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Topological quantum matter with ultracold gases in optical lattices

Nature Physics volume 12, pages 639–645 (2016)Cite this article

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Abstract

Since the discovery of topological insulators, many topological phases have been predicted and realized in a range of different systems, providing both fascinating physics and exciting opportunities for devices. And although new materials are being developed and explored all the time, the prospects for probing exotic topological phases would be greatly enhanced if they could be realized in systems that were easily tuned. The flexibility offered by ultracold atoms could provide such a platform. Here, we review the tools available for creating topological states using ultracold atoms in optical lattices, give an overview of the theoretical and experimental advances and provide an outlook towards realizing strongly correlated topological phases.

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Figure 1: Measurement of the Chern number.
Figure 2: The Peierls substitution and the Aharonov–Bohm effect.
Figure 3: Exploiting a synthetic dimension.
Figure 4: Floquet engineering with cold atoms.

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References

  1. Prange, R. & Girvin, S. The Quantum Hall Effect (Springer, 1990).

    Book  Google Scholar 

  2. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045 (2010).

    Article  ADS  Google Scholar 

  3. Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057 (2011).

    Article  ADS  Google Scholar 

  4. Bergholtz, E. J. & Liu, Z. Topological flat band models and fractional Chern insulators. Int. J. Mod. Phys. B 27, 1330017 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885 (2008).

    Article  ADS  Google Scholar 

  7. Lewenstein, M., Sanpera, A. & Ahufinger, V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems 1st edn (Oxford Univ. Press, 2012).

    Book  MATH  Google Scholar 

  8. Grimm, R., Weidemüller, M. & Ovchinnikov, Y. B. Optical dipole traps for neutral atoms. Adv. At. Mol. Opt. Phys. 42, 95–170 (2000).

    Article  ADS  Google Scholar 

  9. Grynberg, G. & Robilliard, C. Cold atoms in dissipative optical lattices. Phys. Rep. 355, 335–451 (2001).

    Article  ADS  Google Scholar 

  10. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225 (2010).

    Article  ADS  Google Scholar 

  11. Cooper, N. R. Rapidly rotating atomic gases. Adv. Phys. 57, 539–616 (2008).

    Article  ADS  Google Scholar 

  12. Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Colloquium: artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523 (2011).

    Article  ADS  Google Scholar 

  13. Goldman, N., Juzeliūnas, G., Öhberg, P. & Spielman, I. B. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014).

    Article  ADS  Google Scholar 

  14. Bakr, W. S., Gillen, J. I., Peng, A., Fölling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 74–77 (2009).

    Article  ADS  Google Scholar 

  15. Endres, M. et al. Single-site- and single-atom-resolved measurement of correlation functions. Appl. Phys. B 113, 27–39 (2013).

    Article  ADS  Google Scholar 

  16. Islam, R. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77–83 (2015).

    Article  ADS  Google Scholar 

  17. Stamper-Kurn, D. M. et al. Excitation of phonons in a Bose–Einstein condensate by light scattering. Phys. Rev. Lett. 83, 2876–2879 (1999).

    Article  ADS  Google Scholar 

  18. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    Article  ADS  Google Scholar 

  19. Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nature Phys. 11, 162–166 (2015).

    Article  ADS  Google Scholar 

  20. Sørensen, A. S., Demler, E. & Lukin, M. D. Fractional quantum Hall states of atoms in optical lattices. Phys. Rev. Lett. 94, 086803 (2005).

    Article  ADS  Google Scholar 

  21. Eckardt, A., Jinasundera, T., Weiss, C. & Holthaus, M. Analog of photon-assisted tunneling in a Bose–Einstein condensate. Phys. Rev. Lett. 95, 200401 (2005).

    Article  ADS  Google Scholar 

  22. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Article  ADS  Google Scholar 

  23. Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nature Phys. 7, 490–495 (2011).

    Article  ADS  Google Scholar 

  24. Bermudez, A., Schaetz, T. & Porras, D. Synthetic gauge fields for vibrational excitations of trapped ions. Phys. Rev. Lett. 107, 150501 (2011).

    Article  ADS  Google Scholar 

  25. Kolovsky, A. R. Creating artificial magnetic fields for cold atoms by photon-assisted tunneling. Europhys. Lett. 93, 20003 (2011).

    Article  ADS  Google Scholar 

  26. Hauke, P. et al. Non-abelian gauge fields and topological insulators in shaken optical lattices. Phys. Rev. Lett. 109, 145301 (2012).

    Article  ADS  Google Scholar 

  27. Cayssol, J., Dóra, B., Simon, F. & Moessner, R. Floquet topological insulators. Phys. Status Solidi 7, 101–108 (2013).

    Google Scholar 

  28. Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).

    Google Scholar 

  29. Zheng, W. & Zhai, H. Floquet topological states in shaking optical lattices. Phys. Rev. A 89, 061603(R) (2014).

    Article  ADS  Google Scholar 

  30. Bukov, M., D’Alessio, L. & Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139–226 (2015).

    Article  ADS  Google Scholar 

  31. Weitenberg, C. et al. Single-spin addressing in an atomic Mott insulator. Nature 471, 319–324 (2011).

    Article  ADS  Google Scholar 

  32. Juzeliūnas, G. & Öhberg, P. Slow light in degenerate Fermi gases. Phys. Rev. Lett. 93, 033602 (2004).

    Article  ADS  Google Scholar 

  33. Jaksch, D. & Zoller, P. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56 (2003).

    Article  ADS  Google Scholar 

  34. 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).

    Article  ADS  Google Scholar 

  35. Ruseckas, J., Juzeliūnas, G., Öhberg, P. & Fleischhauer, M. Non-abelian gauge potentials for ultracold atoms with degenerate dark states. Phys. Rev. Lett. 95, 010404 (2005).

    Article  ADS  Google Scholar 

  36. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    Article  ADS  Google Scholar 

  37. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Karplus, R. & Luttinger, J. M. Hall effect in ferromagnetics. Phys. Rev. 95, 1154–1160 (1954).

    Article  ADS  MATH  Google Scholar 

  39. Mead, C. A. The geometric phase in molecular systems. Rev. Mod. Phys. 64, 51–85 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  40. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Price, H. M. & Cooper, N. R. Mapping the Berry curvature from semiclassical dynamics in optical lattices. Phys. Rev. A 85, 033620 (2012).

    Article  ADS  Google Scholar 

  42. Duca, L. et al. An Aharonov-Bohm interferometer for determining Bloch band topology. Science 347, 288–292 (2015).

    Article  ADS  Google Scholar 

  43. Alba, E., Fernandez-Gonzalvo, X., Mur-Petit, J., Pachos, J. K. & Garcia-Ripoll, J. J. Seeing topological order in time-of-flight measurements. Phys. Rev. Lett. 107, 235301 (2011).

    Article  ADS  Google Scholar 

  44. Hauke, P., Lewenstein, M. & Eckardt, A. Tomography of band insulators from quench dynamics. Phys. Rev. Lett. 113, 045303 (2014).

    Article  ADS  Google Scholar 

  45. Fläschner, N. et al. Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science 352, 1091–1094 (2016).

    Article  ADS  Google Scholar 

  46. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, 2003).

    MATH  Google Scholar 

  47. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  48. Dauphin, A. & Goldman, N. Extracting the Chern number from the dynamics of a Fermi gas: implementing a quantum Hall bar for cold atoms. Phys. Rev. Lett. 111, 135302 (2013).

    Article  ADS  Google Scholar 

  49. Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Measurement of Chern numbers through center-of-mass responses. Phys. Rev. B 93, 245113 (2016).

    Article  ADS  Google Scholar 

  50. Liu, X.-J., Law, K. T., Ng, T. K. & Lee, P. A. Detecting topological phases in cold atoms. Phys. Rev. Lett. 111, 120402 (2013).

    Article  ADS  Google Scholar 

  51. Wu, Z. et al. Realization of two-dimensional spin-orbit coupling for Bose–Einstein condensates. Preprint at http://arXiv.org/abs/1511.08170 (2015).

  52. Grusdt, F., Yao, N. Y., Abanin, D., Fleischhauer, M. & Demler, E. Interferometric measurements of many-body topological invariants using mobile impurities. Preprint at http://arXiv.org/abs/1512.03407 (2015).

  53. 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, 2185–2190 (1982).

    Article  ADS  Google Scholar 

  54. Goldman, N. et al. Direct imaging of topological edge states in cold-atom systems. Proc. Natl Acad. Sci. USA 110, 6736–6741 (2013).

    Article  ADS  Google Scholar 

  55. Reichl, M. D. & Mueller, E. J. Floquet edge states with ultracold atoms. Phys. Rev. A 89, 063628 (2014).

    Article  ADS  Google Scholar 

  56. Mancini, M. et al. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510–1513 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Stuhl, B. K., Lu, H. I., Aycock, L. M., Genkina, D. & Spielman, I. B. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Atala, M. et al. Observation of chiral currents with ultracold atoms in bosonic ladders. Nature Phys. 10, 588–593 (2014).

    Article  ADS  Google Scholar 

  59. Liu, X.-J., Liu, X., Wu, C. & Sinova, J. Quantum anomalous Hall effect with cold atoms trapped in a square lattice. Phys. Rev. A 81, 033622 (2010).

    Article  ADS  Google Scholar 

  60. Stanescu, T. D., Galitski, V. & Das Sarma, S. Topological states in two-dimensional optical lattices. Phys. Rev. A 82, 013608 (2010).

    Article  ADS  Google Scholar 

  61. Goldman, N., Beugnon, J. & Gerbier, F. Detecting chiral edge states in the Hofstadter optical lattice. Phys. Rev. Lett. 108, 255303 (2012).

    Article  ADS  Google Scholar 

  62. Bermudez, A. et al. Wilson fermions and axion electrodynamics in optical lattices. Phys. Rev. Lett. 105, 190404 (2010).

    Article  ADS  Google Scholar 

  63. Dubcek, T. et al. Weyl points in three-dimensional optical lattices: synthetic magnetic monopoles in momentum space. Phys. Rev. Lett. 114, 225301 (2015).

    Article  ADS  Google Scholar 

  64. Price, H. M., Zilberberg, O., Ozawa, T., Carusotto, I. & Goldman, N. Four-dimensional quantum Hall effect with ultracold atoms. Phys. Rev. Lett. 115, 195303 (2015).

    Article  ADS  Google Scholar 

  65. Luttinger, J. M. The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84, 814–817 (1951).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  66. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    Article  ADS  Google Scholar 

  67. Cooper, N. R. Optical flux lattices for ultracold atomic gases. Phys. Rev. Lett. 106, 175301 (2011).

    Article  ADS  Google Scholar 

  68. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).

    Article  ADS  Google Scholar 

  69. Ruostekoski, J., Dunne, G. V. & Javanainen, J. Manipulating atoms in an optical lattice: fractional fermion number and its optical quantum measurement. Phys. Rev. Lett. 88, 180401 (2002).

    Article  ADS  Google Scholar 

  70. Gerbier, F. & Dalibard, J. Gauge fields for ultracold atoms in optical superlattices. New J. Phys. 12, 033007 (2010).

    Article  ADS  Google Scholar 

  71. Goldman, N. et al. Realistic time-reversal invariant topological insulators with neutral atoms. Phys. Rev. Lett. 105, 255302 (2010).

    Article  ADS  Google Scholar 

  72. Liu, X.-J., Law, K. T. & Ng, T. K. Realization of 2D spin-orbit interaction and exotic topological orders in cold atoms. Phys. Rev. Lett. 112, 086401 (2014).

    Article  ADS  Google Scholar 

  73. Yao, N. Y. et al. Topological flat bands from dipolar spin systems. Phys. Rev. Lett. 109, 266804 (2012).

    Article  ADS  Google Scholar 

  74. Sun, K., Liu, W. V., Hemmerich, A. & Das Sarma, S. Topological semimetal in a fermionic optical lattice. Nature Phys. 8, 67–70 (2012).

    Article  ADS  Google Scholar 

  75. Dauphin, A., Müller, M. & Martin-Delgado, M. A. Rydberg-atom quantum simulation and Chern-number characterization of a topological Mott insulator. Phys. Rev. A 86, 053618 (2012).

    Article  ADS  Google Scholar 

  76. Barbarino, S., Taddia, L., Rossini, D., Mazza, L. & Fazio, R. Magnetic crystals and helical liquids in alkaline-earth fermionic gases. Nature Commun. 6, 8134 (2015).

    Article  ADS  Google Scholar 

  77. Lacki, M. et al. Quantum Hall physics with cold atoms in cylindrical optical lattices. Phys. Rev. A 93, 013604 (2016).

    Article  ADS  Google Scholar 

  78. Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).

    Article  ADS  Google Scholar 

  79. Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C. & Ketterle, W. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302 (2013).

    Article  ADS  Google Scholar 

  80. Kennedy, C. J., Burton, W. C., Chung, W. C. & Ketterle, W. Observation of Bose–Einstein condensation in a strong synthetic magnetic field. Nature Phys. 11, 859–864 (2015).

    Article  ADS  Google Scholar 

  81. Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice. Nature Phys. 12, 350–354 (2016).

    Article  ADS  Google Scholar 

  82. Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nature Phys. 12, 296–300 (2015).

    Article  ADS  Google Scholar 

  83. Lu, H.-I. et al. Geometrical pumping with a Bose–Einstein condensate. Phys. Rev. Lett. 116, 200402 (2016).

    Article  ADS  Google Scholar 

  84. Thouless, D. J. Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  85. Struck, J. et al. Engineering Ising-XY spin-models in a triangular lattice using tunable artificial gauge fields. Nature Phys. 9, 738–743 (2013).

    Article  ADS  Google Scholar 

  86. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  87. Moore, G. & Read, N. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  88. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    Article  ADS  Google Scholar 

  89. Kitaev, A. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  90. Sau, J. D., Lutchyn, R. M., Tewari, S. & Sarma, S. D. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).

    Article  ADS  Google Scholar 

  91. Galitski, V. & Spielman, I. B. Spin-orbit coupling in quantum gases. Nature 494, 49–54 (2013).

    Article  ADS  Google Scholar 

  92. Huang, L. et al. Experimental realization of two-dimensional synthetic spin-orbit coupling in ultracold Fermi gases. Nature Phys. 12, 540–544 (2016).

    Article  ADS  Google Scholar 

  93. Zhang, C., Tewari, S., Lutchyn, R. M. & Das Sarma, S. p x + ip y superfluid from s-wave interactions of fermionic cold atoms. Phys. Rev. Lett. 101, 160401 (2008).

    Article  ADS  Google Scholar 

  94. Massignan, P., Sanpera, A. & Lewenstein, M. Creating p-wave superfluids and topological excitations in optical lattices. Phys. Rev. A 81, 031607 (2010).

    Article  ADS  Google Scholar 

  95. Seo, K., Han, L. & Sá de Melo, C. Emergence of Majorana and dirac particles in ultracold fermions via tunable interactions, spin-orbit effects, and Zeeman fields. Phys. Rev. Lett. 109, 105303 (2012).

    Article  ADS  Google Scholar 

  96. Tewari, S., Das Sarma, S., Nayak, C., Zhang, C. & Zoller, P. Quantum computation using vortices and Majorana zero modes of a p x + ip y superfluid of fermionic cold atoms. Phys. Rev. Lett. 98, 010506 (2007).

    Article  ADS  Google Scholar 

  97. Jiang, L. et al. Majorana fermions in equilibrium and in driven cold-atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011).

    Article  ADS  Google Scholar 

  98. Nascimbène, S. Realizing one-dimensional topological superfluids with ultracold atomic gases. J. Phys. B 46, 134005 (2013).

    Article  ADS  Google Scholar 

  99. Kraus, C. V., Dalmonte, M., Baranov, M. A., Läuchli, A. M. & Zoller, P. Majorana edge states in atomic wires coupled by pair hopping. Phys. Rev. Lett. 111, 173004 (2013).

    Article  ADS  Google Scholar 

  100. Kraus, C. V., Zoller, P. & Baranov, M. A. Braiding of atomic Majorana fermions in wire networks and implementation of the Deutsch–Jozsa algorithm. Phys. Rev. Lett. 111, 203001 (2013).

    Article  ADS  Google Scholar 

  101. Diehl, S. et al. Quantum states and phases in driven open quantum systems with cold atoms. Nature Phys. 4, 878–883 (2008).

    Article  ADS  Google Scholar 

  102. Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation and quantum-state engineering driven by dissipation. Nature Phys. 5, 633–636 (2009).

    Article  ADS  Google Scholar 

  103. Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  104. Diehl, S., Rico, E., Baranov, M. A. & Zoller, P. Topology by dissipation in atomic quantum wires. Nature Phys. 7, 971–977 (2011).

    Article  ADS  Google Scholar 

  105. Bardyn, C.-E. et al. Topology by dissipation. New J. Phys. 15, 085001 (2013).

    Article  ADS  Google Scholar 

  106. Bardyn, C.-E. et al. Majorana modes in driven-dissipative atomic superfluids with a zero Chern number. Phys. Rev. Lett. 109, 130402 (2012).

    Article  ADS  Google Scholar 

  107. Thouless, D. J. Wannier functions for magnetic sub-bands. J. Phys. C 17, L325–L327 (1984).

    ADS  MathSciNet  Google Scholar 

  108. Budich, J. C., Zoller, P. & Diehl, S. Dissipative preparation of Chern insulators. Phys. Rev. A 91, 042117 (2015).

    Article  ADS  Google Scholar 

  109. Budich, J. C. & Diehl, S. Topology of density matrices. Phys. Rev. B 91, 165140 (2015).

    Article  ADS  Google Scholar 

  110. Wen, X.-G. Quantum Field Theory of Many-body Systems (Oxford Univ. Press, 2007).

    Book  Google Scholar 

  111. Daley, A. J., Pichler, H., Schachenmayer, J. & Zoller, P. Measuring entanglement growth in quench dynamics of bosons in an optical lattice. Phys. Rev. Lett. 109, 020505 (2015).

    Article  Google Scholar 

  112. Palmer, R. N. & Jaksch, D. High-field fractional quantum Hall effect in optical lattices. Phys. Rev. Lett. 96, 180407 (2006).

    Article  ADS  Google Scholar 

  113. Hafezi, M., Sørensen, A. S., Demler, E. & Lukin, M. D. Fractional quantum Hall effect in optical lattices. Phys. Rev. A 76, 023613 (2007).

    Article  ADS  Google Scholar 

  114. Moller, G. & Cooper, N. R. Composite Fermion theory for bosonic quantum Hall states on lattices. Phys. Rev. Lett. 103, 105303 (2009).

    Article  ADS  Google Scholar 

  115. Kapit, E. & Mueller, E. Exact parent Hamiltonian for the quantum Hall states in a lattice. Phys. Rev. Lett. 105, 215303 (2010).

    Article  ADS  Google Scholar 

  116. Nielsen, A. E. B., Sierra, G. & Cirac, J. I. Local models of fractional quantum Hall states in lattices and physical implementation. Nature Commun. 4, 2864 (2013).

    Article  ADS  Google Scholar 

  117. Cooper, N. R. & Dalibard, J. Reaching fractional quantum Hall states with optical flux lattices. Phys. Rev. Lett. 110, 185301 (2013).

    Article  ADS  Google Scholar 

  118. Möller, G. & Cooper, N. R. Fractional Chern insulators in Harper-Hofstadter bands with higher Chern number. Phys. Rev. Lett. 115, 126401 (2015).

    Article  ADS  Google Scholar 

  119. Dai, H.-N. et al. Observation of four-body ring-exchange interactions and anyonic fractional statistics. Preprint at http://arXiv.org/abs/1602.05709 (2016).

  120. Kitaev, A. Yu. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  121. Glaetzle, A. W. et al. Designing frustrated quantum magnets with laser-dressed Rydberg atoms. Phys. Rev. Lett. 114, 173002 (2015).

    Article  ADS  Google Scholar 

  122. van Bijnen, R. M. W. & Pohl, T. Quantum magnetism and topological ordering via Rydberg dressing near Förster resonances. Phys. Rev. Lett. 114, 243002 (2015).

    Article  ADS  Google Scholar 

  123. Glaetzle, A. W. et al. Quantum spin-ice and dimer models with Rydberg atoms. Phys. Rev. X 4, 041037 (2014).

    Google Scholar 

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Acknowledgements

The authors would like to acknowledge M. Aidelsburger, I. Bloch, L. Fallani and M. Mancini for providing experimental data. N.G. is financed by the FRS-FNRS Belgium and by the BSPO under PAI Project No. P7/18 DYGEST. This work has also been supported by the ERC synergy grant UQUAM.

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  1. CENOLI, Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), B-1050 Brussels, Belgium

    N. Goldman

  2. Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria

    J. C. Budich & P. Zoller

  3. Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria

    J. C. Budich & P. Zoller

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Goldman, N., Budich, J. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nature Phys 12, 639–645 (2016). https://doi.org/10.1038/nphys3803

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