We have been able to detect topology via several different methods. First, we have directly observed the Berry curvature of ultra cold atoms in an optical lattice, which the indication of the geometry and topology in the system (published in Science, Fläschner et al., 2016). We have extended the study to dynamical evolution of the geometry and topology, by rapidly changing the geometry of the system. This allowed us to observe dynamical topological defects in the evolution of the wave function (Fläschner et. al., Nature Physics (2018)). Moreover, in the static system, we were able to observe the Dirac points characterising the topology of the honeycomb structure (as in graphene). Smoothly deforming the geometry of the system (which would correspond to putting strain on graphene) allowed us to observe and quantify the merging of the topological defects (Tarnowski et al., PRL (2017)). Furthermore, studying the topology of the evolutional path of the dynamical topological defects allowed us to obtain the topology of the underlying Hamiltonian, hence seeing topology with topology (arXiv:1709.01046 (2017)). In order to do the engineering of these shaken systems, one has to know the underlying band structure with great precision and we were able to determine the exact band structure down to the 0.12% level and fully measure the two-dimensional band structure (Fläschner et al., PRA 2018). Recently, we have also been able to observe a completely new phenomena in topological materials, the so-called quantised circular dichroism. In stead of doing drift or transport measurements on topological materials, we have looked at dissipative dynamics of the topological system and extracted the topology of the system from that (Asteria et al., arXiv:1805.11077 (2018)). Currently, we are working on using Machine Learning techniques to determine the topology, which allows us to fully map out the two-dimensional topological phase diagram of the Haldane model. The deep-learning techniques also allow us to map out the superfluid-to-Mott-Insulator transition and might be a feasible method to detect the effect of interactions and spins on topological systems (Rem et al., arXiv:1809.05519).
In January 2017, I was invited to present our results on Berry curvature and dynamical topological defects in Amsterdam. In February 2017, I've presented the results at a workshop in Munich. During the conference Quantum Optics X, I presented a poster in Gdansk in September 2017. In November 2017, I presented the results including linking number at the FOR2414 workshop in Hamburg. I gave a presentation about our results in December 2017 in Stuttgart. In May 2018, I've presented our results on measuring the band structure at the DAMOP meeting in Fort Lauderdale. Klaus Sengstock gave an invited talk at the same conference about circular dichroism.
