title: Bloch state tomography using Wilson lines.
abstract: Topology and geometry are essential to our understanding of modern physics, underlying many foundational concepts from high-energy theories, quantum information, and condensed-matter physics. In condensed-matter systems, a wide range of phenomena stem from the geometry of the band eigenstates, which is encoded in the matrix-valued Wilson line for general multiband systems. Using an ultracold gas of rubidium atoms loaded in a honeycomb optical lattice, we realize strong-force dynamics in Bloch bands that are described by Wilson lines and observe an evolution in the band populations that directly reveals the band geometry. Our technique enables a full determination of band eigenstates, Berry curvature, and topological invariants, including single- and multiband Chern and Z₂ numbers.
authors.affiliation: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstrasse 4, 80799 Munich, Germany. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany.
authors.forename: Tracy
authors.initials: T
authors.lastname: Li
authors.affiliation: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstrasse 4, 80799 Munich, Germany. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany.
authors.forename: Lucia
authors.initials: L
authors.lastname: Duca
authors.affiliation: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstrasse 4, 80799 Munich, Germany. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Reitter
authors.affiliation: Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, Germany. Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany. Department of Physics, Harvard University, Cambridge, MA 02138, USA.
authors.forename: Fabian
authors.initials: F
authors.lastname: Grusdt
authors.affiliation: Department of Physics, Harvard University, Cambridge, MA 02138, USA.
authors.forename: Eugene
authors.initials: E
authors.lastname: Demler
authors.affiliation: Department of Physics, Harvard University, Cambridge, MA 02138, USA. Institute for Quantum Information and Matter, Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA.
authors.forename: Manuel
authors.initials: M
authors.lastname: Endres
authors.affiliation: Department of Physics, Stanford University, Stanford, CA 94305, USA.
authors.forename: Monika
authors.initials: M
authors.lastname: Schleier-Smith
authors.affiliation: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstrasse 4, 80799 Munich, Germany. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany.
authors.forename: Immanuel
authors.initials: I
authors.lastname: Bloch
authors.affiliation: Fakultät für Physik, Ludwig-Maximilians-Universität München, Schellingstrasse 4, 80799 Munich, Germany. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany. Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK. uws20@cam.ac.uk.
authors.forename: Ulrich
authors.initials: U
authors.lastname: Schneider