Abstract

Experimental realization of the Kitaev model is a greatly attractive topic due to the potential applications to build robust qubits against decoherence in topological quantum computation. In this work, we investigate the charged whispering-gallery microcavity array model and simulate the normal Kitaev chain under this mechanism in the first time. We find that the system reveals profound connections with the normal Kitaev chain and its some derivatives, and the topological property of the system depends on effective optomechanical coupling strength deeply. In optomechanically induced Kitaev topologically nontrivial phase, compared to the normal Kitaev chain in the Majorana basis, the novel and distinct structure of charged whispering-gallery microcavity array model leads to controllable photonic and phononic edge localization. Furthermore, we also simulate the extended Kitaev chain and show that two topologically different nontrivial phases of the system allow one to realize more freewheeling controllable photonic and phononic edge localization. Our model offers an alternative approach to correlate with other more complicated one-dimensional noninteracting spinless topological systems relevant to the p-wave superconducting pairing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
[Crossref] [PubMed]

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

2017 (12)

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95, 033820 (2017).
[Crossref]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
[Crossref] [PubMed]

L. L. Wan, X. Y. Lü, J. H. Gao, and Y. Wu, “Controllable photon and phonon localization in optomechanical Lieb lattices,” Opt. Express 25, 17364–17374 (2017).
[Crossref] [PubMed]

J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
[Crossref] [PubMed]

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A 95, 062118 (2017).
[Crossref]

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
[Crossref]

Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
[Crossref]

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
[Crossref]

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

2016 (9)

D. P. Liu, “Topological phase boundary in a generalized Kitaev model,” Chin. Phys. B 25, 057101 (2016).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B 94, 125408 (2016).
[Crossref]

C. Yuce, “Majorana edge modes with gain and loss,” Phys. Rev. A 93, 062130 (2016).
[Crossref]

Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
[Crossref]

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

G. S. Agarwal and S. M. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

2015 (6)

Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B 24, 104208 (2015).
[Crossref]

G. Kells, “Many-body Majorana operators and the equivalence of parity sectors,” Phys. Rev. B 92, 081401 (2015).
[Crossref]

X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
[Crossref]

Y. Xiong and P. Q. Tong, “A NOT operation on Majorana qubits with mobilizable solitons in an extended Su-Schrieffer-Heeger model,” New. J. Phys. 17, 013017 (2015).
[Crossref]

Y. H. Chan, C. K. Chiu, and K. Sun, “Multiple signatures of topological transitions for interacting fermions in chain lattices,” Phys. Rev. B 92, 104514 (2015).
[Crossref]

S. R. Elliott and M. Franz, “Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys. 87, 137 (2015).
[Crossref]

2014 (3)

R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
[Crossref]

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

2013 (3)

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

H. Shi and M. Bhattacharya, “Quantum mechanical study of a generic quadratically coupled optomechanical system,” Phys. Rev. A 87, 043829 (2013).
[Crossref]

C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
[Crossref]

2012 (5)

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
[Crossref] [PubMed]

L. J. Lang and S. Chen, “Majorana fermions in density-modulated p-wave superconducting wires,” Phys. Rev. B 86, 205135 (2012).
[Crossref]

G. Goldstein and C. Chamon, “Exact zero modes in closed systems of interacting fermions,” Phys. Rev. B 86, 115122 (2012).
[Crossref]

F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
[Crossref]

2011 (8)

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
[Crossref]

T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
[Crossref]

S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, “Majorana edge states in interacting one-dimensional systems,” Phys. Rev. Lett. 107, 036801 (2011).
[Crossref] [PubMed]

E. Sela, A. Altland, and A. Rosch, “Majorana fermions in strongly interacting helical liquids,” Phys. Rev. B 84, 085114 (2011).
[Crossref]

E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B 84, 014503 (2011).
[Crossref]

R. M. Lutchyn and M. P. A. Fisher, “Interacting topological phases in multiband nanowires,” Phys. Rev. B 84, 214528 (2011).
[Crossref]

2010 (5)

J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
[Crossref] [PubMed]

J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B 81, 125318 (2010).
[Crossref]

M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B 82, 134521 (2010).
[Crossref]

R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
[Crossref] [PubMed]

Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
[Crossref]

2009 (4)

F. Wilczek, “Majorana returns,” Nat. Phys. 5, 614–618 (2009).
[Crossref]

Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Physics 5, 489–493 (2009).
[Crossref]

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009).
[Crossref]

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

2008 (4)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
[Crossref]

L. Fu and C. L. Kane, “Superconducting proximity effect and Majorana fermions at the surface of a topological insulator,” Phys. Rev. Lett. 100, 096407 (2008).
[Crossref] [PubMed]

2007 (3)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

2006 (1)

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

2001 (1)

A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. 44, 131–136 (2001).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. M. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

Alicea, J.

J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. 75, 076501 (2012).
[Crossref] [PubMed]

E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B 84, 014503 (2011).
[Crossref]

J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B 81, 125318 (2010).
[Crossref]

Altland, A.

E. Sela, A. Altland, and A. Rosch, “Majorana fermions in strongly interacting helical liquids,” Phys. Rev. B 84, 085114 (2011).
[Crossref]

An, C. S.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

Aspelmeyer, M.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bai, C. H.

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
[Crossref] [PubMed]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

Beenakker, C. W. J.

C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. 4, 113–136 (2013).
[Crossref]

Berg, E.

A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
[Crossref]

Bhattacharya, M.

H. Shi and M. Bhattacharya, “Quantum mechanical study of a generic quadratically coupled optomechanical system,” Phys. Rev. A 87, 043829 (2013).
[Crossref]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Bomantara, R. W.

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

Bouwmeester, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

Braunecker, B.

S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, “Majorana edge states in interacting one-dimensional systems,” Phys. Rev. Lett. 107, 036801 (2011).
[Crossref] [PubMed]

Cao, C.

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Cao, J.

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

Cartarius, H.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
[Crossref]

Chamon, C.

G. Goldstein and C. Chamon, “Exact zero modes in closed systems of interacting fermions,” Phys. Rev. B 86, 115122 (2012).
[Crossref]

Chan, Y. H.

Y. H. Chan, C. K. Chiu, and K. Sun, “Multiple signatures of topological transitions for interacting fermions in chain lattices,” Phys. Rev. B 92, 104514 (2015).
[Crossref]

Chen, J. P.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Chen, M. N.

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
[Crossref]

Chen, S.

Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A 95, 062118 (2017).
[Crossref]

Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B 94, 125408 (2016).
[Crossref]

L. J. Lang and S. Chen, “Majorana fermions in density-modulated p-wave superconducting wires,” Phys. Rev. B 86, 205135 (2012).
[Crossref]

Chen, X.

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Chiu, C. K.

Y. H. Chan, C. K. Chiu, and K. Sun, “Multiple signatures of topological transitions for interacting fermions in chain lattices,” Phys. Rev. B 92, 104514 (2015).
[Crossref]

Clerk, A. A.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Das Sarma, S.

T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
[Crossref]

J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
[Crossref] [PubMed]

R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
[Crossref] [PubMed]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
[Crossref]

Dast, D.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
[Crossref]

Duan, Y. W.

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Elliott, S. R.

S. R. Elliott and M. Franz, “Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys. 87, 137 (2015).
[Crossref]

Ezawa, M.

R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
[Crossref]

Fan, L.

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Favero, I.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

Feng, M.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Fidkowski, L.

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

Fisher, M. P. A.

E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B 84, 014503 (2011).
[Crossref]

R. M. Lutchyn and M. P. A. Fisher, “Interacting topological phases in multiband nanowires,” Phys. Rev. B 84, 214528 (2011).
[Crossref]

Flensberg, K.

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Franz, M.

S. R. Elliott and M. Franz, “Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys. 87, 137 (2015).
[Crossref]

Freedman, M.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
[Crossref]

Fu, L.

L. Fu and C. L. Kane, “Superconducting proximity effect and Majorana fermions at the surface of a topological insulator,” Phys. Rev. Lett. 100, 096407 (2008).
[Crossref] [PubMed]

Fujimoto, S.

M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B 82, 134521 (2010).
[Crossref]

Ganeshan, S.

S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
[Crossref]

Gangadharaiah, S.

S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, “Majorana edge states in interacting one-dimensional systems,” Phys. Rev. Lett. 107, 036801 (2011).
[Crossref] [PubMed]

Gao, J. H.

Gao, Y. P.

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

Genes, C.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

Gigan, S.

C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
[Crossref]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Girvin, S. M.

F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics 2, 40 (2009).
[Crossref]

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
[Crossref] [PubMed]

Goldman, N.

F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
[Crossref]

Goldstein, G.

G. Goldstein and C. Chamon, “Exact zero modes in closed systems of interacting fermions,” Phys. Rev. B 86, 115122 (2012).
[Crossref]

Gong, J. B.

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
[Crossref] [PubMed]

Guo, Y. J.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
[Crossref]

Huang, S. M.

G. S. Agarwal and S. M. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A 93, 043844 (2016).
[Crossref]

Jiang, X. X.

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

Jin, H. K.

J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
[Crossref] [PubMed]

Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
[Crossref]

Kane, C. L.

L. Fu and C. L. Kane, “Superconducting proximity effect and Majorana fermions at the surface of a topological insulator,” Phys. Rev. Lett. 100, 096407 (2008).
[Crossref] [PubMed]

Karrai, K.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

Kells, G.

G. Kells, “Many-body Majorana operators and the equivalence of parity sectors,” Phys. Rev. B 92, 081401 (2015).
[Crossref]

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
[Crossref] [PubMed]

Kitaev, A.

L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
[Crossref]

Kitaev, A. Y.

A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. 44, 131–136 (2001).
[Crossref]

Kleckner, D.

D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) 444, 75–78 (2006).
[Crossref]

Klett, M.

M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
[Crossref]

Kong, C.

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95, 033820 (2017).
[Crossref]

Lang, L. J.

L. J. Lang and S. Chen, “Majorana fermions in density-modulated p-wave superconducting wires,” Phys. Rev. B 86, 205135 (2012).
[Crossref]

Law, C. K.

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

Leijnse, M.

M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. 27, 124003 (2012).
[Crossref]

Li, K.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
[Crossref]

Li, Y.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
[Crossref]

Liao, J. Q.

J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
[Crossref]

Liu, D. P.

D. P. Liu, “Topological phase boundary in a generalized Kitaev model,” Chin. Phys. B 25, 057101 (2016).
[Crossref]

Liu, T. T.

X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
[Crossref]

Liu, Y. M.

Loss, D.

S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, “Majorana edge states in interacting one-dimensional systems,” Phys. Rev. Lett. 107, 036801 (2011).
[Crossref] [PubMed]

Lü, R.

Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A 95, 062118 (2017).
[Crossref]

Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B 94, 125408 (2016).
[Crossref]

Lü, X. Y.

Lutchyn, R. M.

T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
[Crossref]

R. M. Lutchyn and M. P. A. Fisher, “Interacting topological phases in multiband nanowires,” Phys. Rev. B 84, 214528 (2011).
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J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
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R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
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P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
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M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
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F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
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Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
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J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
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R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
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C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
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Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
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I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
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Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
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Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Physics 5, 489–493 (2009).
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A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
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J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
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L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
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Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
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L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
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Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
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E. Sela, A. Altland, and A. Rosch, “Majorana fermions in strongly interacting helical liquids,” Phys. Rev. B 84, 085114 (2011).
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S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
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M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B 82, 134521 (2010).
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R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
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J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
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M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
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C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
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T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
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C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083 (2008).
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E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B 84, 014503 (2011).
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M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
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S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. 110, 180403 (2013).
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M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B 82, 134521 (2010).
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R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
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Tewari, S.

J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
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Y. Xiong and P. Q. Tong, “A NOT operation on Majorana qubits with mobilizable solitons in an extended Su-Schrieffer-Heeger model,” New. J. Phys. 17, 013017 (2015).
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X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
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A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
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Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
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R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
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Wan, L. L.

Wang, C.

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
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Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
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J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
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C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
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Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
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C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
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D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
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D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

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Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Physics 5, 489–493 (2009).
[Crossref]

Wang, H. F.

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
[Crossref] [PubMed]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
[Crossref] [PubMed]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

Wang, H. Q.

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
[Crossref]

Wang, T.

Wang, T. J.

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Wang, X. H.

X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
[Crossref]

Wang, Y. C.

Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
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Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B 24, 104208 (2015).
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C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95, 033820 (2017).
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M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
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Xiao, Y.

Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B 24, 104208 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Xing, D. Y.

H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
[Crossref]

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
[Crossref]

Xing, Y.

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
[Crossref] [PubMed]

Xiong, H.

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95, 033820 (2017).
[Crossref]

Xiong, Y.

Y. Xiong and P. Q. Tong, “A NOT operation on Majorana qubits with mobilizable solitons in an extended Su-Schrieffer-Heeger model,” New. J. Phys. 17, 013017 (2015).
[Crossref]

X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
[Crossref]

You, L.

Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
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C. Yuce, “Majorana edge modes with gain and loss,” Phys. Rev. A 93, 062130 (2016).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
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Zeng, Q. B.

Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A 95, 062118 (2017).
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Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
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Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B 94, 125408 (2016).
[Crossref]

Zhang, F. C.

J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
[Crossref] [PubMed]

Zhang, J. Q.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

Zhang, R.

C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. 59, 100315 (2016).
[Crossref]

Zhang, S.

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
[Crossref] [PubMed]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

Zhang, Y.

Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

Zhang, Z. M.

Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B 24, 104208 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
[Crossref]

F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
[Crossref]

Zheng, M. H.

Zhou, Y.

J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
[Crossref] [PubMed]

Zhu, A. D.

Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express 26, 6143–6157 (2018).
[Crossref] [PubMed]

L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
[Crossref]

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z2 topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express 25, 17948–17959 (2017).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

Zhu, B. G.

Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
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Zhu, S. L.

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
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F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
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Zwerger, W.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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Chin. Phys. B (2)

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Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B 24, 104208 (2015).
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J. Phys. Condens. Matter (1)

M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter 29, 035601 (2017).
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Laser Phys. Lett. (1)

J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. 15, 015211 (2018).
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Nat. Photonics (1)

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Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Physics 5, 489–493 (2009).
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Opt. Express (3)

Phys. Rev. A (15)

Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A 96, 043810 (2017).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90, 043825 (2014).
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C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95, 033820 (2017).
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F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z2 topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A 85, 013638 (2012).
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J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A 83, 033820 (2011).
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H. Shi and M. Bhattacharya, “Quantum mechanical study of a generic quadratically coupled optomechanical system,” Phys. Rev. A 87, 043829 (2013).
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Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90, 053841 (2014).
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C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77, 033804 (2008).
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Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A 96, 023826 (2017).
[Crossref]

X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A 92, 012116 (2015).
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C. Yuce, “Majorana edge modes with gain and loss,” Phys. Rev. A 93, 062130 (2016).
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Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A 94, 022119 (2016).
[Crossref]

Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A 95, 062118 (2017).
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M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A 95, 053626 (2017).
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Phys. Rev. B (16)

R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B 90, 014505 (2014).
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Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B 96, 205428 (2017).
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L. J. Lang and S. Chen, “Majorana fermions in density-modulated p-wave superconducting wires,” Phys. Rev. B 86, 205135 (2012).
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T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B 84, 144522 (2011).
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H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B 95, 075136 (2017).
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L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
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A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
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G. Goldstein and C. Chamon, “Exact zero modes in closed systems of interacting fermions,” Phys. Rev. B 86, 115122 (2012).
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G. Kells, “Many-body Majorana operators and the equivalence of parity sectors,” Phys. Rev. B 92, 081401 (2015).
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Y. H. Chan, C. K. Chiu, and K. Sun, “Multiple signatures of topological transitions for interacting fermions in chain lattices,” Phys. Rev. B 92, 104514 (2015).
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E. Sela, A. Altland, and A. Rosch, “Majorana fermions in strongly interacting helical liquids,” Phys. Rev. B 84, 085114 (2011).
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E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B 84, 014503 (2011).
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R. M. Lutchyn and M. P. A. Fisher, “Interacting topological phases in multiband nanowires,” Phys. Rev. B 84, 214528 (2011).
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J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B 81, 125318 (2010).
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M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B 82, 134521 (2010).
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Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B 94, 125408 (2016).
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Phys. Rev. Lett. (10)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007).
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F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98, 030405 (2007).
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J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. 118, 267701 (2017).
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S. R. Elliott and M. Franz, “Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys. 87, 137 (2015).
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L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. 61, 080313 (2018).
[Crossref]

Sci. Rep. (4)

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. 7, 2545 (2017).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. 6, 33404 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6, 24421 (2016).
[Crossref] [PubMed]

D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. 6, 38559 (2016).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of charged whispering-gallery microcavity arrays model including both intercavity couplings shown by the orange arrows and Coulomb interactions shown by the blue arrows. The blue spheres and the red arrow represent the charges embedded in the whispering-gallery microcavities and the mechanical motion of the whispering-gallery microcavity, respectively.
Fig. 2
Fig. 2 The initial charged whispering-gallery microcavity array model can be viewed as two 1D bosonic chains coupled by the effective optomechanical coupling strength G.
Fig. 3
Fig. 3 Energy spectrum for the charged whispering-gallery microcavity array model with parameters Δ = 0.7t and N = 8. The two middle eigenvalues are plotted in magenta and cyan, while other energy eigenvalues are plotted in gray. The twofold-degenerate zero-energy edge modes emerge in the topologically nontrivial regime.
Fig. 4
Fig. 4 Illustrations of controllable photonic and phononic edge localization. For the even number of whispering-gallery microcavities: (a) Photonic edge localization showing two photonic edge states located at ends of the photonic chain; (b) There is a phononic edge state located at each end of the phononic chain for phononic edge localization. (c) and (d) Photonic and phononic common but opposite edge localization with a photonic edge state concentrated at one end of the photonic chain and a phononic edge state centralized at the other end of the phononic chain for the odd number of whispering-gallery microcavities.
Fig. 5
Fig. 5 The populations of photonic (hot colorbar) and phononic (gray colorbar) chains corresponding to different cases of controllable photonic and phononic edge localization: (a) Photonic edge localization; (b) Phononic edge localization; (c) Photonic and phononic common but opposite edge localization.
Fig. 6
Fig. 6 The real and imaginary parts of energy spectrum of the charged whispering-gallery microcavity array model with the dissipation of cavity mode κ = 0.3t, the damping of mechanical mode Γ = 0.1t, and a disorder perturbation added in the intercavity hopping strength and the effective Coulomb interaction strength which is randomly distributed in the range [−0.3Jm, 0.3Jm] and [−0.3λm, 0.3λm]. The magenta and cyan lines and points represent the real and imaginary parts of eigenvalues with the minimum absolute values of real part, respectively.
Fig. 7
Fig. 7 (a) The phase diagram of our model expanded by the parameters G, Δ, and δ. (b) Energy spectrum for the charged whispering-gallery microcavity array model with parameters G = Δ = 0.3t and N = 70. (c) Energy spectrum for the charged whispering-gallery microcavity array model with the same parameters adopted in (b) except N = 71. The two red lines in the energy spectrum represent two phase transition points.
Fig. 8
Fig. 8 Illustrations of controllable photonic and phononic edge localization. (a) Photonic and phononic common bilateral edge localization in the SSH-like TNP when the number of whispering-gallery microcavities is even, which shows that two photonic edge states and two phononic edge states concentrate at ends of the photonic and phononic chains respectively. (b) and (c) When the number of whispering-gallery microcavities is odd, photonic and phononic common unilateral edge localization with a photonic edge state and a phononic edge state located at either one end of the photonic and phononic chains together.

Equations (10)

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H = H om + H c c ,
H om = n = 1 N [ ω c a n a n + p n 2 2 m 0 + 1 2 m 0 ω b 2 q n 2 g 0 a n a n q n + Ω d ( a n e i ω d t + a n e i ω d t ) ] , H c c = m = 1 N 1 [ J m ( a m a m + 1 + a m + 1 a m ) + k e Q m Q m + 1 | r 0 + q m q m + 1 | ] ,
k e Q m Q m + 1 | r 0 + q m q m + 1 | = k e Q m Q m + 1 r 0 [ 1 q m q m + 1 r 0 + ( q m q m + 1 r 0 ) 2 ] ,
H C = χ m q m q m + 1 ,
H = n = 1 N [ c a n a n + ω b b n b n g a n a n ( b n + b n ) + Ω d ( a n + a n ) ] + m = 1 N 1 [ J m ( a m a m + 1 + a m + 1 a m ) + λ m ( b m + b m ) ( b m + 1 + b m + 1 ) ] ,
H L = n = 1 N [ Δ c a n a n + ω b b n b n + G ( a n + a n ) ( b n + b n ) ] + m = 1 N 1 [ J m ( a m a m + 1 + a m + 1 a m ) + λ m ( b m + b m ) ( b m + 1 + b m + 1 ) ] ,
H eff = m = 1 N 1 [ J m ( a m a m + 1 + a m + 1 a m ) + λ m ( b m b m + 1 + b m + 1 b m ) ] + n = 1 N [ G ( a n b n + b n a n ) ] .
J m = { t Δ m odd t + Δ m even , λ m = { t + Δ m odd t Δ m even ,
J m = { t + Δ m odd t Δ m even , λ m = { t Δ m odd t + Δ m even .
J m = { t Δ δ m odd t + Δ + δ m even , λ m = { t + Δ δ m odd t Δ + δ m even ,

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