The tensor structure on the representation category of the Wp\mathcal{W}_p triplet algebra

We study the braided monoidal structure that the fusion product induces on the abelian category $\mathcal{W}_p$-mod, the category of representations of the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of $\mathcal{W}_p$-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of $\mathcal{W}_p$-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective $\mathcal{W}_p$-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.Comment: 58 pages; edit: added references and revisions according to referee reports. Version to appear on J. Phys.

A new three-parameter correlation for gamma-ray bursts with a plateau phase in the afterglow

Gamma ray bursts (GRBs) have great advantages for their huge burst energies, luminosities and high redshifts in probing the Universe. A few interesting luminosity correlations of GRBs have been used to test cosmology models. Especially, for a subsample of long GRBs with known redshifts and a plateau phase in the afterglow, a correlation between the end time of the plateau phase (in the GRB rest frame) and the corresponding X-ray luminosity has been found. In this paper, we re-analyze the subsample and found that a significantly tighter correlation exists when we add a third parameter, i.e. the isotropic $\gamma$-ray energy release, into the consideration. Additionally, both long and intermediate duration GRBs are consistent with the same three-parameter correlation equation. It is argued that the new three-parameter correlation is consistent with the hypothesis that the subsample of GRBs with a plateau phase in the afterglow be associated with the birth of rapidly rotating magnetars, and that the plateau be due to the continuous energy-injection from the magnetar. It is suggested that the newly born millisecond magnetars associated with GRBs might provide a good standard candle in the Universe.Comment: 11 pages, 3 figures, 1 table; A&A, in pres

Optimized Double-well quantum interferometry with Gaussian squeezed-states

A Mach-Zender interferometer with a gaussian number-difference squeezed input state can exhibit sub-shot-noise phase resolution over a large phase-interval. We obtain the optimal level of squeezing for a given phase-interval $\Delta\theta_0$ and particle number $N$, with the resulting phase-estimation uncertainty smoothly approaching $3.5/N$ as $\Delta\theta_0$ approaches 10/N, achieved with highly squeezed states near the Fock regime. We then analyze an adaptive measurement scheme which allows any phase on $(-\pi/2,\pi/2)$ to be measured with a precision of $3.5/N$ requiring only a few measurements, even for very large $N$. We obtain an asymptotic scaling law of $\Delta\theta\approx (2.1+3.2\ln(\ln(N_{tot}\tan\Delta\theta_0)))/N_{tot}$, resulting in a final precision of $\approx 10/N_{tot}$. This scheme can be readily implemented in a double-well Bose-Einstein condensate system, as the optimal input states can be obtained by adiabatic manipulation of the double-well ground state.Comment: updated versio

Self-calibrating threshold detector

A self calibrating threshold detector comprises a single demodulating channel which includes a mixer having one input receiving the incoming signal and another input receiving a local replica code. During a short time interval, an incorrect local code is applied to the mixer to incorrectly demodulate the incoming signal and to provide a reference level that calibrates the noise propagating through the channel. A sample and hold circuit is coupled to the channel for storing a sample of the reference level. During a relatively long time interval, the correct replica code provides an output level which ranges between the reference level and a maximum level that represents incoming signal presence and synchronism with the replica code. A summer substracts the stored sample reference from the output level to provide a resultant difference signal indicative of the acquisition of the expected signal

The combline filter and phase-lock loop. A new technique to improve FM television reception

Development and performance of combline filter phase locked loop combination for television receptio