Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system. Then we measure the Zak phase in a photonic quantum walk of twisted photons, by observing the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general and readily applicable to all present one-dimensional platforms simulating static or Floquet chiral systems.

We describe the polarization topology of the vector beams emerging from a patterned birefringent liquid crystal plate with a topological charge q at its center (q-plate). The polarization topological structures for different q-plates and different input polarization states have been studied experimentally by measuring the Stokes parameters point-by-point in the beam transverse plane. Furthermore, we used a tuned q=1/2-plate to generate cylindrical vector beams with radial or azimuthal polarizations, with the possibility of switching dynamically between these two cases by simply changing the linear polarization of the input beam.

A discrete quantum walk occurs in the orbital angular momentum space of light, both for a single photon and for two simultaneous photons.

Many phenomena in solid-state physics can be understood in terms of their topological properties. Recently, controlled protocols of quantum walk (QW) are proving to be effective simulators of such phenomena. Here we report the realization of a photonic QW showing both the trivial and the non-trivial topologies associated with chiral symmetry in one-dimensional (1D) periodic systems. We find that the probability distribution moments of the walker position after many steps can be used as direct indicators of the topological quantum transition: while varying a control parameter that defines the system phase, these moments exhibit a slope discontinuity at the transition point. Numerical simulations strongly support the conjecture that these features are general of 1D topological systems. Extending this approach to higher dimensions, different topological classes, and other typologies of quantum phases may offer general instruments for investigating and experimentally detecting quantum transitions in such complex systems.

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We study chiral models in one spatial dimension, both static and periodically driven. We demonstrate that their topological properties may be read out through the long time limit of a bulk observable, the mean chiral displacement. The derivation of this result is done in terms of spectral projectors, allowing for a detailed understanding of the physics. We show that the proposed detection converges rapidly and it can be implemented in a wide class of chiral systems. Furthermore, it can measure arbitrary winding numbers and topological boundaries, it applies to all non-interacting systems, independently of their quantum statistics, and it requires no additional elements, such as external fields, nor filled bands.Topological phases of matter constitute a new paradigm by escaping the standard Ginzburg-Landau theory of phase transitions. These exotic phases appear without any symmetry breaking and are not characterized by a local order parameter, but rather by a global topological order. In the last decade, topological insulators have attracted much interest [1]. These systems are insulators in their bulk but exhibit current carrying edge states protected by the topology. A classification of topological insulators in terms of their discrete symmetries and their spatial dimensionality has been obtained in the celebrated periodic table of topological insulators and superconductors [2]. The topological invariant characterizing these models can be derived from the bulk Hamiltonian and allows one to recover the so called bulk-edge correspondence, namely that the number of topologically protected edge states is proportional to the topological invariant. A famous example of this correspondence can be found in the Quantum-Hall effect where the quantization of the Hall conductance is rooted in the current-carrying protected edge states [3][4][5]. The ensemble of (natural and artificial) topological insulators is steadily growing, and these have been by now synthetically engineered in a multitude of physical systems such as atomic [6][7][8][9][10][11], superconducting [12], photonic [13][14][15][16][17] and acoustic platforms [18][19][20].This work focuses on one-dimensional (1D) topological insulators possessing chiral symmetry. As a consequence of the chiral symmetry, the different sites of the unit cell can always be regarded as part of two sublattices. The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system. In the latter case, the topology can be richer than its static counterpart [26][27][28][29]. In both cases, two different approaches to characterize the topology of such systems have been proposed and implemented experimentally. The first one is to look at intrinsic prope...

Abstract:We present a convenient method to generate vector beams of light having polarization singularities on their axis, via partial spin-toorbital angular momentum conversion in a suitably patterned liquid crystal cell. The resulting polarization patterns exhibit a C-point on the beam axis and an L-line loop around it, and may have different geometrical structures such as "lemon", "star", and "spiral". Our generation method allows us to control the radius of L-line loop around the central C-point. Moreover, we investigate the free-air propagation of these fields across a Rayleigh range.

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