Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Direct measurement of the Zak phase in topological Bloch bands

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

References

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  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).

    Article  ADS  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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  33. 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 

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

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

Authors

Contributions

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.

Corresponding author

Correspondence to Immanuel Bloch.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 960 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Atala, M., Aidelsburger, M., Barreiro, J. et al. Direct measurement of the Zak phase in topological Bloch bands. Nature Phys 9, 795–800 (2013). https://doi.org/10.1038/nphys2790

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2790

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing