Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons

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, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of 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, as it can be applied to all one-dimensional platforms simulating static or Floquet chiral systems.Comment: 10 pages, 7 color figures (incl. appendices) Close to the published versio

Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry

MSEL-1L: L'OROLOGIO MOLECOLARE DEL DIFFERENZIAMENTO NEURALE

The gene SEL-1L codifies for a protein involved in the retrotranslocation of misfolded peptides from the lumen on the endoplasmic reticulum to the cytoplasm, where they are degraded by the ubiquitin-proteasome mechanism in the ERAD (Endoplasmic Reticulum Associated Degradation) pathway. Recently, it was found that the murine protein mSEL-1L (murine SEL-1L) plays an essential role during embryonic development, since its homozygous deletion is lethal during mid-gestation, altering the correct organogenesis. This transgenic model allowed us to investigate mSEL-1L contribution in differentiation and brain development in vivo and to study its role in neural stem cell (NSC) biology in vitro, focusing on its relationship with Notch signaling. We have shown that during embryogenesis, mSEL-1L expression is ubiquitous in the developing brain with high levels detected in the ventricular regions, densely populated by neural progenitors. As development proceeds, mSEL-1L protein levels significantly decrease, allowing the detection of rare cells still expressing the protein only in the ventricular zones and in the dentate gyrus, the only adult neurogenic ¡§niches¡¨ in which a population of undifferentiated and quiescent NSCs is retained. It has been subsequently proved that mSEL-1L expression plays as essential role in directing and ensuring a harmonious NSC differentiation, as the protein deletion is associated with an early cellular differentiation that determines the progenitor pull depletion, both in vivo and in vitro. In particular, mSEL-1L absence in vivo compromises the corticogenesis, because it affects the correct sequence of neurogenic and astrogliogenic phases, altering both the residual stem cell population and its differentiated progeny. The similarity of this transgenic model with Notch mutants led us to investigate the possible involvement of mSEL-1L with this pathway. Co-immunoprecipitation analysis has revealed an interaction between these two proteins, while expression studies have shown a specific inhibition of Notch-1 signaling associated to mSEL-1L down-regulation. The expression of the nuclear activated form of the receptor, as well as that of its main effectors HES-1 and HES-5, are significantly inhibited when mSEL-1L is totally or partially depleted, promoting an erroneous up-regulation of the transcriptional factor Neurogenin-2 (NGN-2) and the following increase of the neuronal marker ƒÒIII¡VTubulin expression. The main NSC properties primarily governed by Notch pathway, such as self-renewal, differentiation, proliferation and cell survival, are drastically affected by mSEL-1L misregulation. Moreover, mSEL-1L deficient mouse models show significant vasculogenic and angiogenic defects during embryonic development, probably due to an alteration of Notch signaling. The proper control of mSEL-1L expression is therefore of paramount importance to enable the correct embryonic development, as well as its regulation during differentiation by specific and sophisticated mechanisms. In fact, it has emerged that mSEL-1L is abundantly expressed in mouse embryonic stem cells (ESCs) and is maintained stable through their in vitro differentiation into neural progenitors (NEPs) first, and then into radial glia-like NSCs, while its expression is inhibited during their final maturation into neurons, astrocytes and oligodendrocytes. So, the protein is not affected by passing from a state of pluripotency (ESCs), to multipotency (NEPs), up to tripotency (NSCs), but it must be silenced to ensure IV NSC terminal differentiation. In fact, it has been demonstrated that mSEL-1L expression is regulated in a post-transcriptional way by mmu-miR-183. This microRNA, whose expression is induced during differentiation both in vitro and in vivo, is able to negatively regulate mSEL-1L, as well as other proteins with an important function in stemness, such as ƒÒ-Integrin and Bmi-1. Differently, mmu-miR-183 down-modulation can promote an increase in the protein levels in NSCs derived from the telencephalic cortex of embryos with only one mSEL-1L functional allele (mSEL-1L HET), ensuring a high protein expression, although in presence of a reduced quantity of its messenger. In conclusion, this research has highlighted an ERAD-independent role of mSEL-1L concerning NSC biology, a role that is essential to guarantee the proper embryonic development and the homeostasis of the whole organism. Therefore, the protein appears as a sort of ¡§molecular clock¡¨ that can drive the transition from a stemness state to a differentiate one, only when it is appropriate. The data here presented may provide a good starting point for the development of specific therapeutic strategies for degenerative pathologies and for regenerative medicine applications

Large-scale free-space photonic circuits in two dimensions

Photonic circuits, engineered to couple optical modes according to a specific map, serve as processors for classical and quantum light. The number of components typically scales with that of processed modes, thus correlating system size, circuit complexity, and optical losses. We present a photonic-circuit technology implementing large-scale unitary maps in free space, coupling a single input to hundreds of output modes in a two-dimensional compact layout. The map corresponds to a quantum walk of structured photons, realized through light propagation in three liquid-crystal metasurfaces, having their optic axes artificially patterned. Theoretically, the walk length and the number of connected modes can be arbitrary while keeping losses constant. The patterns can be designed to replicate multiple unitary maps. We also discuss limited reconfigurability by adjusting the overall birefringence and the relative displacement of the optical elements. These results lay the basis for the design of low-loss nonintegrated photonic circuits, primarily for manipulating multiphoton states in quantum regimes