Quantum Information Processing

, Volume 11, Issue 5, pp 1107–1148 | Cite as

Topological phenomena in quantum walks: elementary introduction to the physics of topological phases

Article

Abstract

Discrete quantum walks are dynamical protocols for controlling a single quantum particle. Despite of its simplicity, quantum walks display rich topological phenomena and provide one of the simplest systems to study and understand topological phases. In this article, we review the physics of discrete quantum walks in one and two dimensions in light of topological phenomena and provide elementary explanations of topological phases and their physical consequence, namely the existence of boundary states. We demonstrate that quantum walks are versatile systems that simulate many topological phases whose classifications are known for static Hamiltonians. Furthermore, topological phenomena appearing in quantum walks go beyond what has been known in static systems; there are phenomena unique to quantum walks, being an example of periodically driven systems, that do not exist in static systems. Thus the quantum walks not only provide a powerful tool as a quantum simulator for static topological phases but also give unique opportunity to study topological phenomena in driven systems.

Keywords

Quantum walk Topological phases Periodically driven systems Floquet states Non-equilibrium phenomena Topological phenomena in driven systems Dynamically induced phase Quantum simulator Quantum Hall effect Topological insulator Zero energy state 

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References

  1. 1.
    Aharonov Y., Davidovich L., Zagury N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307 (2003). http://www.tandfonline.com/doi/pdf/10.1080/00107151031000110776 Google Scholar
  3. 3.
    Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Kitagawa T., Rudner M.S., Berg E., Demler E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    Kitagawa, T., et al.: Observation of topologically protected bound states in a one dimensional photonic system. arXiv/1105.5334 (2011)Google Scholar
  6. 6.
    Zähringer F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010)CrossRefGoogle Scholar
  7. 7.
    Karski, M., et al.: Quantum walk in position space with single optically trapped atoms. Science 325, 174 (2009). http://www.sciencemag.org/content/325/5937/174.full.pdf Google Scholar
  8. 8.
    Schreiber A., et al.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Broome M.A. et al.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Biane, P., et al.: Quantum walks. In: Quantum Potential Theory, Lecture Notes in Mathematics, vol. 1954, pp. 309–452. Springer, Berlin, Heidelberg (2008)Google Scholar
  11. 11.
    Grimmett G., Janson S., Scudo P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)ADSCrossRefGoogle Scholar
  12. 12.
    Ryu S., Hatsugai Y.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    Oka T., Konno N., Arita R., Aoki H.: Breakdown of an electric-field driven system: a mapping to a quantum walk. Phys. Rev. Lett. 94, 100602 (2005)ADSCrossRefGoogle Scholar
  14. 14.
    Rudner M.S., Levitov L.S.: Topological transition in a non-hermitian quantum walk. Phys. Rev. Lett. 102, 065703 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    von Klitzing K., Dorda G., Pepper M.: New method for high-accuracy determination of the finestructure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494 (1980)ADSCrossRefGoogle Scholar
  16. 16.
    MacDonald, A.H.: Introduction to the physics of the quantum hall regime: quantized hall conductance in a two-dimensional periodic potential. eprint arXiv:cond-mat/9410047 (1994)Google Scholar
  17. 17.
    Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M.: Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  18. 18.
    Halperin B.I.: Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185 (1982)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Hasan M.Z., Kane C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  20. 20.
    Qi X.-L., Zhang S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    Su W.P., Schrieffer J.R., Heeger A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979)ADSCrossRefGoogle Scholar
  22. 22.
    Jackiw R., Rebbi C.: Solitons with fermion number. Phys. Rev. D 13, 3398 (1976)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Jackiw R., Schrieffer J.: Solitons with fermion number 12 in condensed matter and relativistic field theories. Nuclear Physics B 190, 253 (1981)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Kane C.L., Mele E.J.: Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    Bernevig, B.A., Hughes, T.L., Zhang, S.-C.: Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757 (2006). http://www.sciencemag.org/content/314/5806/1757.full.pdf
  26. 26.
    Konig, M., et al.: Quantum spin hall insulator state in HgTe quantum wells. Science 318, 766 (2007). http://www.sciencemag.org/content/318/5851/766.full.pdf
  27. 27.
    Fu L., Kane C.L., Mele E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    Chen, Y.L., et al.: Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science 325, 178 (2009). http://www.sciencemag.org/content/325/5937/178.full.pdf
  29. 29.
    Xia Y. et al.: Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398 (2009)CrossRefGoogle Scholar
  30. 30.
    Schnyder A.P., Ryu S., Furusaki A., Ludwig A.W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Qi X.-L., Hughes T.L., Zhang S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    Kitaev, A.: Periodic table for topological insulators and superconductors. In: Lebedev V., Feigel’Man, M. (eds.) American Institute of Physics Conference Series, vol. 1134, pp. 22–30 (2009)Google Scholar
  33. 33.
    Wang Z., Chong Y., Joannopoulos J.D., Soljacic M.: Observation of unidirectional backscattering–immune topological electromagnetic states. Nature 461, 772 (2009)ADSCrossRefGoogle Scholar
  34. 34.
    Kitagawa T., Berg E., Rudner M., Demler E.: Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010)ADSCrossRefGoogle Scholar
  35. 35.
    Jiang L. et al.: Majorana Fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Sørensen A.S., Demler E., Lukin M.D.: Fractional quantum hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005)ADSCrossRefGoogle Scholar
  37. 37.
    Zhu S.-L., Fu H., Wu C.-J., Zhang S.-C., Duan L.-M.: Spin hall effects for cold atoms in a light-induced gauge potential. Phys. Rev. Lett. 97, 240401 (2006)ADSCrossRefGoogle Scholar
  38. 38.
    Jaksch D., Zoller P.: Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. N. J. Phys. 5, 56 (2003)CrossRefGoogle Scholar
  39. 39.
    Osterloh K., Baig M., Santos L., Zoller P., Lewenstein M.: Cold atoms in non-abelian gauge potentials: from the hofstadter “Moth” to lattice gauge theory. Phys. Rev. Lett. 95, 010403 (2005)ADSCrossRefGoogle Scholar
  40. 40.
    Kitagawa, T.: Wolfram Demonstration: topological phases with quantum walks. http://demonstrations.wolfram.com/TopologicalPhasesWithQuantumWalks/
  41. 41.
    Thouless D.J.: Quantization of particle transport. Phys. Rev. B 27, 6083 (1983)MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Obuse H., Kawakami N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84, 195139 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    Kitagawa, T., Oka, T., Demler, E.: Photo control of transport properties in disorderd wire; average conductance, conductance statistics, and time-reversal symmetry. WolframDemonstration: topological phases with quantum walks. ArXiv e-prints (2012). 1201.0521Google Scholar
  44. 44.
    Moore J.E., Ran Y., Wen X.-G.: Topological surface states in three-dimensional magnetic insulators. Phys. Rev. Lett. 101, 186805 (2008)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Physics DepartmentHarvard UniversityCambridgeUSA

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