Spectral phase conjugation with cross-phase modulation compensation

Spectral phase conjugation with short pump pulses in a third-order nonlinear material is analyzed in depth. It is shown that if signal amplification is considered, the conversion efficiency can be significantly higher than previously considered, while the spectral phase conjugation operation remains accurate. A novel method of compensating for cross-phase modulation, the main parasitic effect, is also proposed. The validity of our theory and the performance of the spectral phase conjugation scheme are studied numerically

Complex conjugation supermap of unitary quantum maps and its universal implementation protocol

A complex conjugation of unitary quantum map is a second-order map (supermap) that maps a unitary operator $U$ to its complex conjugate $U^*$. First, we present a deterministic quantum protocol that universally implements the complex conjugation supermap when we are given a blackbox quantum circuit, guaranteed to implement some unitary operation, whose only known description is its dimension. We then discuss the complex conjugation supermap in the context of entanglement theory and derive a conjugation-based expression of the $G$-concurrence. Finally, we present a physical process involving identical fermions from which the complex conjugation protocol is derived as a simulation of the process using qudits.Comment: ver.5: published version, 5 pages, 2 figures, double-colum

Conjugation in Semigroups

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.Comment: 41 pages, 14 figure

[n]cycloparaphenylenes with charges

Oligophenylenes (polyphenylenes) are constituted by an array of conjugated benzenes where inter-ring electron delocalization tends to extend over the whole chain (linear conjugation) being intrinsically limited, among other factors, by terminal effects. Alternatively, cyclic conjugation is envisaged as the unlimited free-boundary versionofconjugation which will impact the structure of molecules in rather unknown ways. The cyclic version of oligophenylenes, cycloparaphenylenes ([n]CPPs with n the number of phenyl rings) were first synthesized in 2008 by Beztozzi and Jasti.1 Today the whole [n]CPP series from [5]CPP to [18]CPP has been prepared. [n]CPPs represent ideal models to investigate new insights of the electronic structure of molecules and cyclic conjugation when electrons or charges circulate in a closed circuit without boundaries. Radical cations and dications of [n]CPP from n=5 to n=12 have been prepared and studied by Raman spectroscopy.2 Small [n]CPP dications own their stability to the closed-shell electronic configuration imposed by cyclic conjugation. However, in large [n]CPP dications cyclic conjugation is minimal and these divalent species form open-shell biradicals. The Raman spectra reflect the effect of cyclic conjugation in competition with cyclic strain and biradicaloid aromatic stabilization. Cyclic conjugation provokes the existence of a turning point or V-shape behavior of the frequencies of the G bands as a function of n. In this communication we will show the vibrational spectroscopic fingerprint of this rare form of conjugation. [1] R. Jasti, J. Bhattacharjee, J. B. Neaton, C. R. Bertozzi, “Synthesis, Characterization, and Theory of [9]-, [12]-, and [18]Cycloparaphenylene: Carbon Nanohoop Structures”, J. Am. Chem. Soc. 130 (2008), 17646–17647. [2] M. P. Alvarez, P. M. Burrezo, M. Kertesz, T. Iwamoto, S. Yamago, J. Xia, R. Jasti, J. T. L. Navarrete, M. Taravillo, V. G. Baonza, J. Casado, “Properties of Sizeable [n]CycloParaPhenylenes As Molecular Models of Single-Wall Carbon Nanotubes By Raman Spectroscopy: Structural and Electron-Transfer Responses Under Mechanical Stress”, Angew. Chem. Int. Ed. 53, (2014), 7033−7037.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Quaternionic Hyperbolic Fenchel-Nielsen Coordinates

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs of hyperbolic elements in $Sp(2,1)$ up to conjugation. A hyperbolic element of $Sp(2,1)$ is called `loxodromic' if it has no real eigenvalue. We show that the set of $Sp(2,1)$ conjugation orbits of irreducible loxodromic pairs is a $(\mathbb C {\mathbb P}^1)^4$-bundle over a topological space that is locally a semi-analytic subspace of ${\mathbb R}^{13}$. We use the above classification to show that conjugation orbits of `geometric' representations of a closed surface group (of genus $g \geq 2$) into $Sp(2,1)$ can be determined by a system of $42g-42$ real parameters. Further, we consider the groups $Sp(1,1)$ and $GL(2, {\mathbb H})$. These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of `geometric' surface group representations into these groups.Comment: major structural revision. Restructured the exposition. Introduction re-writte