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In recent years, two independent research fields, i.e. non-Hermitian andstrongly correlated systems have been merged, forming an important researchfield in physics. The progress of relevant theories and experiments hasreshaped our understanding of matter. In this field, the research object isnot limited to the influence of non-Hermiticity on the energy spectrum andthe eigenstate properties of many-body systems. Researchers have paid more attentionto the manipulation of quantum states. It is universally received that the exceptional point is the most significant featurethat distinguishes non-Hermitian quantum mechanics from Hermitian quantum mechanics. In addition to the recent advances in non-Hermitian topological band theory and quantum sensing around the exceptional points, this paper concentrates on the non-Hermitian critical dynamical phenomenon and its application to the quantum many-body system. When the system has an exceptional point, an arbitrary initial state belonging to the coalescent subspace will be projected on the coalescent state. Based on the directionality of the evolved quantum state, this paper reviews our several representative researches in recent years, including local-field-induced dynamical magnetization, quantum phase transition in transverse field, Ising model at non-zero temperature, quantum mold casting in the center-environment system, as well as superconducting state preparation in the non-Hermitian strongly correlated system. We also focus on the new preparation methods and detection schemes of non-equilibrium quantum states related to exception points.
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Keywords:
- non-Hermitian system /
- quantum many-body system /
- exceptional point
[1] Gamow G 1928 Z. Phys. 51 204
Google Scholar
[2] Siegert A J F 1939 Phys. Rev. 56 750
Google Scholar
[3] Majorana E 2006 EJTP 3 293
[4] Feshbach H 1958 Ann. Phys. 5 357
Google Scholar
[5] Feshbach H 1958 Ann. Phys. 19 287
[6] Schrödinger E 1926 Ann. Phys. 384 489
Google Scholar
[7] Ashida Y, Gong Z P, Ueda M 2020 Adv. Phys. 69 249
[8] Cohen-Tannoudji C, Dupnot-Roc J, Grynberg G 1998 Atom-photon Interactions: Basic Processes and Applications (Berlin: Wiley-VCH)
[9] Anderson P W 1972 Science 177 393
Google Scholar
[10] Lee T D, Yang C N 1952 Phys. Rev. 87 410
Google Scholar
[11] Zhou Y H, Shen H Z, Yi X X 2018 Phys. Rev. A 97 043819
Google Scholar
[12] Song F, Yao S Y, Wang Z 2019 Phys. Rev. Lett. 123 170401
Google Scholar
[13] Pan L, Chen X, Chen Y, Zhai H 2020 Nat. Phys. 1 6
[14] Longhi S 2020 Phys. Rev. B 102 201103
Google Scholar
[15] Liu T, He J J, Yoshida T, Xiang Z L, Nori F 2020 Phys. Rev. B 102 235151
Google Scholar
[16] Xu Z H, Chen S 2020 Phys. Rev. B 102 035153
Google Scholar
[17] Tang L Z, Zhang G Q, Zhang L F, Zhang D W 2021 Phys. Rev. A 103 033325
Google Scholar
[18] Mao L, Deng T S, Zhang P F 2021 Phys. Rev. B 104 125435
Google Scholar
[19] Li J X, Xu L, Zhao Y H, He Z, Wang Q 2021 Laser Phys. 31 075202
Google Scholar
[20] Ohlsson T, Zhou S 2021 Phys. Rev. A 103 022218
Google Scholar
[21] Pan J S, Li L H, Gong JB 2021 Phys. Rev. B 103 205425
Google Scholar
[22] Xue W T, Hu Y M, Song F, Wang Z 2022 Phys. Rev. Lett. 128 120401
Google Scholar
[23] Barontini G, Labouvie R, Stubenrauch F, Vogler A, Guarrera V, Ott H 2013 Phys. Rev. Lett. 110 035302
Google Scholar
[24] Beige A, Braun D, Tregenna B, Knight P L 2000 Phys. Rev. Lett. 85 1762
Google Scholar
[25] Zanardi P, Campos Venuti L 2014 Phys. Rev. Lett. 113 240406
Google Scholar
[26] Militello B, Napoli A 2020 Phys. Lett. A 384 126355
Google Scholar
[27] Gong Z, Yoshioka N, Shibata N, Hamazaki R 2020 Phys. Rev. A 101 052122
Google Scholar
[28] Ashida Y, Furukawa S, Ueda M 2016 Phys. Rev. A 94 053615
Google Scholar
[29] Tomita T, Nakajima S, Danshita I, Takasu Y, Takahashi Y 2017 Sci. Adv. 3 1701513
Google Scholar
[30] Yan B, Moses A S, Gadway B, Covey J P, Hazzard K R, Rey A M, Jin D S, Ye J 2013 Nature 501 521
Google Scholar
[31] Zhu B, Gadway B, Foss-Feig M, Schachenmayer J, Wall M L, Hazzard K R A, Yan B, Moses S A, Covey J P, Jin D S, Ye J, Holland M, Rey A M 2014 Phys. Rev. Lett. 112 070404
Google Scholar
[32] Daley A J, Taylor J M, Diehl S, Baranov M, Zoller P 2009 Phys. Rev. Lett. 102 040402
Google Scholar
[33] Paredes B, Keilmann T, Cirac J I 2007 Phys. Rev. A 75 053611
Google Scholar
[34] Diehl S, Baranov M, Daley A J, Zoller P 2010 Phys. Rev. Lett. 104 165301
Google Scholar
[35] Moroz S, Schmidt R 2010 Ann. Phys. 325 491
Google Scholar
[36] Zhou Z, Wang Z, Zhong H, Luo Y, Chen H, Tan J 2020 Phys. Lett. A 384 126197
Google Scholar
[37] Nakagawa M, Tsuji N, Kawakami N, Ueda M 2020 Phys. Rev. Lett. 124 147203
Google Scholar
[38] Buca B, Booker C, Medenjak M, Jaksch D 2020 New J. Phys. 22 123040
Google Scholar
[39] Pan L, Wang X, Cui X, Chen S 2020 Phys. Rev. A 102 023306
[40] Berry M V 2004 Czech. J. Phys. 54 1039
Google Scholar
[41] Rotter I 2009 J. Phys. A: Math. Theor. 42 153001
Google Scholar
[42] Zhang X Z, Jin L, Song Z 2012 Phys. Rev. A 85 042116
Google Scholar
[43] Mostafazadeh A 2009 Phys. Rev. Lett. 102 220402
[44] Longhi S 2010 Phys. Rev. A 81 022102
Google Scholar
[45] Guo C X, Wang X R, Wang C, Kou S P 2020 Phys. Rev. B 101 144439
Google Scholar
[46] Sun G Y, Tang J C, Kou S P 2022 Front. Phys. 17 33502
[47] Heiss W D 2012 J. Phys. A: Math. Theor. 45 444016
Google Scholar
[48] Jin L, Song Z 2009 Phys. Rev. A 81 032109
[49] Jin L, Song Z 2009 Phys. Rev. A 80 052107
Google Scholar
[50] Jin L, Song Z 2013 Ann. Phys. 330 142
[51] Jin L 2018 Phys. Rev. A 97 012121
Google Scholar
[52] Zhang S M, Zhang X Z, Jin L, Song Z 2020 Phys. Rev. A 101 033820
Google Scholar
[53] Jin L, Wang P, Song Z 2017 Sci. Rep. 7 5903
Google Scholar
[54] Lin S, Jin L, Song Z 2019 Phys. Rev. B 99 165148
Google Scholar
[55] Zhang X Z, Jin L, Song Z 2012 Phys. Rev. A 85 012106
Google Scholar
[56] Wang P, Jin L, Zhang G, Song Z 2016 Phys. Rev. A 94 012106
Google Scholar
[57] Daley A J 2014 Adv. Phys. 6 3
[58] Zhang X Z, Jin L, Song Z 2020 Phys. Rev. B 101 224301
Google Scholar
[59] Zhang K L, Song Z 2021 Phys. Rev. Lett. 126 116401
Google Scholar
[60] Yang X M, Song Z 2021 Phys. Rev. B 103 094307
[61] Zhang X Z, Song Z 2020 Phys. Rev. B 102 174303
Google Scholar
[62] Mostafazadeh A 2009 J. Phys. A: Math. Theor. 42 125303
Google Scholar
[63] Wang P, Zhang K L, Song Z 2021 Phys. Rev. B 104 245406
[64] Eismann U, Khaykovich L, Laurent S, Ferrier-Barbut I, Rem B S, Grier A T, Delehaye M, Chevy F, Salomon C, Ha L C, Chin C 2016 Phys. Rev. X 6 021025
[65] Patil Y S, Chakram S, Vengalattore M 2015 Phys. Rev. Lett. 115 140402
Google Scholar
[66] Kitaev A Y 2001 Phys. Usp. 44 131
Google Scholar
[67] Zhang X Z, Jin L, Song Z 2017 Phys. Rev. A 95 052122
Google Scholar
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图 1 自旋在(a)局域磁场、(b)各向同性全局复数磁场及(c)粒子与粒子相互作用与局域复数磁场共同作用下的动力学演化示意图. 复数磁场通过绿色阴影部分标注. 自旋与自旋的非各项同性相互作用
$ J_{ij} $ 通过不同的颜色来区分. 根据非厄米临界动力学的理论, 图(a) 中含有一个二阶例外点, 对应两个简并态合并. 图(b)和图(c)有N个简并态合并对应N阶例外点. 从图中可以看到, 局域的复数磁场可以通过与相互作用的协作来影响系统自旋的整体取向Figure 1. Schematics of spins subjected to (a) a local complex field, (b) a global complex field, and (c) a local complex field and interaction. The complex magnetic field is shaded green. The couplings between different spins are denoted by different colors representing inhomogeneous coupling
$ J_{ij} $ . Two states coalesce in panel (a) and N states coalesce in panels (b) and (c). Local complex field only affects local spin without interaction, but can affect globally with interaction
图 2 (a) 对于初始态
$ \left\vert \varPsi _{\mathrm{I}}\left( 0\right) \right\rangle $ 在时间域上的磁滞回线; (b), (c) 对于初始态$ \left\vert \varPsi _{\mathrm{II}}\left( 0\right) \right\rangle $ 的磁滞回线. 这里局域的复数外场被施加到格点1上. 其强度在图(a)中为$ 0.02 $ , 在图(b)和图(c)中为$ 0.1 $ . 弛豫时间分别为(a), (b)$ t_{{\rm{f}}} = 2\times 10^{3}J^{-1} $ 及(c)$ t_{{\rm{f}}} = 3\times 10^{3}J^{-1} $ Figure 2. Hysteresis loops for the initial state
$ \left\vert \varPsi _{\mathrm{I}}\left( 0\right) \right\rangle $ in (a) and$ \left\vert \varPsi _{\mathrm{II}}\left( 0\right) \right\rangle $ in (b), (c). The critical local complex field$ g_{1} $ is$ 0.02 $ in panels (a) and (b), and$ 0.1 $ in panel (c). The relaxation time is$ t_{{\rm{f}}} = 2\times 10^{3}J^{-1} $ in panels (a) and (b), and$ t_{{\rm{f}}} = 3\times 10^{3}J^{-1} $ in panel (c)
图 3 (a) 通过施密特回波所给出的相图; (b) 关联函数所给出的相图, 这里
$ \beta ^{-1} $ 是温度,$ g_{{\rm{c}}} $ 是量子相变点; (c) 有限横场Ising模型低能激发谱随着参数g的变化.$ E_{{\rm{g}}} $ 代表基态能量. 其他系统参数为$ N = 50 $ 和$ J = 1 $ . 系统的不同相通过两种不同颜色来区分. 通过图(c)可以发现, 当$ g < 1 $ 时, 系统的能谱都变为二重简并Figure 3. (a) Phase diagram detected from the Loschmidt echoes in this work. (b) Phase diagram studied in term of correlation function in the work of Sachdev et al.. Here
$ \beta ^{-1} $ is the temperature and$ g_{{\rm{c}}} $ is the quantum critical point. (c) Spectrum of the low-lying states for a finite quantum Ising chain as a function of g, obtained numerically through exact diagonalization.$ E_{{\rm{g}}} $ is the ground-state energy. System parameters:$ N = 50 $ and$ J = 1 $ . The energy gap closes at a quasicritical point, indicated by the boundary of the two shaded areas. Notably, all energy levels become twofold degeneracy simultaneously at one point, protected by the symmetry of the quasi-zero-mode operator D.
图 4 施密特回波随时间变化曲线. 线和点分别代表不同的温度, 即
$ \beta = 5 $ 和$ \beta = 10 $ . 其他系统参数为$ N = 10 $ ,$\kappa = $ $ 0.1$ 及$ J = 1 $ . 施密特回波在不同物质相内的动力学行为不同, 最终趋近于$ 1.0 $ 和$ 0.5 $ 两个定值. 需要注意的是这个结果独立于初始热态的温度. 因为D依赖于参数g, 并且其在$ g > 1 $ 时发散, 所以在这里的数值模拟中, 非厄米外场只取D的主导项, 即$ {\cal{H}}=H+\kappa D_{1} $ Figure 4. Loschmidt echoes of different g values. The lines and dots represent the Loschmidt echoes for
$ \beta = 5 $ and$ \beta = 10 $ , respectively. Other parameters:$ N = 10 $ ,$ \kappa = 0.1 $ , and$ J = 1 $ . The profiles of the Loschmidt echoes in the two regions are distinct, independent of the temperature of the initial thermal states, and converge to$ 1.0 $ and$ 0.5 $ , respectively.
图 5 量子铸模系统示意图 (a)系统由两部分组成, 中心系统
$ H_{{\rm{c}}} $ 和源系统$ H_{{\rm{s}}} $ .$ H_{{\rm{in}}} $ 为非厄米项, 表示$ H_{{\rm{c}}} $ 和$ H_{{\rm{s}}} $ 之间的连接, 并承担从$ H_{{\rm{s}}} $ 向$ H_{{\rm{c}}} $ 单向传输费米子的任务. (b)该方案的紧束缚模型, 包含三种格点A, B和D. 嵌入阴影区域的晶格A和B (红色和蓝色实心圆)是拓扑绝缘体, 而晶格D (黄色实心圆)是一个无跳跃的平带系统, 但具有振荡的化学势. 绿色箭头表示从D点到B点的单向跳跃. 本工作的目的是通过时间演化实现以下过程: 初始时刻D格点填充满粒子, 而A和B格点无粒子; 最终末态是$ H_{{\rm{c}}} $ 半满填充的基态. (c)动力学过程的基本机制. 在瞬时$ t_{k} $ ,$ H_{{\rm{s}}} $ 的化学势和$ H_{{\rm{c}}} $ 的能级共振导致例外点. 相应的例外点动力学允许费米子在简并能级之间完全转移, 并且在长时间极限下, 这样的动力学过程发生在每个$ {\boldsymbol{k}} $ 子空间.$ H_{{\rm{c}}} $ 的能带颜色表示能带反转, 说明能带可以是拓扑绝缘带. 预计$ H_{{\rm{c}}} $ 的下带可以完全填充Figure 5. Schematics for the system and process of quantum mold casting: (a) The system consists of two parts, central system
$ H_{\rm{c}} $ and source system$ H_{\rm{s}} $ . The target state is the ground state of$ H_{\rm{c}} $ , which can be topologically non-trivial or not.$ H_{\rm{s}} $ is a topologically trivial system, providing the supply of fermions. Both$ H_{\rm{c}} $ and$ H_{\rm{s}} $ are Hermitian, while$ H_{\rm{in}} $ is non-Hermitian, representing the connection between$ H_{\mathrm{c}} $ and$ H_{\mathrm{s}} $ , and taking the role to transport fermions unidirectionally from$ H_{\mathrm{s}} $ to$ H_{\mathrm{c}} $ . (b) A tight-binding model for the scheme, which contains three sets, A, B, and D. Lattices A and B (red and blue filled circles) embedded in the shadow area is topological insulator, while lattice D (yellow filled circle) is a flat-band (hopping-free) system but with oscillating chemical potential. Green arrows represent unidirectional hopping from D to B lattices. The aim of this work is to realize the following process via time evolution. Initially, D lattice is fully filled, while A and B are empty. The final state is expected to a half-filled ground state of$ H_{\rm{c}} $ . (c) The underlying mechanism of the dynamic process. At instant$ t_{{\boldsymbol{k}}} $ , the chemical potential and energy levels of$ H_{\rm{c}} $ are resonant, leading to exceptional points. The corresponding (EP) dynamics allows a complete transfer of fermions between the degenerate energy levels. In the long-time limit, such dynamics occurs at each$ {\boldsymbol{k}} $ sector again and again. The band color of$ H_{\rm{c}} $ illustrates the band inversion, indicating that the energy band can be topological insulating band. It is expected the lower band of$ H_{\rm{c}} $ can be completely filled
图 6 (a)—(d) 4格点Hubbard模型中关联函数
$ |\langle \varPhi \left( t\right) \vert \eta _{i}^{+}\left\vert \varPhi \left( t\right) \right\rangle | $ 以及$ \langle \varPhi \left( t\right) \vert \eta _{i}^{+}\eta _{i+r}^{-}\left\vert \varPhi \left( t\right) \right\rangle $ 随时间的变化图. 初始态被制备在$ H_{0} $ 的真空态$\left\vert {V_{{\rm{vac}}}}\right\rangle$ 中, 其相互作用强度$U = 2J$ . 随后其运动由外加局域虚数磁场来驱动. 对于图(a)和图(c), 局域虚数磁场为$g_{1}=g = 0.2J$ ; 对于图(b)和图(d), 系统受到一个各向同性的磁场驱动, 其强度为$g_{j}=g = 0.2J$ $ (j = 1,\cdots, N) $ . 需要注意的是此时外场$ H_{{\rm{I}}} $ 处于例外点上. (e), (f) 稳态关联函数与相对距离之间的函数曲线. 对于图(e), 弛豫时间为$ t_{{\rm{f}}} = 400/J $ , 而对于图(f),$ t_{{\rm{f}}} = 100/J $ . 从图中可以看到当$ i\neq j $ 时,$ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/4 $ ; 当$ i=j $ 时,$ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/2 $ , 与正文中的结论一致Figure 6. (a)–(d) Evolution of the correlators
$ |\langle \varPhi \left( t\right) \vert \eta _{i}^{+}\left\vert \varPhi \left( t\right) \right\rangle | $ and$ \langle \varPhi \left( t\right) \vert \eta _{i}^{+}\eta _{i+r}^{-}\left\vert \varPhi \left( t\right) \right\rangle $ , averaged over all sites for the$ 4 $ -site Hubbard model. The initial state is prepared in vacuum state$\left\vert {V_{{\rm{vac}}}}\right\rangle$ of$ H_{0} $ with interaction$U = 2J$ , and then it is driven by the system with the local imaginary field$g_{1}=g = 0.2J$ for panels (a) and (c), and homogeneous dissipation$g_{j}=g = 0.2J$ $(j = 1,\cdots, N)$ for panels (b) and (d), respectively. Notice that$ H_{{\rm{I}}} $ is at EP such that$ \lambda /\gamma = 1 $ . (e), (f) The correlation values of steady state for different relative distance ($ \langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle $ ) at relaxation time$ t_{{\rm{f}}} = 400/J $ for panel (e) and$ t_{{\rm{f}}} = 100/J $ for panel (f). It is shown that$\langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/4$ for$ i\neq j $ and$\langle \varPsi \left( t_{{\rm{f}}}\right) \vert \eta _{i}^{+}\eta _{j}^{-}\left\vert \varPsi \left( t_{{\rm{f}}}\right) \right\rangle = 1/2$ for$ i=j $ , which confirms the understanding in the main text -
[1] Gamow G 1928 Z. Phys. 51 204
Google Scholar
[2] Siegert A J F 1939 Phys. Rev. 56 750
Google Scholar
[3] Majorana E 2006 EJTP 3 293
[4] Feshbach H 1958 Ann. Phys. 5 357
Google Scholar
[5] Feshbach H 1958 Ann. Phys. 19 287
[6] Schrödinger E 1926 Ann. Phys. 384 489
Google Scholar
[7] Ashida Y, Gong Z P, Ueda M 2020 Adv. Phys. 69 249
[8] Cohen-Tannoudji C, Dupnot-Roc J, Grynberg G 1998 Atom-photon Interactions: Basic Processes and Applications (Berlin: Wiley-VCH)
[9] Anderson P W 1972 Science 177 393
Google Scholar
[10] Lee T D, Yang C N 1952 Phys. Rev. 87 410
Google Scholar
[11] Zhou Y H, Shen H Z, Yi X X 2018 Phys. Rev. A 97 043819
Google Scholar
[12] Song F, Yao S Y, Wang Z 2019 Phys. Rev. Lett. 123 170401
Google Scholar
[13] Pan L, Chen X, Chen Y, Zhai H 2020 Nat. Phys. 1 6
[14] Longhi S 2020 Phys. Rev. B 102 201103
Google Scholar
[15] Liu T, He J J, Yoshida T, Xiang Z L, Nori F 2020 Phys. Rev. B 102 235151
Google Scholar
[16] Xu Z H, Chen S 2020 Phys. Rev. B 102 035153
Google Scholar
[17] Tang L Z, Zhang G Q, Zhang L F, Zhang D W 2021 Phys. Rev. A 103 033325
Google Scholar
[18] Mao L, Deng T S, Zhang P F 2021 Phys. Rev. B 104 125435
Google Scholar
[19] Li J X, Xu L, Zhao Y H, He Z, Wang Q 2021 Laser Phys. 31 075202
Google Scholar
[20] Ohlsson T, Zhou S 2021 Phys. Rev. A 103 022218
Google Scholar
[21] Pan J S, Li L H, Gong JB 2021 Phys. Rev. B 103 205425
Google Scholar
[22] Xue W T, Hu Y M, Song F, Wang Z 2022 Phys. Rev. Lett. 128 120401
Google Scholar
[23] Barontini G, Labouvie R, Stubenrauch F, Vogler A, Guarrera V, Ott H 2013 Phys. Rev. Lett. 110 035302
Google Scholar
[24] Beige A, Braun D, Tregenna B, Knight P L 2000 Phys. Rev. Lett. 85 1762
Google Scholar
[25] Zanardi P, Campos Venuti L 2014 Phys. Rev. Lett. 113 240406
Google Scholar
[26] Militello B, Napoli A 2020 Phys. Lett. A 384 126355
Google Scholar
[27] Gong Z, Yoshioka N, Shibata N, Hamazaki R 2020 Phys. Rev. A 101 052122
Google Scholar
[28] Ashida Y, Furukawa S, Ueda M 2016 Phys. Rev. A 94 053615
Google Scholar
[29] Tomita T, Nakajima S, Danshita I, Takasu Y, Takahashi Y 2017 Sci. Adv. 3 1701513
Google Scholar
[30] Yan B, Moses A S, Gadway B, Covey J P, Hazzard K R, Rey A M, Jin D S, Ye J 2013 Nature 501 521
Google Scholar
[31] Zhu B, Gadway B, Foss-Feig M, Schachenmayer J, Wall M L, Hazzard K R A, Yan B, Moses S A, Covey J P, Jin D S, Ye J, Holland M, Rey A M 2014 Phys. Rev. Lett. 112 070404
Google Scholar
[32] Daley A J, Taylor J M, Diehl S, Baranov M, Zoller P 2009 Phys. Rev. Lett. 102 040402
Google Scholar
[33] Paredes B, Keilmann T, Cirac J I 2007 Phys. Rev. A 75 053611
Google Scholar
[34] Diehl S, Baranov M, Daley A J, Zoller P 2010 Phys. Rev. Lett. 104 165301
Google Scholar
[35] Moroz S, Schmidt R 2010 Ann. Phys. 325 491
Google Scholar
[36] Zhou Z, Wang Z, Zhong H, Luo Y, Chen H, Tan J 2020 Phys. Lett. A 384 126197
Google Scholar
[37] Nakagawa M, Tsuji N, Kawakami N, Ueda M 2020 Phys. Rev. Lett. 124 147203
Google Scholar
[38] Buca B, Booker C, Medenjak M, Jaksch D 2020 New J. Phys. 22 123040
Google Scholar
[39] Pan L, Wang X, Cui X, Chen S 2020 Phys. Rev. A 102 023306
[40] Berry M V 2004 Czech. J. Phys. 54 1039
Google Scholar
[41] Rotter I 2009 J. Phys. A: Math. Theor. 42 153001
Google Scholar
[42] Zhang X Z, Jin L, Song Z 2012 Phys. Rev. A 85 042116
Google Scholar
[43] Mostafazadeh A 2009 Phys. Rev. Lett. 102 220402
[44] Longhi S 2010 Phys. Rev. A 81 022102
Google Scholar
[45] Guo C X, Wang X R, Wang C, Kou S P 2020 Phys. Rev. B 101 144439
Google Scholar
[46] Sun G Y, Tang J C, Kou S P 2022 Front. Phys. 17 33502
[47] Heiss W D 2012 J. Phys. A: Math. Theor. 45 444016
Google Scholar
[48] Jin L, Song Z 2009 Phys. Rev. A 81 032109
[49] Jin L, Song Z 2009 Phys. Rev. A 80 052107
Google Scholar
[50] Jin L, Song Z 2013 Ann. Phys. 330 142
[51] Jin L 2018 Phys. Rev. A 97 012121
Google Scholar
[52] Zhang S M, Zhang X Z, Jin L, Song Z 2020 Phys. Rev. A 101 033820
Google Scholar
[53] Jin L, Wang P, Song Z 2017 Sci. Rep. 7 5903
Google Scholar
[54] Lin S, Jin L, Song Z 2019 Phys. Rev. B 99 165148
Google Scholar
[55] Zhang X Z, Jin L, Song Z 2012 Phys. Rev. A 85 012106
Google Scholar
[56] Wang P, Jin L, Zhang G, Song Z 2016 Phys. Rev. A 94 012106
Google Scholar
[57] Daley A J 2014 Adv. Phys. 6 3
[58] Zhang X Z, Jin L, Song Z 2020 Phys. Rev. B 101 224301
Google Scholar
[59] Zhang K L, Song Z 2021 Phys. Rev. Lett. 126 116401
Google Scholar
[60] Yang X M, Song Z 2021 Phys. Rev. B 103 094307
[61] Zhang X Z, Song Z 2020 Phys. Rev. B 102 174303
Google Scholar
[62] Mostafazadeh A 2009 J. Phys. A: Math. Theor. 42 125303
Google Scholar
[63] Wang P, Zhang K L, Song Z 2021 Phys. Rev. B 104 245406
[64] Eismann U, Khaykovich L, Laurent S, Ferrier-Barbut I, Rem B S, Grier A T, Delehaye M, Chevy F, Salomon C, Ha L C, Chin C 2016 Phys. Rev. X 6 021025
[65] Patil Y S, Chakram S, Vengalattore M 2015 Phys. Rev. Lett. 115 140402
Google Scholar
[66] Kitaev A Y 2001 Phys. Usp. 44 131
Google Scholar
[67] Zhang X Z, Jin L, Song Z 2017 Phys. Rev. A 95 052122
Google Scholar
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