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Direct measurement of the Zak phase in topological Bloch bands

Nature Physics volume 9, pages 795–800 (2013)Cite this article

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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.

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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.

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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.

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Author notes

  1. Marcos Atala and Monika Aidelsburger: These authors contributed equally to this work

Authors and Affiliations

  1. Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 Munich, Germany

    Marcos Atala, Monika Aidelsburger, Julio T. Barreiro & Immanuel Bloch

  2. Max-Planck Institute of Quantum Optics, Hans-Kopfermann Straße 1, 85748 Garching, Germany

    Julio T. Barreiro & Immanuel Bloch

  3. Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA

    Dmitry Abanin, Takuya Kitagawa & Eugene Demler

  4. Rakuten.Inc, Shinagawa Seaside Rakuten Tower, 140-0002 Tokyo, 4-12-3, Japan

    Takuya Kitagawa

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  1. Marcos Atala
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  4. Dmitry Abanin
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  5. Takuya Kitagawa
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  7. Immanuel Bloch
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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.

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Correspondence to Immanuel Bloch.

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The authors declare no competing financial interests.

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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). https://doi.org/10.1038/nphys2790

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