Measuring Chern numbers in Cold Gases: 2D and 4D Quantum Hall Physics in the Lab
Prof. Dr. Nathan Goldman (Université Libre de Bruxelles, Belgien)

Optical-lattice experiments have recently succeeded in probing the geometry of 2D Bloch bands with cold neutral atoms. Beyond these local geometrical effects, which are captured by the Berry curvature, 2D Bloch bands may also display non-trivial topology, a global property captured by a topological invariant (e.g. the first Chern number). Such topological properties have dramatic consequences on the transport of non-interacting atoms, such as quantized responses whenever the bands are uniformly populated. In this talk, I will start with the first experimental demonstration of topological transport in a gas of neutral particles, which revealed the Chern number through a cold-atom analogue of quantum-Hall measurements [1]. This measurement is based on a center-of-mass response, which I will describe in some detail [2]. I will then show how this Chern-number measurement could be extended in order to probe the topology of higher-dimensional systems. In particular, I will show how the "second Chern number" - an emblematic topological invariant associated with 4D Bloch bands - could be extracted from an atomic gas, using a 3D optical lattice extended by a synthetic dimension [3].

[1] Measuring the Chern Number of Hofstadter Bands with Ultracold Bosonic Atoms, M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbene, N. R. Cooper, I. Bloch, N. Goldman, Nature Physics 11, 162 (2015).

[2] On the measurement of Chern numbers through center-of-mass responses, H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, N. Goldman, In preparation.

[3] Four-Dimensional Quantum Hall Effect with Ultracold Atoms, H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, N. Goldman, Phys. Rev. Lett. 115, 195303 (2015).