A note on an unusual type of polar decomposition

AbstractMotivated by applications in the theory of unitary congruence, we introduce the factorization of a square complex matrix A of the form A=SU, where S is complex symmetric and U is unitary. We call this factorization a symmetric–unitary polar decomposition or an SUPD. It is shown that an SUPD exists for every matrix A and is always nonunique. Even the symmetric factor S can be chosen in infinitely many ways. Nevertheless, we show that many properties of the conventional polar decomposition related to normal matrices have their counterparts for the SUPD, provided that normal matrices are replaced with conjugate–normal ones

Freestanding metasurfaces for optical frequencies

We present freestanding metasurfaces operating at optical frequencies with a total thickness of only 40$\,$nm. The metasurfaces are fabricated by focused ion beam milling of nanovoids in a carbon film followed by thermal evaporation of gold and plasma ashing of the carbon film. As a first example, we demonstrate a metasurface lens based on resonant V-shaped nanovoids with a focal length of 1$\,$mm. The second example is a metasurface phase-plate consisting of appropriately oriented rectangular nanovoids that transforms a Gaussian input beam into a Laguerre-Gaussian ${LG_{-1,0}}$ mode

Evidence for Kosterlitz-Thouless type orientational ordering of CF3_3Br monolayers physisorbed on graphite

Monolayers of the halomethane CF$_3$Br adsorbed on graphite have been investigated by x-ray diffraction. The layers crystallize in a commensurate triangular lattice. On cooling they approach a three-sublattice antiferroelectric pattern of the in-plane components of the dipole moments. The ordering is not consistent with a conventional phase transition, but points to Kosterlitz-Thouless behavior. It is argued that the transition is described by a 6-state clock model on a triangular lattice with antiferromagnetic nearest neighbor interactions which is studied with Monte-Carlo simulations. A finite-size scaling analysis shows that the ordering transition is indeed in the KT universality class.Comment: 4 pages, 5 figure

Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem

The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. We establish the equivalence between Bethe-Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe-Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm-Dancoff approximation are overestimated. In order to solve large scale problems of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms

7th Workshop on Matrix Equations and Tensor Techniques

No abstract availabl

Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page

A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange

Parametric wavelength exchange (PWE) has been demonstrated as a versatile device in providing different functionalities. In this paper, we will concentrate, numerically and experimentally, on one of these functionalities, namely, all-optical time demultiplexing of 160-Gb/s return-to-zero (RZ) signals based on a pulsed-pump PWE in a 400 m highly nonlinear dispersion-shifted fiber. Experimental results show power penalties < 2.7 dB at bit-error rate of 10-9 for all demultiplexed 10-Gb/s RZ signals. We also derive theoretical expressions for the conversion/residual efficiencies and investigate the impact of pump pulse width and phase mismatch on these efficiencies. Furthermore, the impacts of pulsed-pump wavelength and power level on the characteristics of the switching window are investigated numerically. As a result, the demultiplexer can be easily upgraded to an add-drop multiplexer because of the complete exchange nature of PWE, which is justified by the surviving channels' waveform performance. © 2009 IEEE.published_or_final_versio