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Abelian Gauge Potentials on Cubic Lattices

  • M. Burrello
  • L. Lepori
  • S. Paganelli
  • A. Trombettoni
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

The study of the properties of quantum particles in a periodic potential subjected to a magnetic field is an active area of research both in physics and mathematics, and it has been and is yet deeply investigated. In this chapter we discuss how to implement and describe tunable Abelian magnetic fields in a system of ultracold atoms in optical lattices. After reviewing two of the main experimental schemes for the physical realization of synthetic gauge potentials in ultracold set-ups, we study cubic lattice tight-binding models with commensurate flux. We finally discuss applications of gauge potentials in one-dimensional rings.

Keywords

Abelian gauge potentials Lattice models Quantum entanglement Quantum transfer Ultracold atoms 

References

  1. 1.
    I. Affleck, J.B. Marston, Phys. Rev. B 37, 3774 (1988)CrossRefGoogle Scholar
  2. 2.
    A. Agazzi, J.-P. Eckmann, G.M. Graf, J. Stat. Phys. 156, 417 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, I. Bloch, Phys. Rev. Lett. 107, 255301 (2011)CrossRefGoogle Scholar
  5. 5.
    M. Aidelsburger, M. Atala, M. Lohse, J.T. Barreiro, B. Paredes, I. Bloch, Phys. Rev. Lett. 111, 185301 (2013)CrossRefGoogle Scholar
  6. 6.
    M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J.T. Barreiro, S. Nascimbéne, N.R. Cooper, I. Bloch, N. Goldman, Nat. Phys. 11, 162 (2015)CrossRefGoogle Scholar
  7. 7.
    T.J.G. Apollaro, S. Lorenzo, A. Sindona, S. Paganelli, G.L. Giorgi, F. Plastina, Phys. Scripta T165, 014036 (2015)CrossRefGoogle Scholar
  8. 8.
    M. Atala, M. Aidelsburger, M. Lohse, J.T. Barreiro, B. Paredes, I. Bloch, Nat. Phys. 10, 588 (2014)CrossRefGoogle Scholar
  9. 9.
    J.E. Avron, Colored Hofstadter Butterflies, in Multiscale Methods in Quantum Mechanics: Theory and Experiment, ed. by P. Blanchard, G. Dell’​Antonio, pp. 11–22 (Birkhauser, Boston, 2004)Google Scholar
  10. 10.
    J. Avron, I. Herbst, B. Simon, Duke Math. J. 45, 847 (1978)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.E. Avron, D. Osadchy, R. Seiler, Phys. Today 56, 38 (2003)CrossRefGoogle Scholar
  12. 12.
    M.Ya. Azbel, Sov. Phys. JETP 19, 634 (1964)Google Scholar
  13. 13.
    J. Bellissard, A. van Elst, H. SchulzBaldes, J. Math. Phys. 35, 5373 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    B.A. Bernevig, T.L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013)CrossRefMATHGoogle Scholar
  15. 15.
    I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)CrossRefGoogle Scholar
  16. 16.
    C. Brouder, G. Panati, M. Calandra, C. Mourougane, N. Marzari, Phys. Rev. Lett. 98, 046402 (2007)CrossRefGoogle Scholar
  17. 17.
    J. Brüning, V.V. Demidov, V.A. Geyler, Phys. Rev. B 69, 033202 (2004)CrossRefGoogle Scholar
  18. 18.
    F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001)CrossRefGoogle Scholar
  19. 19.
    C.-K. Chiu, J.C.Y. Teo, A.P. Schnyder, S. Ryu, Rev. Mod. Phys. 88, 035005 (2016)CrossRefGoogle Scholar
  20. 20.
    J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011)CrossRefGoogle Scholar
  21. 21.
    C.R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K.L. Shepard, J. Hone, P. Kim, Nature 497, 598 (2013)CrossRefGoogle Scholar
  22. 22.
    G. De Nittis, G. Panati, arXiv:1007.4786 Google Scholar
  23. 23.
    T. Dubcek, C.J. Kennedy, L. Lu, W. Ketterle, M. Soljacic, H. Buljan, Phys. Rev. Lett. 114, 225301 (2015)CrossRefGoogle Scholar
  24. 24.
    A. Eckardt, C. Weiss, M. Holthaus, Phys. Rev. Lett. 95, 260404 (2005)CrossRefGoogle Scholar
  25. 25.
    A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, E. Arimondo, Phys. Rev. A 79, 013611 (2009)CrossRefGoogle Scholar
  26. 26.
    J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)CrossRefGoogle Scholar
  27. 27.
    T. Feil, K. Výborný, L. Smrčka, C. Gerl, W. Wegscheider, Phys. Rev. B 75, 075303 (2007)CrossRefGoogle Scholar
  28. 28.
    S. Freund, S. Teufel, Anal. PDE 9, 773 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    M.C. Geisler, J.H. Smet, V. Umansky, K. von Klitzing, B. Naundorf, R. Ketzmerick, H. Schweizer, Phys. Rev. Lett. 92, 256801 (2004)CrossRefGoogle Scholar
  30. 30.
    D. Gioev, I. Klich, Phys. Rev. Lett. 96, 100503 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    N. Goldman, J. Dalibard, Phys. Rev. X 4, 031027 (2014)Google Scholar
  32. 32.
    N. Goldman, G. Juzeliunas, P. Ohberg, I.B. Spielman, Rep. Prog. Phys. 77, 126401 (2014)CrossRefGoogle Scholar
  33. 33.
    N. Goldman, J. Dalibard, M. Aidelsburger, N.R. Cooper, Phys. Rev. A 91, 033632 (2015)CrossRefGoogle Scholar
  34. 34.
    G. Gori, S. Paganelli, A. Sharma, P. Sodano, A. Trombettoni, Phys. Rev. B 91, 245138 (2015)CrossRefGoogle Scholar
  35. 35.
    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)CrossRefGoogle Scholar
  36. 36.
    P.G. Harper, Proc. Phys. Soc. A 68, 874 (1955)CrossRefGoogle Scholar
  37. 37.
    Y. Hasegawa, J. Phys. Soc. Jap. 59 4384 (1990); Physica C 185–189, 1541 (1991)Google Scholar
  38. 38.
    D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976)CrossRefGoogle Scholar
  39. 39.
    J.K. Jain, Composite Fermions (Cambridge University Press, Cambridge, 2007)CrossRefMATHGoogle Scholar
  40. 40.
    D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). MoreCrossRefGoogle Scholar
  41. 41.
    G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, T. Esslinger, Nature 515, 237 (2014)CrossRefGoogle Scholar
  42. 42.
    M.V. Karasev, T.A. Osborn, J. Math. Phys. 43, 756 (2002)MathSciNetCrossRefGoogle Scholar
  43. 43.
    E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, M.K. Oberthaler, Phys. Rev. Lett. 100, 190405 (2008)CrossRefGoogle Scholar
  44. 44.
    T. Kimura, Prog. Theor. Exp. Phys. 2014(10), 103B05 (2014). doi:10.1093/ptep/ptu144Google Scholar
  45. 45.
    W. Kohn, Phys. Rev. 115, 1460 (1959)MathSciNetCrossRefGoogle Scholar
  46. 46.
    M. Koshino, H. Aoki, K. Kuroki, S. Kagoshima, T. Osada, Phys. Rev. Lett. 86, 1062 (2001)CrossRefGoogle Scholar
  47. 47.
    L.D. Landau, E.M. Lifschitz, Quantum Mechanics (Pergamon Press, New York, 1965)MATHGoogle Scholar
  48. 48.
    L. Lepori, G. Mussardo, A. Trombettoni, Europhys. Lett. 92, 50003 (2010)CrossRefGoogle Scholar
  49. 49.
    L. Lepori, I.C. Fulga, A. Trombettoni, M. Burrello, Phys. Rev. B. 94, 085107 (2016)CrossRefGoogle Scholar
  50. 50.
    E.M. Lifschitz, L.P. Pitaevskii, Statistical Physics, Part 2 (Pergamon Press, New York, 1980)Google Scholar
  51. 51.
    H. Lignier C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, E. Arimondo, Phys. Rev. Lett. 99, 220403 (2007)CrossRefGoogle Scholar
  52. 52.
    Y.-L. Lin, F. Nori, Phys. Rev. B 53, 13374 (1996)CrossRefGoogle Scholar
  53. 53.
    J.M. Luttinger, Phys. Rev. 84, 814 (1951)CrossRefGoogle Scholar
  54. 54.
    M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, L. Fallani, Science 349, 1510 (2015)MathSciNetCrossRefGoogle Scholar
  55. 55.
    N. Marzari, D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)CrossRefGoogle Scholar
  56. 56.
    L. Mazza, M. Aidelsburger, H.-H. Tu, N. Goldman, M. Burrello, New J. Phys. 17, 105001 (2015)CrossRefGoogle Scholar
  57. 57.
    M. Mǎntoiu, R. Purice, The Mathematical Formalism of a Particle in a Magnetic Field, in Mathematical Physics of Quantum Mechanics: Selected and Refereed Lectures, ed. by J. Asch, A. Joye. Lecture Notes in Physics, vol. 690, pp. 417–434 (Springer, Berlin, 2006)Google Scholar
  58. 58.
    M. Mǎntoiu, R. Purice, S. Richard, J. Funct. Anal. 250, 42 (2007)MathSciNetCrossRefGoogle Scholar
  59. 59.
    S. Melinte, M. Berciu, C. Zhou, E. Tutuc, S.J. Papadakis, C. Harrison, E.P. De Poortere, M. Wu, P.M. Chaikin, M. Shayegan, R.N. Bhatt, R.A. Register, Phys. Rev. Lett. 92, 036802 (2004)CrossRefGoogle Scholar
  60. 60.
    H. Miyake, G.A. Siviloglou, C.J. Kennedy, W.C. Burton, W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013)CrossRefGoogle Scholar
  61. 61.
    T. Oka, H. Aoki, Phys. Rev. B 79, 081406 (2009)CrossRefGoogle Scholar
  62. 62.
    S. Paganelli, G.L. Giorgi, F. de Pasquale, Fortschr. Phys. 57, 1094 (2009)CrossRefGoogle Scholar
  63. 63.
    R. Peierls, Z. Phys. 80, 763 (1933)CrossRefGoogle Scholar
  64. 64.
    L. Pitaevskii, S. Stringari, Bose-Einstein Condensation and Superfluidity (Oxford University Press, Oxford, 2016)CrossRefMATHGoogle Scholar
  65. 65.
    L.A. Ponomarenko, R.V. Gorbachev, G.L. Yu, D.C. Elias, R. Jalil, A.A. Patel, A. Mishchenko, A.S. Mayorov, C.R. Woods, J.R. Wallbank, M. Mucha-Kruczynski, B.A. Piot, M. Potemski, I.V. Grigorieva, K.S. Novoselov, F. Guinea, V.I. Falko, A. K. Geim, Nature 497, 594 (2013)CrossRefGoogle Scholar
  66. 66.
    R. Prange, S.M. Girvin (eds.), The Quantum Hall Effect (Springer-Verlag, New York, 1990)Google Scholar
  67. 67.
    E. Prodan, H. Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics (Springer, Cham, 2016)CrossRefMATHGoogle Scholar
  68. 68.
  69. 69.
    J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, P. Windpassinger, Phys. Rev. Lett. 108, 225304 (2012)CrossRefGoogle Scholar
  70. 70.
    J. Struck, M. Weinberg, C. Ölschläger, P. Windpassinger, J. Simonet, K. Sengstock, R. Höppner, P. Hauke, A. Eckardt, M. Lewenstein, L. Mathey, Nat. Phys. 9, 738 (2013)CrossRefGoogle Scholar
  71. 71.
    D.J. Thouless, The quantum hall effect and the Schrödinger equation with competing periods, in Number Theory and Physics, ed. by J.M. Luck, P. Moussa, M. Waldschmidt. Springer Proceedings in Physics, vol. 47, pp. 170–176 (Springer-Verlag, Berlin, 1990)Google Scholar
  72. 72.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)CrossRefGoogle Scholar
  73. 73.
    A. Trombettoni, A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001)CrossRefGoogle Scholar
  74. 74.
    A. Trombettoni, A. Smerzi, P. Sodano, New J. Phys. 7, 57 (2005)CrossRefGoogle Scholar
  75. 75.
    D. van Oosten, P. van der Straten, H.T.C. Stoof, Phys. Rev. A 63, 053601 (2001)CrossRefGoogle Scholar
  76. 76.
    M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006)CrossRefGoogle Scholar
  77. 77.
    D. Yoshioka, The Quantum Hall Effect (Springer, Berlin, 2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • M. Burrello
    • 1
  • L. Lepori
    • 2
    • 3
  • S. Paganelli
    • 2
  • A. Trombettoni
    • 4
    • 5
  1. 1.Niels Bohr International Academy and Center for Quantum DevicesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Dipartimento di Scienze Fisiche e ChimicheUniversitá dell’AquilaCoppito-L’AquilaItaly
  3. 3.INFN, Laboratori Nazionali del Gran SassoAssergiItaly
  4. 4.CNR-IOM DEMOCRITOS Simulation CenterTriesteItaly
  5. 5.SISSA and INFN, Sezione di TriesteTriesteItaly

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