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Sixty years ago, Karplus and Luttinger pointed out that quantum particles moving on a lattice could acquire an anomalous transverse velocity in response to a force, providing an explanation for the unusual Hall effect in ferromagnetic metals. A striking manifestation of this transverse transport was then revealed in the quantum Hall effect, where the plateaus depicted by the Hall conductivity were attributed to a topological invariant characterizing Bloch bands: the Chern number. Until now, topological transport associated with non-zero Chern numbers has only been revealed in electronic systems. Here we use studies of an atomic clouds transverse deflection in response to an optical gradient to measure the Chern number of artificially generated Hofstadter bands. These topological bands are very flat and thus constitute good candidates for the realization of fractional Chern insulators. Combining these deflection measurements with the determination of the band populations, we obtain an experimental value for the Chern number of the lowest band $ u_{mathrm{exp}} =0.99(5)$. This result, which constitutes the first Chern-number measurement in a non-electronic system, is facilitated by an all-optical artificial gauge field scheme, generating uniform flux in optical superlattices.
We demonstrate the experimental implementation of an optical lattice that allows for the generation of large homogeneous and tunable artificial magnetic fields with ultracold atoms. Using laser-assisted tunneling in a tilted optical potential we engi
We introduce a scheme by which flat bands with higher Chern number $vert Cvert>1$ can be designed in ultracold gases through a coherent manipulation of Bloch bands. Inspired by quantum-optics methods, our approach consists in creating a dark Bloch ba
The present Chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gas
There have been significant recent advances in realizing bandstructures with geometrical and topological features in experiments on cold atomic gases. We provide an overview of these developments, beginning with a summary of the key concepts of geome
We employ the exact diagonalization method to analyze the possibility of generating strongly correlated states in two-dimensional clouds of ultracold bosonic atoms which are subjected to a geometric gauge field created by coupling two internal atomic