Exact results for the star lattice chiral spin liquid

We examine the star lattice Kitaev model whose ground state is a a chiral spin liquid. We fermionize the model such that the fermionic vacua are toric code states on an effective Kagome lattice. This implies that the Abelian phase of the system is inherited from the fermionic vacua and that time reversal symmetry is spontaneously broken at the level of the vacuum. In terms of these fermions we derive the Bloch-matrix Hamiltonians for the vortex free sector and its time reversed counterpart and illuminate the relationships between the sectors. The phase diagram for the model is shown to be a sphere in the space of coupling parameters around the triangles of the lattices. The abelian phase lies inside the sphere and the critical boundary between topologically distinct Abelian and non-Abelian phases lies on the surface. Outside the sphere the system is generically gapped except in the planes where the coupling parameters are zero. These cases correspond to bipartite lattice structures and the dispersion relations are similar to that of the original Kitaev honeycomb model. In a further analysis we demonstrate the three-fold non-Abelian groundstate degeneracy on a torus by explicit calculation.Comment: 7 pages, 8 figure

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

First observation of the quantized exciton-polariton field and effect of interactions on a single polariton.

© 2018 The Authors. Published by Science. This is an open access article available under a Creative Commons licence. The published version can be accessed at the following link on the publisher’s website: https://doi.org/10.1126/sciadv.aao6814Polaritons are quasi-particles that originate from the coupling of light with matter and that demonstrate quantum phenomena at the many-particle mesoscopic level, such as Bose-Einstein condensation and superfluidity. A highly sought and long-time missing feature of polaritons is a genuine quantum manifestation of their dynamics at the single-particle level. Although they are conceptually perceived as entangled states and theoretical proposals abound for an explicit manifestation of their single-particle properties, so far their behavior has remained fully accounted for by classical and mean-field theories. We report the first experimental demonstration of a genuinely quantum state of the microcavity polariton field, by swapping a photon for a polariton in a two-photon entangled state generated by parametric downconversion. When bringing this single-polariton quantum state in contact with a polariton condensate, we observe a disentangling with the external photon. This manifestation of a polariton quantum state involving a single quantum unlocks new possibilities for quantum information processing with interacting bosons

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

The emergence of health inequalities in early adulthood: evidence on timing and mechanisms from a West of Scotland cohort

Background Evidence is inconsistent as to whether or not there are health inequalities in adolescence according to socio-economic position (SEP) and whether or when they emerge in early adulthood. Despite the large health inequalities literature, few studies have simultaneously compared the relative importance of ?health selection? versus ?social causation? at this life-stage. This study followed a cohort through the youth-adult transition to: (1) determine whether, and if so, when, health inequalities became evident according to both class of origin and current SEP; (2) compare the importance of health selection and social causation mechanisms; and (3) investigate whether these phenomena vary by gender. Methods Data are from a West-of-Scotland cohort, surveyed five times between age 15 (in 1987, N=1,515, response=85%) and 36. Self-reported physical and mental health were obtained at each survey. SEP was based on parental occupational class at 15, a combination of own education or occupational status at 18 and own occupational class (with an additional non-employment category) at older ages. In respect of when inequalities emerged, we used the relative index of inequality to examine associations between both parental and own current SEP and health at each age. In respect of mechanisms, path models, including SEP and health at each age, investigated both inter and intra-generational paths from SEP to health (?causation?) and from health to SEP (?selection?). Analyses were conducted separately for physical and mental health, and stratified by gender. Results Associations between both physical and mental health and parental SEP were non-significant at every age. Inequalities according to own SEP emerged for physical health at 24 and for mental health at 30. There was no evidence of selection based on physical health, but some evidence of associations between mental health in early adulthood and later SEP (intra-generational selection). Paths indicated intra-generational (males) and inter-generational (females) social causation of physical health inequalities, and intra-generational (males and females) and inter-generational (females) social causation of mental health inequalities. Conclusions The results suggest complex and reciprocal relationships between SEP and health and highlight adolescence and early adulthood as a sensitive period for this process, impacting on future life-chances and health