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Direct measurement of the Zak phase in topological Bloch bands


Geometric phases that characterize the topological properties of Bloch bands play a fundamental role in the band theory of solids. Here we report on the measurement of the geometric phase acquired by cold atoms moving in one-dimensional optical lattices. Using a combination of Bloch oscillations and Ramsey interferometry, we extract the Zak phase—the Berry phase gained during the adiabatic motion of a particle across the Brillouin zone—which can be viewed as an invariant characterizing the topological properties of the band. For a dimerized lattice, which models polyacetylene, we measure a difference of the Zak phase δ φZak = 0.97(2)π for the two possible polyacetylene phases with different dimerization. The two dimerized phases therefore belong to different topological classes, such that for a filled band, domain walls have fractional quantum numbers. Our work establishes a new general approach for probing the topological structure of Bloch bands inoptical lattices.

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Figure 1: Energy bands and topology of dimerized lattice model.
Figure 2: Experimental sequence and spin-dependent Bloch oscillations.
Figure 3: Determination of the Zak phase.
Figure 4: Fractional Zak phase.


  1. Jackiw, R. & Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  2. Goldstone, J. & Wilczek, F. Fractional quantum numbers on solitons. Phys. Rev. Lett. 47, 986–989 (1981).

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  4. Bell, J. S. & Rajaraman, R. On states, on a lattice, with half-integer charge. Nucl. Phys. B 220, 1–12 (1983).

    ADS  Article  Google Scholar 

  5. 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).

    ADS  Article  Google Scholar 

  6. Xiao, D., Chang, M-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  Article  Google Scholar 

  7. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  Article  Google Scholar 

  8. Qi, X. & Zhang, S. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    ADS  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  10. Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).

    ADS  Article  Google Scholar 

  11. Ryu, S., Schneider, A., Furusaki, A. & Ludwig, A. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).

    ADS  Article  Google Scholar 

  12. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    ADS  Article  Google Scholar 

  13. Ryu, S. & Hatsugai, Y. Topological origin of zero-energy edge states in particle–hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002).

    ADS  Article  Google Scholar 

  14. Delplace, P., Ullmo, D. & Montambaux, G. Zak phase and the existence of edge states in graphene. Phys. Rev. B 84, 195452 (2011).

    ADS  Article  Google Scholar 

  15. Niemi, A. J. & Semenoff, G. W. Spectral asymmetry on an open space. Phys. Rev. D 30, 809–818 (1984).

    ADS  MathSciNet  Article  Google Scholar 

  16. Rice, M. J. & Mele, E. J. Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455–1459 (1982).

    ADS  Article  Google Scholar 

  17. Alba, E. et al. Seeing topological order in time-of-flight measurements. Phys. Rev. Lett. 107, 235301 (2011).

    ADS  Article  Google Scholar 

  18. Zhao, E. et al. Chern numbers hiding in time-of-flight images. Phys. Rev. A 84, 063629 (2011).

    ADS  Article  Google Scholar 

  19. Goldman, N. et al. Measuring topology in a laser-coupled honeycomb lattice: From Chern insulators to topological semi-metals. New J. Phys. 15, 013025 (2013).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  21. Fölling, S. et al. Direct observation of second-order atom tunnelling. Nature 448, 1029–1032 (2007).

    ADS  Article  Google Scholar 

  22. Wannier, G. H. Dynamics of band electrons in electric and magnetic fields. Rev. Mod. Phys. 34, 645–655 (1962).

    ADS  MathSciNet  Article  Google Scholar 

  23. King-Smith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651–1654 (1993).

    ADS  Article  Google Scholar 

  24. Ben Dahan, M., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508–4511 (1996).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  26. Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nature Commun. 3, 882 (2012).

    ADS  Article  Google Scholar 

  27. Kraus, Y. E. et al. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012).

    ADS  Article  Google Scholar 

  28. Ruostekoski, J., Dunne, G. & Javanainen, J. Particle number fractionalization of an atomic Fermi–Dirac gas in an optical lattice. Phys. Rev. Lett. 88, 180401 (2002).

    ADS  Article  Google Scholar 

  29. Grusdt, F., Hoening, M. & Fleischhauer, M. Topological edge states in the one-dimensional super-lattice Bose–Hubbard model. Phys. Rev. Lett. 110, 260405 (2013).

    ADS  Article  Google Scholar 

  30. Abanin, D. et al. Interferometric approach to measuring band topology in 2D optical lattices. Phys. Rev. Lett. 110, 165304 (2013).

    ADS  Article  Google Scholar 

  31. Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    ADS  Article  Google Scholar 

  32. Grifoni, M. & Hänggi, P. Driven quantum tunneling. Phys. Rep. 304, 229–354 (1998).

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  35. Volovik, G. E. The Universe in a Helium Droplet (Oxford Univ. Press, 2003).

    MATH  Google Scholar 

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We acknowledge helpful discussions with B. Paredes. We thank Y-A. Chen and S. Nascimbène for their help in setting up the experiment and for their comments in the early stages of the experiment. This work was supported by the DFG (FOR635, FOR801), NIM, DARPA (OLE program), Harvard-MIT CUA, the ARO-MURI on Atomtronics, and the ARO MURI Quism program. M. Aidelsburger was further supported by the Deutsche Telekom Stiftung.

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M. Atala, M. Aidelsburger and J.T.B. carried out the experiments and the data analysis. D.A., T.K. and E.D. carried out the theoretical analysis and derived the measurement protocol. I.B. and E.D. supervised the work and developed the general measurement idea. All authors contributed extensively to the analysis and the writing of the manuscript.

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Correspondence to Immanuel Bloch.

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Atala, M., Aidelsburger, M., Barreiro, J. et al. Direct measurement of the Zak phase in topological Bloch bands. Nature Phys 9, 795–800 (2013).

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