When physicists first observed quantised steps of conductance in a two-dimensional electron gas, one of the most important insights from these experiments was that topology is at work here. Topology, originally a branch of mathematics, has since then transformed our understanding of condensed matter physics, linking physically measurable observables to geometric properties of matter. Geometric (global) characteristics of a material are often robust to small deformations and physical phenomena associated with topology can thus also be robust. Taking into account the intrinsic fragility of quantum systems, harnessing the robustness of topological phenomena may revolutionise many quantum devices, as well as quantum technology as a whole.
Topology with ultracold atoms in optical lattices
An attractive route towards better understanding topological matter is to build controllable model systems. Ultracold atoms in optical lattices represent models of condensed matter systems, roughly at a scale of 1:1000. Here, neutral atoms mimic the electrons, allowing the observation of microscopic processes in lattices on natural timescales. In 2014-2016, researchers were first able to create topological bandstructures and observe several topological phenomena in optical lattices [1,2], including topological charge pumping [3,4]. A topological or 'Thouless' charge pump can be viewed as the one-dimensional analogue of the bulk quantum Hall effect, in which the time axis represents a second dimension.
The milestone experiments mentioned above [1-4] were operating in a regime in which inter-particle interactions did not play an important role. Yet, many-body interactions are ubiquitously present in real solids and understanding many-body phenomena constitutes one of the most active areas in condensed matter physics. An advantage of ultracold atoms is the ability to tune interactions via Feshbach resonances  and directly observe phenomena in response to varying interaction strength.
Observation of quantisation and breakdown of topological pumping
In our work, we employ a novel dynamical superlattice to realise a topological pump of ultracold fermionic potassium-40 atoms (Fig. 1). Ramping the phase φ of one of the laser beams, leads to a pumping motion in the x-direction, whose efficiency we record by taking successive photos of the moving cloud of atoms.
In a first experiment, we confirm the presence of topological pumping for non-interacting atoms. The measured efficiency is 0.95(3) unit cells per period, very close to the quantised value of unity. We then repeat the measurement for a range of Hubbard U, keeping the lattice filling and other experimental parameters fixed. The results of this measurement are shown below in Fig. 2.
We measure near-quantised efficiencies for weak and intermediate interactions, as well as strongly attractive interactions (negative U). Yet, for strongly repulsive interactions (positive U) the efficiency drops down. The data shows that topological pumping indeed remains robust against Hubbard interactions, as long as their absolute values are smaller than the non-interacting energy gap to excited states (which in our case is roughly 2Δ0 in the units of the x-axis in Fig. 2). For strong interactions, there is a striking asymmetry between repulsive and attractive interactions. At first, we were surprised by the asymmetry, since the Hubbard model behaves symmetrically between U and -U, in certain regimes. After performing additional measurement, including time-resolved double occupancy detection and measurements of adiabaticity timescale for pumping, it became clear that pairs of two fermions are responsible for quantised transport for very strong attractive interactions. Conversely, a strongly repulsive Hubbard U precludes the formation of double occupancies, preventing quantum transport.
Our measurements set the stage for further investigation of topological phases with ultracold atoms. One possible avenue is the investigation of interaction-induced topological phases. Rather than leading to a breakdown, could interactions induce or even stabilise topological behaviour?
Another largely open question is how topological edge states respond to many-body interactions. In many topological phenomena, such as the paradigmatic quantum Hall effect in two-dimensional electron gases, edge states play a key role in the quantisation. So far, we investigated the 'bulk' response of an entire atomic cloud and its transport properties. In the future, we plan to study the role of edge states by increasing the atomic confinement.
 Jotzu et al. Experimental realization of the topological Haldane model with ultracold fermions, Nature 515, 237 (2014)
 Aidelsburger et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nature Physics 11, 162 (2015)
 Lohse et al. A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice Nature Physics 12, 350 (2016)
 Nakajima et al. Topological Thouless pumping of ultracold fermions Nature Physics 12, 296 (2016)
 Bloch, Dalibard, Zwerger Many-body physics with ultracold gases Reviews of Modern Physics 80, 885 (2008)