Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in }\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence $\eta_k$ converges to the function $\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta =(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|$

Quantum Cryptography Based on the Time--Energy Uncertainty Relation

A new cryptosystem based on the fundamental time--energy uncertainty relation is proposed. Such a cryptosystem can be implemented with both correlated photon pairs and single photon states.Comment: 5 pages, LaTex, no figure

Entangled photon pairs produced by a quantum dot strongly coupled to a microcavity

We show theoretically that entangled photon pairs can be produced on demand through the biexciton decay of a quantum dot strongly coupled to the modes of a photonic crystal. The strong coupling allows to tune the energy of the mixed exciton-photon (polariton) eigenmodes, and to overcome the natural splitting existing between the exciton states coupled with different linear polarizations of light. Polariton states are moreover well protected against dephasing due to their lifetime ten to hundred times shorter than that of a bare exciton. Our analysis shows that the scheme proposed can be achievable with the present technology

Geometry of the 3-Qubit State, Entanglement and Division Algebras

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere $S^{15}$, which allows for a natural (last) Hopf fibration with $S^8$ as base and $S^7$ as fiber. A striking feature is, as in the case of 1 and 2 qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling 3 qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the 3-qubit state. Conjectures on the possibility to generalize the construction for higher qubit states are also discussed.Comment: 12 pages, 2 figures, final versio

Entanglement of electrons in interacting molecules

Quantum entanglement is a concept commonly used with reference to the existence of certain correlations in quantum systems that have no classical interpretation. It is a useful resource to enhance the mutual information of memory channels or to accelerate some quantum processes as, for example, the factorization in Shor's Algorithm. Moreover, entanglement is a physical observable directly measured by the von Neumann entropy of the system. We have used this concept in order to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electronic correlation is not directly observable, since it is defined as the difference between the exact ground state energy of the many--electrons Schroedinger equation and the Hartree--Fock energy. We have calculated the correlation energy and compared with the entanglement, as functions of the nucleus--nucleus separation using, for the hydrogen molecule, the Configuration Interaction method. Then, in the same spirit, we have analyzed a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules, in order to obtain the configuration corresponding to maximum entanglement.Comment: 15 pages, 7 figures, standard late

Digging the optimum pit: Antlions, spirals and spontaneous stratification

Most animal traps are constructed from self-secreted silk, so antlions are rare among trap builders because they use only materials found in the environment. We show how antlions exploit the properties of the substrate to produce very effective structures in the minimum amount of time. Our modelling demonstrates how antlions: (i) exploit self-stratification in granular media differentially to expose deleterious large grains at the bottom of the construction trench where they can be ejected preferentially, and (ii) minimize completion time by spiral rather than central digging. Both phenomena are confirmed by our experiments. Spiral digging saves time because it enables the antlion to eject material initially from the periphery of the pit where it is less likely to topple back into the centre. As a result, antlions can produce their pits—lined almost exclusively with small slippery grains to maximize powerful avalanches and hence prey capture—much more quickly than if they simply dig at the pit’s centre. Our demonstration, for the first time to our knowledge, of an animal using self-stratification in granular media exemplifies the sophistication of extended phenotypes even if they are only formed from material found in the animal’s environment

Using Tau Polarization for Charged Higgs Boson and SUSY Searches at LHC

The $\tau$ polarization can be easily measured at LHC in the 1-prong hadronic $\tau$ decay channel by measuring what fraction of the $\tau$-jet momentum is carried by the charged track. A simple cut requiring this fraction to be >0.8 retains most of the polarization of $\tau=$+1 $\tau$-jet signal while suppressing the polarization of $\tau=$-1 $\tau$-jet background and practically eliminating the fake $\tau$ background. This can be utilized to extract the charged Higgs signal. It can be also utilized to extract the SUSY signal in the stau NLSP region, and in particular the stau co-annihilaton region.Comment: 8 pages, 9 figures; Fig.8 and Fig.9 are replaced, published in "Physics at the Large Hadron Collider", A Platinum Jubilee Special Issue of the Indian National Science Academy, Springer (2009) p 20

Multiphoton localization and propagating quantum gap solitons in a frequency gap medium

The many-particle spectrum of an isotropic frequency gap medium doped with impurity resonance atoms is studied using the Bethe ansatz technique. The spectrum is shown to contain pairs of quantum correlated ``gap excitations'' and their heavy bound complexes (``gap solitons''), enabling the propagation of quantum information within the classically forbidden gap. In addition, multiparticle localization of the radiation and the medium polarization occurs when such a gap soliton is pinned to the impurity atom.Comment: 8 pages, RevTEX, to appear in Phys. Rev. Let

Entanglement and the SU(2) phase states in atomic systems

We show that a system of 2n identical two-level atoms interacting with n cavity photons manifests entanglement and that the set of entangled states coincides with the so-called SU(2) phase states. In particular, violation of classical realism in terms of the GHZ and GHSH conditions is proved. We discuss a new property of entanglement expressed in terms of local measurements. We also show that generation of entangled states in the atom-photon systems under consideration strongly depends on the choice of initial conditions and that the parasitic influence of cavity detuning can be compensated through the use of Kerr medium.Comment: 10 pages, 1 figur