Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Topological features are usually detected by means of transport measurements in systems with filled bands, or by direct imaging of the edge states.
Here, we focus on two specific classes of models: one-dimensional chiral systems , and two-dimensional Hofstadter stripes , and show that in both cases the value of the topological invariants may be read out by performing a bulk measurement in real time.
For one-dimensional chiral systems, we introduce the mean chiral displacement, and we show analytically that for a localized initial condition this observable rapidly approaches a multiple of the Zak phase. Then we measure experimentally the two Zak phases characterizing a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. The two windings are then combined to characterize the complete topological phase diagram of this Floquet system. This detection method is extremely general, as it can be applied to all one-dimensional platforms simulating static or periodically-driven chiral systems.
We switch then to consider Hofstadter stripes (lattices elongated along one direction, pierced by a uniform magnetic flux) which are readily realized by means of synthetic dimensions. We show that following the free evolution of an initially localized wavepacket in presence of a weak force allows to extract the value of the Chern number of the various bands of the model, and that this measure is resistant to weak disorder and confining potentials.
 "Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons" F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, Nature Communications 8, 15516 (2017)
 "Measuring Chern numbers in Hofstadter strips" S. Mugel, A. Dauphin, P. Massignan, L. Tarruell, M. Lewenstein, C. Lobo, and A. Celi, arXiv:1705.04676