Phase transitions, entanglement and quantum noise interferometry in cold atoms

We show that entanglement monotones can characterize the pronounced enhancement of entanglement at a quantum phase transition if they are sensitive to long-range high order correlations. These monotones are found to develop a sharp peak at the critical point and to exhibit universal scaling. We demonstrate that similar features are shared by noise correlations and verify that these experimentally accessible quantities indeed encode entanglement information and probe separability.Comment: 4 pages 4 figure

A new basis for eigenmodes on the Sphere

The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space ${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the Laplacian on the sphere, with eigenvalue $\lambda_{\ell} = -\ell ~(\ell +1)$. Here we show the existence of a different basis $\Phi ^{\ell}_j$ for ${\cal V} ^{\ell}$, where $\Phi ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell ^{th}$ power of the scalar product of the current point with a specific null vector $N_j$. We give explicitly the transformation properties between the two bases. The simplicity of calculations in the new basis allows easy manipulations of the harmonic functions. In particular, we express the transformation rules for the new basis, under any isometry of the sphere. The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and back) allows to derive new properties for the $Y_{\ell m}$. In particular, this leads to a new relation for the $Y_{\ell m}$, which is a finite version of the well known integral representation formula. It provides also new development formulae for the Legendre polynomials and for the special Legendre functions.Comment: 6 pages, no figure; new version: shorter demonstrations; new references; as will appear in Journal of Physics A. Journal of Physics A, in pres

Measuring the irreversibility of numerical schemes for reversible stochastic differential equations

Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th