The Euclidean distance degree of an algebraic variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic

Adiabatic Splitting, Transport, and Self-Trapping of a Bose-Einstein Condensate in a Double-Well Potential

We show that the adiabatic dynamics of a Bose-Einstein condensate (BEC) in a double well potential can be described in terms of a dark variable resulting from the combination of the population imbalance and the spatial atomic coherence between the two wells. By means of this dark variable, we extend, to the non-linear matter wave case, the recent proposal by Vitanov and Shore [Phys. Rev. A 73, 053402 (2006)] on adiabatic passage techniques to coherently control the population of two internal levels of an atom/molecule. We investigate the conditions to adiabatically split or transport a BEC as well as to prepare an adiabatic self trapping state by the optimal delayed temporal variation of the tunneling rate via either the energy bias between the two wells or the BEC non-linearity. The emergence of non-linear eigenstates and unstable stationary solutions of the system as well as their role in the breaking down of the adiabatic dynamics is investigated in detail.Comment: 8 pages, 7 figure

Quantum theory of a polarization phase-gate in an atomic tripod configuration

We present the quantum theory of a polarization phase-gate that can be realized in a sample of ultracold rubidium atoms driven into a tripod configuration. The main advantages of this scheme are in its relative simplicity and inherent symmetry. It is shown that the conditional phase shifts of order $\pi$ can be attained.Comment: X International Conference on Quantum Optics, Minsk, Belaru

The curve of lines on a prime Fano threefold of genus 8

We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\Gamma,L)$, where $\Gamma$ is its Fano curve of lines and $L=O_{\Gamma}(1)$ is the theta-characteristic which gives a natural embedding \Gamma \subset \matbb{P}^5.Comment: 24 pages, misprints corrected, to appear in International Journal of Mathematic

Fundamental limits of repeaterless quantum communications

Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed ‘teleportation stretching’, we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters

Hybrid Superconducting Neutron Detectors

A new neutron detection concept is presented that is based on superconductive niobium (Nb) strips coated by a boron (B) layer. The working principle of the detector relies on the nuclear reaction 10B+n $\rightarrow$ $\alpha$+ 7Li , with $\alpha$ and Li ions generating a hot spot on the current-biased Nb strip which in turn induces a superconducting-normal state transition. The latter is recognized as a voltage signal which is the evidence of the incident neutron. The above described detection principle has been experimentally assessed and verified by irradiating the samples with a pulsed neutron beam at the ISIS spallation neutron source (UK). It is found that the boron coated superconducting strips, kept at a temperature T = 8 K and current-biased below the critical current Ic, are driven into the normal state upon thermal neutron irradiation. As a result of the transition, voltage pulses in excess of 40 mV are measured while the bias current can be properly modulated to bring the strip back to the superconducting state, thus resetting the detector. Measurements on the counting rate of the device are presented and the future perspectives leading to neutron detectors with unprecedented spatial resolutions and efficiency are highlighted.Comment: 8 pages 6 figure

Swiss mice CD1 fed on mussels contaminated by okadaic acid and yessotoxins: effects on thymus and spleen

The toxicity of okadaic acid (OA) and yessotoxins (YTXs) was studied in mice orally fed on (i) OA (17.80 +/- 2.41 mu g/kg) for 24 h and mouse feed for 24 h; (ii) CA (17.2 +/- 2.13 mu g/kg) plus YTXs (1.30 +/- 0.12 mg/kg)_for 24 h and mouse feed for 24 h; (iii) OA (18.88 +/- 1.86 Ig/kg) plus YTXs (1.45 +/- 0.12 mg/kg) for 24 h. After toxin treatments the thymus and spleen were examined. More severe morpho-functional modifications were found in the thymus, which presented atrophy, a significant depletion in the lymphoid compartment and angiogenesis. In spite of the impairment, a number of inflammatory cells, reactive to anti-cytokine antibodies, were recruited. Moreover, greater expression of matrix metalloproteinase-9, particularly in cells located near new blood vessels, was observed. Thymus injury was still observed after 48 h. Histopathological changes to the spleen were more evident in mice orally treated for 24 h and immediately sacrificed. The organ showed a significant loss of volume and a fibrous component invaded regions involved in immune functions. In white pulp the marginal zones were reduced, lymphoid nodules contained large germinal centres and the periarteriolar lymphoid sheaths showed cellular depletion. An inflammatory cell response was activated by the recruitment of granulocytes, an increased number of active macrophages and increased immunoreactivity to cytokines. Unlike in the thymus, some evidence of recovery was seen in the spleen. The data suggest that low oral doses of OA alone or OA plus YTXs are able to provoke immunostimulation and systemic immunotoxicity, thus also indicative of tumorigenic properties

Polarization phase gate with a tripod atomic system

We analyze the nonlinear optical response of a four-level atomic system driven into a tripod configuration. The large cross-Kerr nonlinearities that occurr in such a system are shown to produce nonlinear phase shift of order $\pi$. Such a substantial shift may be observed in a cold atomic gas in a magneto-optical trap where it coupl be fasibly exploited towards the realization of a polarization quantum phase gate. The experimental feasibility of such a gate is here examined in detail.Comment: Corrected versio