Xia Y, Hsieh D, Fedorov A V, SanHor Y, Cava R J, Bansil A, Lin H, Hasan M Z 2011 Nat. Phys. 7 32
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[10] Malki M, Uhrig G S 2017 Phys. Rev. B 95 235118
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[11] Chitov G Y 2018 Phys. Rev. B 97 085131
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[12] Agrapidis C E, van den Brink J, Nishimoto S 2019 Phys. Rev. B 99 224418
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[17] Martin I, Shnirman A, Lin T, Zoller P 2004 Phys. Rev. B 69 125339
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[18] Huang S M, Agarwal G S 2010 Phys. Rev. A 81 033830
Google Scholar
[19] Wang K, Yu Y F, Zhang Z M 2019 Phys. Rev. A 100 053832
Google Scholar
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Google Scholar
[21] Xiao Y, Yu Y F, Zhang Z M 2014 Opt. Express 22 17979
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Google Scholar
[25] Teufel J D, Li D, Allman M S, Cicak K, Sirois A J, Whittaker J D, Simmonds R W 2011 Nature 471 204
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[26] Zhang Z C, Wang Y P, Yu Y F, Zhang Z M 2019 Ann. Phys. 531 1800461
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[27] Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2019 Laser Phys. Lett. 16 015205
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[28] Devoret M H, Schoelkopf R J 2013 Science 339 1169
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[29] Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2018 J. Phys. B: At. Mol. Opt. Phys. 51 175504
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[30] Roque T F, Peano V, Yevtushenko O M, Marquardt F 2017 New J. Phys. 19 013006
Google Scholar
[31] Wan L L, Lü X Y, Gao J H, Wu Y 2017 Opt. Express 25 017364
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Google Scholar
[33] Qi L, Yan Y, Wang G L, Zhang S, Wang H F 2019 Phys. Rev. B 100 062323
Google Scholar
[34] Xu X W, Zhao Y J, Wang H, Chen A X, Liu Y X 2022 Front. Phys. 9 813801
Google Scholar
[35] 刘浪, 王一平 2022 71 224202
Google Scholar
Liu L, Wang Y P 2022 Acta Phys. Sin. 71 224202
Google Scholar
[36] Mei F, Xue Z Y, Zhang D W, Tian L, Lee C, Zhu S L 2016 Quantum Sci. Technol. 1 015006
Google Scholar
[37] Koch J, Houck A A, Le Hur K, Girvin S M 2010 Phys. Rev. A 82 043811
Google Scholar
[38] Mei F, You J B, Nie W, Fazio R, Zhu S L, Kwek L C 2015 Phys. Rev. A 92 041805
Google Scholar
[39] Cao J, Yi X X, Wang H F 2020 Phys. Rev. A 102 032619
Google Scholar
[40] Cai W, Han J, Mei F, Yuan X Z, Sun L Y 2019 Phys. Rev. Lett. 123 080501
Google Scholar
[41] Chatterjee P, Pradhan S, Nandy A K, Saha A 2023 Phys. Rev. B 107 085423
Google Scholar
[42] Tong X, Meng Y M, Jiang X, Lee C, de Moraes Neto G D, Gao X L 2021 Phys. Rev. B 103 104202
Google Scholar
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图 1 (a)基于超导微波腔组成的一维晶格系统, $Q_{1}$($Q_{2}$)是晶胞之间的耦合磁通量子比特, $g_{1}$($g_{2}$)表示$a_{n}$($b_{n}$)和$b_{n}$($a_{n+1}$)的耦合参数, T表示$a_{n}$和$a_{n+1}$($b_{n}$和$b_{n+1}$)的耦合参数; (b) $a_{n}$和$b_{n}$耦合在一个频率可调的控制场上, $g_{1}$($g_{2}$) 可以通过磁通量子比特外部磁通调控, T通过电容C耦合调制
Figure 1. (a) Schematic of the 1D superconducting microwave cavity lattice system, $Q_{1}$($Q_{2}$) is the coupling flux qubit between the unit cell, $a_n$ and $b_n$ ($b_n $ and $a_{n+1}$) coupling coefficient is $g_{1}$($g_{2}$), $a_n$ and $a_{n+1}$ ($b_n$ and $b_{n+1}$) coupling coefficient is T; (b) $a_{n}$ and $b_{n}$ are connected in a tunable frequency field, $g_{1}$($g_{2}$) can be modulated by the external flux of qubits, T is modulated by the capacitance C coupling.
图 2 (a)系统能谱与晶格数目的关系; (b)蓝色和(c)红色边缘态的概率分布图; 其中$\widetilde{G}_{12}=0.15, \widetilde{G}_{24}=0.3$和晶格数$N=100$
Figure 2. (a) Energy spectrum of the system via the lattice numbers; (b), (c) probability distributions of (b) blue and (c) red edge states. $\widetilde{G}_{12}=0.15, \widetilde{G}_{24}=0.3$ and lattice size $N=100$.
图 3 系统能谱与晶格数目的关系 (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.003$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.03$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.06$; (d) $2\widetilde{G}_{11}=$ $ \widetilde{G}_{23}=0.15 $; (e) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.165$; (f) $2\widetilde{G}_{11}=\widetilde{G}_{23}=2.1$; 其他参数为$\widetilde{G}_{12}=0.15, \widetilde{G}_{24}=0.3$和晶格数$N=200$
Figure 3. Energy spectrum of the system via the lattice numbers: (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.003$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.03$; (c) $2\widetilde{G}_{11}= $$\widetilde{G}_{23}=0.06 $; (d) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.15$; (e) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.165$; (f) $2\widetilde{G}_{11}=\widetilde{G}_{23}=2.1$. Other parameters are $ \widetilde{G}_{12}=0.15,$ $ \widetilde{G}_{24}=0.3$ and lattice size $N=200$.
图 4 4个不同边缘态的概率分布图 (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.003$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.006$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.009$; (d) $2\widetilde{G}_{11}= $ $ \widetilde{G}_{23}=0.015$; (e) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.021$; (f) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.027$; 其他参数为$\widetilde{G}_{12}=0.15$, $\widetilde{G}_{24}=0.3$和晶格数$N=200$
Figure 4. State distributions of four different edge states: (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.003$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.006$; (c) $2\widetilde{G}_{11}= $ $ \widetilde{G}_{23}=0.009$; (d) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.015$; (e) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.021$; (f) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.027$. Other parameters are $\widetilde{G}_{12}=0.15$, $\widetilde{G}_{24}=0.3$ and lattice size $N=200$.
图 5 系统能谱与相位的关系 (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1,\; T=0$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1,\; T=0.05$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08(1+ $$ \cos\theta),\; T=0$; 其他参数为$\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$和晶格数$N=200$
Figure 5. Energy spectrum of the system via the phase: (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1,\; T=0$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}= 0.1,\; T=0.05$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08(1+\cos\theta),\; T=0$. Other parameters are $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$ and lattice size $N=200$.
图 6 系统能谱与相位的关系 (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.16$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.24$; (d) $2\widetilde{G}_{11}= $ $\widetilde{G}_{23}=0.32$; 其他参数为$\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $T=0.1$和晶格数$N=200$
Figure 6. Energy spectrum of the system via the phase: (a) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08$; (b) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.16$; (c) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.24$; (d) $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.32$. Other parameters are $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $T=0.1$ and lattice size $N=200$.
图 7 系统能谱与相位的关系 (a) $T=0.1$; (b) $T=0.2$; (c) $T=0.3$; (d) $T=3$; 其他参数为$\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $2\widetilde{G}_{11}= \widetilde{G}_{23}=0.04$和晶格数$N=200$
Figure 7. Energy spectrum of the system via the phase: (a) $T=0.1$; (b) $T=0.2$; (c) $T=0.3$; (d) $T=3$. Other parameters are $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.04$ and lattice size $N=200$.
图 8 4个不同边缘态的分布图 (a) $\theta=\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08$; (b) $\theta=\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1$; (c) $\theta=3\pi/2$, $2\widetilde{G}_{11}= $ $ \widetilde{G}_{23}=0.08$; (d) $\theta=3\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1$. 其他参数为$T=0.1$, $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$和晶格数$N=200$
Figure 8. State distributions of four different edge states: (a) $\theta=\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08$; (b) $\theta=\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}= $$ 0.1$; (c) $\theta=3\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.08$; (d) $\theta=3\pi/2$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.1$. Other parameters are $T=0.1$, $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$ and lattice size $N=200$.
图 9 系统能谱与随机缺陷的关系图 (a) $\omega=0.1$, $\nu=\tau=0$; (b) $\omega=0.3$, $\nu=\tau=0$; (c) $\omega=0.5$, $\nu=\tau=0$; (d) $\omega=0.7$, $\nu=\tau=0$; (e) $\nu=0.1$, $\omega=\tau=0$; (f) $\nu=0.2$, $\omega=\tau=0$; (g) $\nu=0.3$, $\omega=\tau=0$; (h) $\nu=0.4$, $\omega=\tau=0$; (i) $\tau=0.1$, $\omega=\nu=0$; (j) $\tau=0.2$, $\omega=\nu=0$; (k) $\tau=0.3$, $\omega=\nu=0$; (l) $\tau=0.4$, $\omega=\nu=0$; 其他参数为$\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.04$, $T=0.1$和晶格数$N=200$
Figure 9. Energy spectrum of the system via the random defects: (a) $\omega=0.1$, $\nu=\tau=0$; (b) $\omega=0.3$, $\nu=\tau=0$; (c) $\omega=0.5$, $\nu=\tau=0$; (d) $\omega=0.7$, $\nu=\tau=0$; (e) $\nu=0.1$, $\omega=\tau=0$; (f) $\nu=0.2$, $\omega=\tau=0$; (g) $\nu=0.3$, $\omega=\tau=0$; (h) $\nu=0.4$, $\omega=\tau=0$; (i) $\tau=0.1$, $\omega=\nu=0$; (j) $\tau=0.2$, $\omega=\nu=0$; (k) $\tau=0.3$, $\omega=\nu=0$; (l) $\tau=0.4$, $\omega=\nu=0$. Other parameters are $\widetilde{G}_{12}=0.2$, $\widetilde{G}_{24}=0.4$, $2\widetilde{G}_{11}=\widetilde{G}_{23}=0.04$, $T=0.1$ and lattice size $N=200$.
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[1] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[2] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[3] Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004
Google Scholar
[4] Chiu C K, Teo J C Y, Schnyder A P, Ryu S 2016 Rev. Mod. Phys. 88 035005
Google Scholar
[5] Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606
Google Scholar
[6] Li L, Xu Z, Chen S 2014 Phys. Rev. B 89 085111
Google Scholar
[7] Li L, Chen S 2015 Phys. Rev. B 92 085118
Google Scholar
[8] Mei F, Zhu S L, Zhang Z M, Oh C H, Goldman N 2012 Phys. Rev. A 85 013638
Google Scholar
[9] Wray L A, Xu V, Xia Y, Hsieh D, Fedorov A V, SanHor Y, Cava R J, Bansil A, Lin H, Hasan M Z 2011 Nat. Phys. 7 32
Google Scholar
[10] Malki M, Uhrig G S 2017 Phys. Rev. B 95 235118
Google Scholar
[11] Chitov G Y 2018 Phys. Rev. B 97 085131
Google Scholar
[12] Agrapidis C E, van den Brink J, Nishimoto S 2019 Phys. Rev. B 99 224418
Google Scholar
[13] Braginskii V B, Manukin A B 1967 Sov. Phys. JETP 25 653
[14] Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys 86 1391
Google Scholar
[15] Liu Y L, Wang C, Zhang J, Liu Y X 2018 Chin. Phys. B 27 024204
Google Scholar
[16] Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520
Google Scholar
[17] Martin I, Shnirman A, Lin T, Zoller P 2004 Phys. Rev. B 69 125339
Google Scholar
[18] Huang S M, Agarwal G S 2010 Phys. Rev. A 81 033830
Google Scholar
[19] Wang K, Yu Y F, Zhang Z M 2019 Phys. Rev. A 100 053832
Google Scholar
[20] Wei W Y, Yu Y F, Zhang Z M 2018 Chin. Phys. B 27 034204
Google Scholar
[21] Xiao Y, Yu Y F, Zhang Z M 2014 Opt. Express 22 17979
Google Scholar
[22] Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2019 Chin. Phys. B 28 014202
Google Scholar
[23] You J Q, Nori F 2011 Nature 474 589
Google Scholar
[24] Massel F, Heikkil T T, Pirkkalainen J M, Cho S U, Saloniemi H, Hakonen P J, Sillanpää M A 2011 Nature 480 351
Google Scholar
[25] Teufel J D, Li D, Allman M S, Cicak K, Sirois A J, Whittaker J D, Simmonds R W 2011 Nature 471 204
Google Scholar
[26] Zhang Z C, Wang Y P, Yu Y F, Zhang Z M 2019 Ann. Phys. 531 1800461
Google Scholar
[27] Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2019 Laser Phys. Lett. 16 015205
Google Scholar
[28] Devoret M H, Schoelkopf R J 2013 Science 339 1169
Google Scholar
[29] Wang Y P, Zhang Z C, Yu Y F, Zhang Z M 2018 J. Phys. B: At. Mol. Opt. Phys. 51 175504
Google Scholar
[30] Roque T F, Peano V, Yevtushenko O M, Marquardt F 2017 New J. Phys. 19 013006
Google Scholar
[31] Wan L L, Lü X Y, Gao J H, Wu Y 2017 Opt. Express 25 017364
Google Scholar
[32] Wang W, Wang Y P 2022 Acta Phys. Sin. 71 194203
Google Scholar
[33] Qi L, Yan Y, Wang G L, Zhang S, Wang H F 2019 Phys. Rev. B 100 062323
Google Scholar
[34] Xu X W, Zhao Y J, Wang H, Chen A X, Liu Y X 2022 Front. Phys. 9 813801
Google Scholar
[35] 刘浪, 王一平 2022 71 224202
Google Scholar
Liu L, Wang Y P 2022 Acta Phys. Sin. 71 224202
Google Scholar
[36] Mei F, Xue Z Y, Zhang D W, Tian L, Lee C, Zhu S L 2016 Quantum Sci. Technol. 1 015006
Google Scholar
[37] Koch J, Houck A A, Le Hur K, Girvin S M 2010 Phys. Rev. A 82 043811
Google Scholar
[38] Mei F, You J B, Nie W, Fazio R, Zhu S L, Kwek L C 2015 Phys. Rev. A 92 041805
Google Scholar
[39] Cao J, Yi X X, Wang H F 2020 Phys. Rev. A 102 032619
Google Scholar
[40] Cai W, Han J, Mei F, Yuan X Z, Sun L Y 2019 Phys. Rev. Lett. 123 080501
Google Scholar
[41] Chatterjee P, Pradhan S, Nandy A K, Saha A 2023 Phys. Rev. B 107 085423
Google Scholar
[42] Tong X, Meng Y M, Jiang X, Lee C, de Moraes Neto G D, Gao X L 2021 Phys. Rev. B 103 104202
Google Scholar
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