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
- Solitons with fermion number 1/2. Phys. Rev. D 13, 3398–3409 (1976). &
- Fractional quantum numbers on solitons. Phys. Rev. Lett. 47, 986–989 (1981). &
- Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979). , &
- On states, on a lattice, with half-integer charge. Nucl. Phys. B 220, 1–12 (1983). &
- Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982). , , &
- Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010). , &
- Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010). &
- Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). &
- Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).
- Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009).
- Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010). , , &
- Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).
- Topological origin of zero-energy edge states in particle–hole symmetric systems. Phys. Rev. Lett. 89, 077002 (2002). &
- Zak phase and the existence of edge states in graphene. Phys. Rev. B 84, 195452 (2011). , &
- Spectral asymmetry on an open space. Phys. Rev. D 30, 809–818 (1984). &
- Elementary excitations of a linearly conjugated diatomic polymer. Phys. Rev. Lett. 49, 1455–1459 (1982). &
- Seeing topological order in time-of-flight measurements. Phys. Rev. Lett. 107, 235301 (2011). et al.
- Chern numbers hiding in time-of-flight images. Phys. Rev. A 84, 063629 (2011). et al.
- Measuring topology in a laser-coupled honeycomb lattice: From Chern insulators to topological semi-metals. New J. Phys. 15, 013025 (2013). et al.
- Mapping the Berry curvature from semiclassical dynamics in optical lattices. Phys. Rev. A 85, 033620 (2012). &
- Direct observation of second-order atom tunnelling. Nature 448, 1029–1032 (2007). et al.
- Dynamics of band electrons in electric and magnetic fields. Rev. Mod. Phys. 34, 645–655 (1962).
- Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651–1654 (1993). &
- Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508–4511 (1996). , , , &
- Single-spin addressing in an atomic Mott insulator. Nature 471, 319–324 (2011). et al.
- Observation of topologically protected bound states in photonic quantum walks. Nature Commun. 3, 882 (2012). et al.
- Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012). et al.
- Particle number fractionalization of an atomic Fermi–Dirac gas in an optical lattice. Phys. Rev. Lett. 88, 180401 (2002). , &
- Topological edge states in the one-dimensional super-lattice Bose–Hubbard model. Phys. Rev. Lett. 110, 260405 (2013). , &
- Interferometric approach to measuring band topology in 2D optical lattices. Phys. Rev. Lett. 110, 165304 (2013). et al.
- Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005). &
- Driven quantum tunneling. Phys. Rep. 304, 229–354 (1998). &
- Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010). , , &
- Floquet topological insulator in semiconductor quantum wells. Nature Phys. 7, 490–495 (2011). , &
- 2003). The Universe in a Helium Droplet (Oxford Univ. Press,
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