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

Quantum Information Processing

Volumn 11, Issue 5, 2012, Pages 1107-1148

  Kitagawa, Takuya ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

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

Indexed keywords

DRIVEN SYSTEM; DYNAMICALLY INDUCED PHASE; FLOQUET STATE; NON-EQUILIBRIUM PHENOMENA; QUANTUM SIMULATORS; QUANTUM WALK; TOPOLOGICAL INSULATORS; TOPOLOGICAL PHASIS; ZERO ENERGY STATE;

HAMILTONIANS; QUANTUM HALL EFFECT; TOPOLOGY;

QUANTUM THEORY;

EID: 84870912213     PISSN: 15700755     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11128-012-0425-4     Document Type: Article

References (44)

1. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687 (1993)
2. Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307 (2003). http://www.tandfonline.com/doi/pdf/10.1080/00107151031000110776
3. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)
4. Kitagawa, T., Rudner, M.S., Berg, E., Demler, E.: Exploring topological phases with quantum walks. Phys. Rev. A 82, 033429 (2010)
5. Kitagawa, T., et al.: Observation of topologically protected bound states in a one dimensional photonic system. arXiv/1105.5334 (2011)
6. Zähringer, F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010)
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
8. Schreiber, A., et al.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104, 050502 (2010)
9. Broome, M.A., et al.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010)
10. Biane, P., et al.: Quantum walks. In: Quantum Potential Theory, Lecture Notes in Mathematics, vol. 1954, pp. 309-452. Springer, Berlin, Heidelberg (2008)
11. Grimmett, G., Janson, S., Scudo, P.F.: Weak limits for quantum random walks. Phys. Rev. E 69, 026119 (2004)
12. Ryu, S., Hatsugai, Y.: Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002)
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)
14. Rudner, M.S., Levitov, L.S.: Topological transition in a non-hermitian quantum walk. Phys. Rev. Lett. 102, 065703 (2009)
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)
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)
17. Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized hall conductance in a twodimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)
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)
19. Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010)
20. Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Rev.Mod. Phys. 83, 1057 (2011)
21. Su, W.P., Schrieffer, J.R., Heeger, A.J.: Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979)
22. Jackiw, R., Rebbi, C.: Solitons with fermion number. Phys. Rev. D 13, 3398 (1976)
23. Jackiw, R., Schrieffer, J.: Solitons with fermion number 12 in condensed matter and relativistic field theories. Nuclear Physics B 190, 253 (1981)
24. Kane, C.L., Mele, E.J.: Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005)
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. 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. Fu, L., Kane, C.L., Mele, E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)
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. 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)
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)
31. Qi, X.-L., Hughes, T.L., Zhang, S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)
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)
33. Wang, Z., Chong, Y., Joannopoulos, J.D., Soljacic, M.: Observation of unidirectional backscattering- immune topological electromagnetic states. Nature 461, 772 (2009)
34. Kitagawa, T., Berg, E., Rudner, M., Demler, E.: Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010)
35. Jiang, L., et al.:Majorana Fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011)
36. Sørensen, A.S., Demler, E., Lukin, M.D.: Fractional quantum hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005)
37. Zhu, S.-L., Fu, H., Wu, C.-J., Zhang, S.-C., Duan, L.-M.: Spin hall effects for cold atoms in a lightinduced gauge potential. Phys. Rev. Lett. 97, 240401 (2006)
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)
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)
40. Kitagawa, T.:Wolfram Demonstration: topological phases with quantum walks. http://demonstrations. wolfram.com/TopologicalPhasesWithQuantumWalks/
41. Thouless, D.J.: Quantization of particle transport. Phys. Rev. B 27, 6083 (1983)
42. Obuse, H., Kawakami, N.: Topological phases and delocalization of quantum walks in random environments. Phys. Rev. B 84, 195139 (2011)
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). 1201.0521
44. Moore, J.E., Ran, Y., Wen, X.-G.: Topological surface states in three-dimensional magnetic insulators. Phys. Rev. Lett. 101, 186805 (2008)