We present the realisation of a photonic quantum walk based on the Orbital Angular Momentum of light [1,2], showing topological phases that characterise periodically modulated 1D systems (Floquet topological insulators). These phases of matter are typically investigated through the analysis of protected edges states. Following a different approach, here we rather focus our attention to bulk observables. In particular, we find that important information about the system topology can be extracted from the statistical moments associated with photons OAM spectrum; while varying a control parameter that determines the value of the invariants, these show marked differences in distinct phases and exhibit abrupt variations at the transition points . Quite remarkably, specific combinations of such moments are quantised and proportional to the topological invariants , hence can be used to detect topological phases. Such quantisation arises in absence of any external force, and it is robust to perturbations that preserve the system symmetries. We confirm experimentally these results in our photonic platform, and report the measurement of complete topological invariants characterising this class of systems [3, 4]. We demonstrate that the method we propose is widely applicable in a variety of different 1D topological systems, and readily implementable in experimental platforms that are presently used to simulate these exotic phases.
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