Direct measurement of the Zak phase in topological Bloch bands

Journal name:
Nature Physics
Volume:
9,
Pages:
795–800
Year published:
DOI:
doi:10.1038/nphys2790
Received
Accepted
Published online

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 δφZak=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.

At a glance

Figures

  1. Energy bands and topology of dimerized lattice model.
    Figure 1: Energy bands and topology of dimerized lattice model.

    a, Schematic illustration of the optical superlattice potential used in the experiment to realize the SSH model (yellow box denotes the unit cell of size d=λs). b, Exemplary curves for the lower and upper energy bands (red and blue lines) and phase θk for dimerization D1 and D2 (solid and dashed line) as a function of quasimomentum k. c, Pseudo-spin representation of the eigenstates  of the upper and lower energy bands for the two dimerization configurations D1 and D2. The pseudo-spin vectors point in opposite directions and exhibit the same sense of rotation (winding) with quasimomentum k. In the phase D1 (D2) evolve (anti-) clockwise and therefore exhibit opposite winding.

  2. Experimental sequence and spin-dependent Bloch oscillations.
    Figure 2: Experimental sequence and spin-dependent Bloch oscillations.

    a, Energy band, MW pulses and state evolution of a single atom in a superposition of two spin-states with opposite magnetic moment (brown and green balls) during the three-step echo sequence described in the text. The winding of the state vector with k is given by θk (solid line dimerization D1, dashed line dimerization D2). b,c, Time-of-flight momentum distributions taken for different evolution times of the spin-dependent Bloch oscillations in the lower (b) and upper energy band (c) used in the experiment. Each momentum point is an average of three identical measurements.

  3. Determination of the Zak phase.
    Figure 3: Determination of the Zak phase.

    a, Following the sequence described in the text, the atom number in the two spin states, N↑,↓, is measured and the fraction of atoms in the |↑right fence spin state, n=N/(N+N), is plotted as a function of the phase of the final microwave π/2-pulse. Blue (black) circles correspond to the fringe in which the dimerization was (not) swapped, and the corresponding solid lines are sinusoidal fits to the data, where the free parameters were the amplitude, offset and initial phase. The difference in phase of the two fits to the Ramsey fringes yields the Zak phase difference δφZak=φZakD1φZakD2. To reduce the effect of fluctuations, every data point is an average of five individual measurements and the error bars show the standard deviation of the mean. The phase of the reference fringe (black) is determined by a small detuning of the microwave pulse (Supplementary Information). b, Measured relative phase for 14 identical experimental runs (left), which give an average value of δφZak=0.97(2)π. The corresponding histogram is shown on the right with a binning of 0.05π. The 1σ-width of the resulting distribution is σ=0.07π.

  4. Fractional Zak phase.
    Figure 4: Fractional Zak phase.

    a, Lattice potential without and with an on-site energy staggering Δ. When Δ=0 the Zak phase is φZak(Δ=0)=π/2. As Δ increases, the pseudo-spin vectors of the lower (blue vector) and upper (red vector) band move away from the equatorial plane and the value of φZak(Δ) decays rapidly to zero. b, Measured phase difference φZakφZak(Δ) as a function of Δ. Each individual point was obtained from four individual measurements. The vertical error bars represent the standard error of the mean. The green line is the theoretical prediction and the shaded area represents the uncertainties in the calibration of the energy offset Δ. The insets show a typical Ramsey fringe for Δ/J=−1.2 (left) and Δ/J=1.2 (right), which were used to extract the relative phase δφ. The blue (black) fringes correspond to measurements with (without) staggering (Supplementary Information).

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Author information

  1. These authors contributed equally to this work

    • Marcos Atala &
    • Monika Aidelsburger

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, 4-12-3, 140-0002 Tokyo, Japan

    • Takuya Kitagawa

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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The authors declare no competing financial interests.

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