Propagation in media as a probe for topological properties

The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete classification of topological phases remains elusive. In this context, experimental studies are key, both for studying the interplay between topology and complex effects and for identifying new forms of topological order. It is therefore crucial to find complementary means to measure topological properties in order to reach a fundamental understanding of topological phases. In one dimensional chiral systems, we suggest a new way to construct and identify topologically protected bound states, which are the smoking gun of these materials. In two dimensional Hofstadter strips (i.e: systems which are very short along one dimension), we suggest a new way to measure the topological invariant directly from the atomic dynamics.In one dimensional optical lattices, topological bound states are difficult to generate due to the absence of sharp boundaries, and harder still to identify unambiguously. By periodically driving a one dimensional dilute gas of atoms with a pair of Raman lasers, we find that a system analogous to the two-step quantum walk can be realised. This system can host two flavours of topologically protected bound states, meaning that it escapes the standard classification of topological phases. This study details the considerations and many of the relevant experimental tools to design a topologically non-trivial system. In particular, we show that we can build a topological boundary by using the lasers’ finite beam width, and that the topologically protected states which live at this boundary can be identified, and differentiated, by studying their spin distribution.The bulk-boundary correspondence states that a system’s bulk and edge properties are indissociable. It is unclear, however, how this principle extends to systems with vanishingly small bulks, as for instance the Hofstadter strip, which was recently realised using a one dimensional gas of spinful atoms. We define a topological invariant for this system which accurately counts the number of topological bound states. This suggests that, even in such an extreme situation, the bulk-boundary correspondence applies. We suggest a method for experimentally measuring this invariant from the atomic dynamics which relies on three main ingredients: the adiabatic loading of a well localised wavepacket in the ground state of the lattice, the application of a weak force along the axis of the strip, and the measurement of the centre of mass position after a Bloch oscillation