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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.

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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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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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