A series of algebras generalizing the octonions and Hurwitz-Radon identity

International audienceWe study non-associative twisted group algebras over (ℤ2)n with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: (a21+⋯+a2N)(b21+⋯+b2ρ(N))=c21+⋯+c2N , where c k are bilinear functions of the a i and b j and where ρ(N) is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop

Effect of a noisy driving field on a bistable polariton system

International audienceWe report on the effect of noise on the characteristics of the bistable polariton emission system. The present experiment provides a time-resolved access to the polariton emission intensity. We evidence the noise-induced transitions between the two stable states of the bistable polaritons. It is shown that the external noise specifications, intensity and correlation time, can efficiently modify the polariton Kramers time and residence time. We find that there is a threshold noise strength that provokes the collapse of the hysteresis loop. The experimental results are reproduced by numerical simulations using Gross-Pitaevskii equation driven by a stochastic excitation

Spin-to-Orbital Angular Momentum Conversion in Semiconductor Microcavities

We experimentally demonstrate a technique for the generation of optical beams carrying orbital angular momentum using a planar semiconductor microcavity. Despite being isotropic systems, the transverse electric - transverse magnetic (TE-TM) polarization splitting featured by semiconductor microcavities allows for the conversion of the circular polarization of an incoming laser beam into the orbital angular momentum of the transmitted light field. The process implies the formation of topological entities, a pair of optical half-vortices, in the intracavity field

Scanning Tunneling Microscope-Induced Luminescence Spectroscopy on Semiconductor Heterostructures

Scanning tunneling microscope (STM)-induced luminescence is explored as a technique for the characterization of semiconductor quantum wells and quantum wire heterostructures. By injecting minority carriers into the cleaved cross section of these structures, luminescence excitation on a nanometer scale is demonstrated. Using spectrally resolved STM-induced luminescence for the tip placed at various positions across the cleaved heterostructure, it is possible to obtain local spectroscopic information on closely spaced quantum structures

Frieze Patterns of Integers

The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics.Comment: 9 pages, 6 figure