Direct Measurement of the Zak phase in Topological Bloch Bands
  Geometric phases that characterize the topological properties of Bloch bands play a fundamental role in the modern band theory of solids. Here we report on the direct 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 acquired during an 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 optical lattice, which models polyacetylene, we measure a difference of the Zak phase equal to phi_Zak=0.97(2)pi for the two possible polyacetylene phases with different dimerization. This indicates that the two dimerized phases 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 in optical lattices. 
Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin,   Takuya Kitagawa, Eugene Demler and Immanuel Bloch
cond-mat.quant-gas cond-mat.mes-hall quant-ph