Skip to main content
SpringerLink
Log in

Floquet Spectrum and Dynamics for Non-Hermitian Floquet One-Dimension Lattice Model

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Periodically driven non-Hermitian quantum systems have become the center of interest in recent years due to their rich physical phenomena. In this work, we consider a one-dimensional non-Hermitian lattice model induced by partially asymmetric coupling with time-periodic and spatially periodic modulations upon on-site potentials. Within Floquet theorem, we obtain the Floquet quasienergy spectrum of this one-dimensional non-Hermitian system. We show that the robust zero-energy modes exist in the band gap from the real parts of quasienergy spectrum. Compared with the case of no modulations, two pairs of the conjugate imaginary parts are added, which can be attributed to the combined modulations upon the on-site potentials. With the increase of non-Hermitian degree, the imaginary parts of the spectra are enlarged. We also observe the dynamical characteristics that, for different kinds of tunneling amplitudes between lattices, the amplitude of evolution gradually either decays to zero or eventually stabilizes at one particular value. Our protocol, possible to realize in photonic lattices, may facilitate the engineering of novel Floquet topological phases in non-Hermitian systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Tong, Q.J., An, J.H., Gong, J., Luo, H.G., Oh, C.H.: Generating many majorana modes via periodic driving: A superconductor model. Phys. Rev. B 87, 201109 (2013)

    Article  ADS  Google Scholar 

  2. Vorberg, D., Wustmann, W., Ketzmerick, R., Eckardt, A.: Generalized bose-einstein condensation into multiple states in driven-dissipative systems. Phys. Rev. Lett. 111, 240405 (2013)

    Article  ADS  Google Scholar 

  3. Harper, F., Roy, R.: Floquet topological order in interacting systems of bosons and fermions. Phys. Rev. Lett. 118, 115301 (2017)

    Article  ADS  Google Scholar 

  4. Zhou, L., Gong, J.: Recipe for creating an arbitrary number of floquet chiral edge states. Phys. Rev. B 97, 245430 (2018)

    Article  ADS  Google Scholar 

  5. Sajid, M., Asbóth, J. K., Meschede, D., Werner, R.F., Alberti, A.: Creating anomalous floquet chern insulators with magnetic quantum walks. Phys. Rev. B 99, 214303 (2019)

    Article  ADS  Google Scholar 

  6. Wu, H., An, J.H.: Floquet topological phases of non-hermitian systems. Phys. Rev. B 102, 041119 (2020)

    Article  ADS  Google Scholar 

  7. Rechtsman, M.C., Zeuner, J.M., Plotnik, Y., Lumer, Y., Podolsky, D., Dreisow, F., Nolte, S., Segev, M., Szameit, A.: Photonic floquet topological insulators. Nature 496(7444), 196–200 (2013)

    Article  ADS  Google Scholar 

  8. Meinert, F., Mark, M.J., Lauber, K., Daley, A.J., Nägerl, H. C.: Floquet engineering of correlated tunneling in the bose-hubbard model with ultracold atoms. Phys. Rev. Lett. 116, 205301 (2016)

    Article  ADS  Google Scholar 

  9. Eckardt, A.: Colloquium: Atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys. 89, 011004 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  10. Cheng, Q., Pan, Y., Wang, H., Zhang, C., Yu, D., Gover, A., Zhang, H., Li, T., Zhou, L., Zhu, S.: Observation of anomalous π modes in photonic floquet engineering. Phys. Rev. Lett. 122, 173901 (2019)

    Article  ADS  Google Scholar 

  11. Bermudez, A., Schmidt, P.O., Plenio, M.B., Retzker, A.: Robust trapped-ion quantum logic gates by continuous dynamical decoupling. Phys. Rev. A 85, 040302 (2012)

    Article  ADS  Google Scholar 

  12. Lemmer, A., Bermudez, A., Plenio, M.B.: Driven geometric phase gates with trapped ions. J. Phys. 15(8), 083001 (2013)

    Google Scholar 

  13. Creffield, C.E.: Quantum control and entanglement using periodic driving fields. Phys. Rev. Lett. 99, 110501 (2007)

    Article  ADS  Google Scholar 

  14. Lignier, H., Sias, C., Ciampini, D., Singh, Y., Zenesini, A., Morsch, O., Arimondo, E.: Dynamical control of matter-wave tunneling in periodic potentials. Phys. Rev. Lett. 99, 220403 (2007)

    Article  ADS  Google Scholar 

  15. Zhou, L., Yang, S., Liu, Y.x., Sun, C.P., Nori, F.: Quantum zeno switch for single-photon coherent transport. Phys. Rev. A 80, 062109 (2009)

    Article  ADS  Google Scholar 

  16. Gong, J., Wang, Q.h.: Stabilizing non-hermitian systems by periodic driving. Phys. Rev. A 91, 042135 (2015)

    Article  ADS  Google Scholar 

  17. Nathan, F., Rudner, M.S.: Topological singularities and the general classification of floquet–bloch systems. J. Phys. 17(12), 125014 (2015)

    Google Scholar 

  18. Maczewsky, L.J., Zeuner, J.M., Nolte, S., Szameit, A.: Observation of photonic anomalous floquet topological insulators. Nat. Commun. 8(1), 13756 (2017)

    Article  ADS  Google Scholar 

  19. Rudner, M.S., Lindner, N.H., Berg, E., Levin, M.: Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013)

    Google Scholar 

  20. Dal Lago, V., Atala, M., Foa Torres, L.E.F.: Floquet topological transitions in a driven one-dimensional topological insulator. Phys. Rev. A 92, 023624 (2015)

    Article  ADS  Google Scholar 

  21. Lee, T.E.: Anomalous edge state in a non-hermitian lattice. Phys. Rev. Lett. 116, 133903 (2016)

    Article  ADS  Google Scholar 

  22. Leykam, D., Bliokh, K.Y., Huang, C., Chong, Y.D., Nori, F.: Edge modes, degeneracies, and topological numbers in non-hermitian systems. Phys. Rev. Lett. 118, 040401 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  23. Xiong, Y.: Why does bulk boundary correspondence fail in some non-hermitian topological models. J. Phys. Commun. 2(3), 035043 (2018)

    Article  Google Scholar 

  24. Shen, H., Zhen, B., Fu, L.: Topological band theory for non-hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  25. Gong, Z., Ashida, Y., Kawabata, K., Takasan, K., Higashikawa, S., Ueda, M.: Topological phases of non-hermitian systems. Phys. Rev. X 8, 031079 (2018)

    Google Scholar 

  26. Zhang, Q., Wu, B.: Non-hermitian quantum systems and their geometric phases. Phys. Rev. A 99, 032121 (2019)

    Article  ADS  Google Scholar 

  27. Kawabata, K., Shiozaki, K., Ueda, M., Sato, M.: Symmetry and topology in non-hermitian physics. Phys. Rev. X 9, 041015 (2019)

    Google Scholar 

  28. Borgnia, D.S., Kruchkov, A.J., Slager, R.J.: Non-hermitian boundary modes and topology. Phys. Rev. Lett. 124, 056802 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  29. Zeng, Q.B., Yang, Y.B., Xu, Y.: Topological phases in non-hermitian Aubry-André-harper models. Phys. Rev. B 101, 020201 (2020)

    Article  ADS  Google Scholar 

  30. Zeuner, J.M., Rechtsman, M.C., Plotnik, Y., Lumer, Y., Nolte, S., Rudner, M.S., Segev, M., Szameit, A.: Observation of a topological transition in the bulk of a non-hermitian system. Phys. Rev. Lett. 115, 040402 (2015)

    Article  ADS  Google Scholar 

  31. St-Jean, P., Goblot, V., Galopin, E., Lemaître, A., Ozawa, T., Le Gratiet, L., Sagnes, I., Bloch, J., Amo, A.: Lasing in topological edge states of a one-dimensional lattice. Nat. Photonics 11(10), 651–656 (2017)

    Article  ADS  Google Scholar 

  32. Bahari, B., Ndao, A., Vallini, F., El Amili, A., Fainman, Y., Kanté, B.: Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358(6363), 636–640 (2017)

    Article  ADS  Google Scholar 

  33. Xiao, L., Zhan, X., Bian, Z.H., Wang, K.K., Zhang, X., Wang, X.P., Li, J., Mochizuki, K., Kim, D., Kawakami, N., Yi, W., Obuse, H., Sanders, B.C., Xue, P.: Observation of topological edge states in parity–time-symmetric quantum walks. Nat. Phys. 13(11), 1117–1123 (2017)

    Article  Google Scholar 

  34. Parto, M., Wittek, S., Hodaei, H., Harari, G., Bandres, M.A., Ren, J., Rechtsman, M.C., Segev, M., Christodoulides, D.N., Khajavikhan, M.: Edge-mode lasing in 1d topological active arrays. Phys. Rev. Lett. 120, 113901 (2018)

    Article  ADS  Google Scholar 

  35. Li, J., Harter, A.K., Liu, J., de Melo, L., Joglekar, Y.N., Luo, L.: Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms. Nat. Commun. 10(1), 855 (2019)

    Article  ADS  Google Scholar 

  36. Wang, K., Qiu, X., Xiao, L., Zhan, X., Bian, Z., Sanders, B.C., Yi, W., Xue, P.: Observation of emergent momentum–time skyrmions in parity–time-symmetric non-unitary quench dynamics. Nat. Commun. 10(1), 2293 (2019)

    Article  ADS  Google Scholar 

  37. Kunst, F.K., Edvardsson, E., Budich, J.C., Bergholtz, E.J.: Biorthogonal bulk-boundary correspondence in non-hermitian systems. Phys. Rev. Lett. 121, 026808 (2018)

    Article  ADS  Google Scholar 

  38. Yao, S., Wang, Z.: Edge states and topological invariants of non-hermitian systems. Phys. Rev. Lett. 121, 086803 (2018)

    Article  ADS  Google Scholar 

  39. Yao, S., Song, F., Wang, Z.: Non-hermitian chern bands. Phys. Rev. Lett. 121, 136802 (2018)

    Article  ADS  Google Scholar 

  40. Ghatak, A., Das, T.: New topological invariants in non-hermitian systems. J. Phys.: Condensed Matter 31(26), 263001 (2019)

    ADS  Google Scholar 

  41. Longhi, S.: Probing non-hermitian skin effect and non-bloch phase transitions. Phys. Rev. Res. 1, 023013 (2019)

    Article  Google Scholar 

  42. Yuce, C.: Non-hermitian anomalous skin effect. Phys. Lett. A 384 (4), 126094 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. Berry, M.V.: Physics of nonhermitian degeneracies. Czechoslov. J. Phys. 54(10), 1039–1047 (2004)

    Article  ADS  Google Scholar 

  44. Hu, W., Wang, H., Shum, P.P., Chong, Y.D.: Exceptional points in a non-hermitian topological pump. Phys. Rev. B 95, 184306 (2017)

    Article  ADS  Google Scholar 

  45. Zhang, X.L., Wang, S., Hou, B., Chan, C.T.: Dynamically encircling exceptional points: In situ control of encircling loops and the role of the starting point. Phys. Rev. X 8, 021066 (2018)

    Google Scholar 

  46. Kawabata, K., Bessho, T., Sato, M.: Classification of exceptional points and non-hermitian topological semimetals. Phys. Rev. Lett. 123, 066405 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  47. Rivero, J.D.H., Ge, L.: Origin of robust exceptional points: Restricted bulk zero mode. Phys. Rev. A 101, 063823 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  48. Feng, L., Wong, Z.J., Ma, R.M., Wang, Y., Zhang, X.: Single-mode laser by parity-time symmetry breaking. Science 346(6212), 972–975 (2014)

    Article  ADS  Google Scholar 

  49. Hodaei, H., Miri, M.A., Heinrich, M., Christodoulides, D.N., Khajavikhan, M.: Parity-time–symmetric microring lasers. Science 346(6212), 975–978 (2014)

    Article  ADS  Google Scholar 

  50. Wiersig, J.: Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection. Phys. Rev. Lett. 112, 203901 (2014)

    Article  ADS  Google Scholar 

  51. Chen, W., Kaya Özdemir, S.̧, Zhao, G., Wiersig, J., Yang, L.: Exceptional points enhance sensing in an optical microcavity. Nature 548 (7666), 192–196 (2017)

    Article  ADS  Google Scholar 

  52. Lin, Z., Ramezani, H., Eichelkraut, T., Kottos, T., Cao, H., Christodoulides, D.N.: Unidirectional invisibility induced by \(\mathcal {P}\mathcal {T}\)-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011)

    Article  ADS  Google Scholar 

  53. Regensburger, A., Bersch, C., Miri, M.A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity–time synthetic photonic lattices. Nature 488 (7410), 167–171 (2012)

    Article  ADS  Google Scholar 

  54. Ozcakmakli Turker, Z., Yuce, C.: Open and closed boundaries in non-hermitian topological systems. Phys. Rev. A 99, 022127 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  55. Ganeshan, S., Sun, K., Das Sarma, S.: Topological zero-energy modes in gapless commensurate Aubry-André-harper models. Phys. Rev. Lett. 110, 180403 (2013)

    Article  ADS  Google Scholar 

  56. Padavić, K., Hegde, S.S., DeGottardi, W., Vishveshwara, S.: Topological phases, edge modes, and the hofstadter butterfly in coupled su-schrieffer-heeger systems. Phys. Rev. B 98, 024205 (2018)

    Article  ADS  Google Scholar 

  57. Halpern, D., Li, H., Kottos, T.: Floquet protocols of adiabatic state flips and reallocation of exceptional points. Phys. Rev. A 97, 042119 (2018)

    Article  ADS  Google Scholar 

  58. Zhou, L., Gong, J.: Non-hermitian floquet topological phases with arbitrarily many real-quasienergy edge states. Phys. Rev. B 98, 205417 (2018)

    Article  ADS  Google Scholar 

  59. Turker, Z., Tombuloglu, S., Yuce, C.: PT symmetric Floquet topological phase in SSH model. Phys. Lett. A 382(30), 2013–2016 (2018)

    Article  ADS  Google Scholar 

  60. Gong, J., Wang, Q.h.: Piecewise adiabatic following in non-hermitian cycling. Phys. Rev. A 97, 052126 (2018)

    Article  ADS  Google Scholar 

  61. Zhou, L.: Dynamical characterization of non-hermitian floquet topological phases in one dimension. Phys. Rev. B 100, 184314 (2019)

    Article  ADS  Google Scholar 

  62. Li, C.F., Luan, L.N., Wang, L.C.: Topological properties of an extend su-schrieffer-heeger model under periodic kickings. Int. J. Theor. Phys. 59 (9), 2852–2866 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhang, X., Gong, J.: Non-hermitian floquet topological phases: Exceptional points, coalescent edge modes, and the skin effect. Phys. Rev. B 101, 045415 (2020)

    Article  ADS  Google Scholar 

  64. Shirley, J.H.: Solution of the schrödinger equation with a hamiltonian periodic in time. Phys. Rev. 138, B979–B987 (1965)

    Article  ADS  Google Scholar 

  65. Sambe, H.: Steady states and quasienergies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7, 2203–2213 (1973)

    Article  ADS  Google Scholar 

  66. Eckardt, A., Anisimovas, E.: High-frequency approximation for periodically driven quantum systems from a floquet-space perspective. J. Phys. 17(9), 093039 (2015)

    MATH  Google Scholar 

  67. Yao, S., Yan, Z., Wang, Z.: Topological invariants of floquet systems: General formulation, special properties, and floquet topological defects. Phys. Rev. B 96, 195303 (2017)

    Article  ADS  Google Scholar 

  68. Su, W.P., Schrieffer, J.R., Heeger, A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979)

    Article  ADS  Google Scholar 

  69. Oroszlány, JK.AL., Asbóth László, A.P.a.: A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions. Springer International Publishing, Berlin (2016)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC) (Grants No. 11875103 and No. 11775048).

Author information

Authors and Affiliations

  1. Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun, 130024, China

    Ya-Nan Zhang, Hao-Di Liu & Xue-Xi Yi

  2. College of Sciences, Northeastern University, Shenyang, 110819, China

    Shuang Xu

Authors
  1. Ya-Nan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao-Di Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xue-Xi Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xue-Xi Yi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, YN., Xu, S., Liu, HD. et al. Floquet Spectrum and Dynamics for Non-Hermitian Floquet One-Dimension Lattice Model. Int J Theor Phys 60, 355–365 (2021). https://doi.org/10.1007/s10773-020-04699-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-020-04699-4

Keywords

Search

Navigation