Abstract:
We reconsider the Euclidean version of the photon number integral introduced in ref 1. This integral is well defined for any smooth non-self-intersecting curve in $\R^N$. Besides studying general features of this integral (including it s conformal invariance), we evaluate it explicitly for the ellipse. The result is $n_{ellipse}=(\xi^{-1}+\xi)\pi^2$, where $\xi$ is the ratio of the minor and major axes. This is in agreement with the previous result $n_{circle}=2\pi^2$ and also with the conjecture that the minimum value of $n$ for any plane curve occurs for the circle.

Abstract:
The scheme of photon-number tomography is discussed in the framework of star-product quantization. The connection of dual quantization scheme and observables is reviewed. The quantizer and dequantizer operators and kernels of star product of tomograms in photon-number tomography scheme and its dual one are presented in explicit form. The fidelity and state purity are discussed in photon{number tomographic scheme, and the expressions for fidelity and purity are obtained in the form of integral of the product of two photon-number tomograms with integral kernel which is presented in explicit form. The properties of quantumness are discussed in terms of inequalities on state photon{number tomograms.

Abstract:
The application of the Lorentz integral transform (LIT) method to photon scattering off nuclei is presented in general. As an example, elastic photon scattering off the deuteron in the unretarded dipole approximation is considered using the LIT method. The inversion of the integral transform is discussed in detail paying particular attention to the high-energy contributions in the resonance term. The obtained E1-polarizabilities are compared to results from the literature. The corresponding theoretical cross section is confronted with experimental results confirming, as already known from previous studies, that the E1-contribution is the most important one at lower energies.

Abstract:
The Lorentz integral transform method is briefly reviewed. The issue of the inversion of the transform, and in particular its ill-posedness, is addressed. It is pointed out that the mathematical term ill-posed is misleading and merely due to a historical misconception. In this connection standard regularization procedures for the solution of the integral transform problem are presented. In particular a recent one is considered in detail and critical comments on it are provided. In addition a general remark concerning the concept of the Lorentz integral transform as a method with a controlled resolution is made.

Abstract:
The Visible Light Photon Counter (VLPC) has the capability to discriminate photon number states, in contrast to conventional photon counters which can only detect the presence or absence of photons. We use this capability, along with the process of parametric down-conversion, to generate photon number states. We experimentally demonstrate generation of states containing 1,2,3 and 4 photons with high fidelity. We then explore the effect the detection efficiency of the VLPC has on the generation rate and fidelity of the created states.

Abstract:
The Lorentz Integral Transform approach allows microscopic calculations of electromagnetic reaction cross sections without explicit knowledge of final state wave functions. The necessary inversion of the transform has to be treated with great care, since it constitutes a so-called ill-posed problem. In this work new inversion techniques for the Lorentz Integral Transform are introduced. It is shown that they all contain a regularization scheme, which is necessary to overcome the ill-posed problem. In addition it is illustrated that the new techniques have a much broader range of application than the present standard inversion method of the Lorentz Integral Transform.

Abstract:
We demonstrate an integral gated mode single photon detector at telecom wavelengths. The charge number of an avalanche pulse rather than the peak current is monitored for single-photon detection. The transient spikes in conventional gated mode operation are canceled completely by integrating, which enables one to improve the performance of single photon detector greatly with the same avalanche photodiode. This method has achieved a detection efficiency of 29.9% at the dark count probability per gate equal to 5.57E-6/gate (1.11E-6/ns) at 1550nm.

Abstract:
The group theoretical aspect of the description of passive lossless optical four-ports (beam splitters) is revisited. It is shown through an example, that this approach can be useful in understanding interferometric schemes where a low number of photons interfere. The formalism is extended to passive lossless optical six-ports, their SU(3)-theory is outlined.

Abstract:
A Lorentz invariant positive definite expression for photon number density is derived as the absolute square of the invariant scalar product of a polarization sensitive position eigenvector and the photon wave function. It is found that this scalar product is independent of the form chosen for the wave function and that the normalized positive frequency vector potential-electric field pair is a convenient choice of wave function in the presence of matter. The number amplitude describing a localized state is a delta-function at the instant at which localization and detection are seen as simultaneous.

Abstract:
We consider the set of alternating paths on a fixed fully packed loop of size n. This set is in bijection with the set of fully packed loops of size n. Furthermore, for a special choice of fully packed loop, we demonstrate that the set of alternating paths are nested osculating loops, which we call Dyck islands. Dyck islands can be constructed as a union of lattice Dyck paths, and we use this structure to give a simple graphical formula for the calculation of the inversion number of an alternating sign matrix.