Modulation of topological phase transitions and topological quantum states by counter-rotating wave effect in one-dimensional superconducting microwave cavity lattice

Zheng Zhi-Yong; Chen Li-Jie; Xiang Lü; Wang He; Wang Yi-Ping

doi:10.7498/aps.72.20231321

Abstract

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.

Keywords:

Authors and contacts

College of Science, Northwest A&F University, Yangling 712100, China

Corresponding author: Wang Yi-Ping, ypwang2019@nwafu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No.12004312), the Undergraduate Innovation Program of Northwest A&F University, China (Grant No. X202310712503), and the Fundamental Research Fund for the Central Universities, China (Grant Nos. 2452022027, 2452022169).

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References

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Cited By

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.

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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 $N=100$.

DownLoad: Full-Size Img PowerPoint

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 $N=200$.

DownLoad: Full-Size Img PowerPoint

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 $N=200$.

DownLoad: Full-Size Img PowerPoint

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 $N=200$.

DownLoad: Full-Size Img PowerPoint

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 $N=200$.

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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 $N=200$.

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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 $N=200$.

DownLoad: Full-Size Img PowerPoint

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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[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
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[15]	Liu Y L, Wang C, Zhang J, Liu Y X 2018 Chin. Phys. B 27 024204 Google Scholar
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[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
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[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 物理学报 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
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[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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