Macroscopic objects in quantum mechanics: A combinatorial approach

Why we do not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic: the coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty to observe a large scale entanglement. As the number of particles n grows we need an ever more precise knowledge of the state, and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness W distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness W to satisfy |tr(W\rho)| \leq 1 for all separable states \rho, then the efficiency of W depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm ||W||. It is known that there are witnesses on the space of n qbits for which ||W|| is exponential in n. However, we conjecture that for a large majority of n-qbit witnesses ||W|| \leq O(\sqrt{n logn}). Thus, in a non ideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf (Phys. Rev. A 64, 032112 (2001)).Comment: RevTeX, 14 pages, some additions to the published version: A second conjecture added, discussion expanded, and references adde

Complex organics in IRAS 4A revisited with ALMA and PdBI: Striking contrast between two neighbouring protostellar cores

We used the Atacama Large (sub-)Millimeter Array (ALMA) and the IRAM Plateau de Bure Interferometer (PdBI) to image, with an angular resolution of 0.5$''$ (120 au) and 1$''$ (235 au), respectively, the emission from 11 different organic molecules in the protostellar binary NGC1333 IRAS 4A. We clearly disentangled A1 and A2, the two protostellar cores present. For the first time, we were able to derive the column densities and fractional abundances simultaneously for the two objects, allowing us to analyse the chemical differences between them. Molecular emission from organic molecules is concentrated exclusively in A2 even though A1 is the strongest continuum emitter. The protostellar core A2 displays typical hot corino abundances and its deconvolved size is 70 au. In contrast, the upper limits we placed on molecular abundances for A1 are extremely low, lying about one order of magnitude below prestellar values. The difference in the amount of organic molecules present in A1 and A2 ranges between one and two orders of magnitude. Our results suggest that the optical depth of dust emission at these wavelengths is unlikely to be sufficiently high to completely hide a hot corino in A1 similar in size to that in A2. Thus, the significant contrast in molecular richness found between the two sources is most probably real. We estimate that the size of a hypothetical hot corino in A1 should be less than 12 au. Our results favour a scenario in which the protostar in A2 is either more massive and/or subject to a higher accretion rate than A1, as a result of inhomogeneous fragmentation of the parental molecular clump. This naturally explains the smaller current envelope mass in A2 with respect to A1 along with its molecular richness.Comment: Accepted in Astronomy and Astrophysic

A spectral line survey in the 2 mm and 1.3 mm windows toward the carbon rich envelope of IRC +10216

We present the results of our spectral line surveys in the 2 mm and 1.3 mm windows toward the carbon rich envelope of IRC +10216. Totally 377 lines are detected, among which 360 lines are assigned to 57 known molecules (including 29 rare isotopomers and 2 cyclic isomers). Only 17 weak lines remain unidentified. Rotational lines of isotopomers 13CCH and HN13C are detected for the first time in IRC +10216. The detection of the formaldehyde lines in this star is also confirmed. Possible abundance difference among the three 13C substituted isotopic isomers of HC3N is reported. Isotopic ratios of C and O are confirmed to be non-solar while those of S and Si to be nearly solar. Column densities have been estimated for 15 molecular species. Modified spectroscopic parameters have been calculated for NaCN, Na13CN, KCN and SiC2. Transition frequencies from the present observations were used to improve the spectroscopic parameters of Si13CC, 29SiC2 and 30SiC2.Comment: 17 pages of text, 18 pages of 14 tables, 35 pages of 4 figures, a typo corrected in Abstrac

Hubble Space Telescope Survey of Interstellar ^12CO/^13CO in the Solar Neighborhood

We examine 20 diffuse and translucent Galactic sight lines and extract the column densities of the ^12CO and ^13CO isotopologues from their ultraviolet A--X absorption bands detected in archival Space Telescope Imaging Spectrograph data with lambda/Deltalambda geq 46,000. Five more targets with Goddard High-Resolution Spectrograph data are added to the sample that more than doubles the number of sight lines with published Hubble Space Telescope observations of ^13CO. Most sight lines have 12-to-13 isotopic ratios that are not significantly different from the local value of 70 for ^12C/^13C, which is based on mm-wave observations of rotational lines in emission from CO and H_2CO inside dense molecular clouds, as well as on results from optical measurements of CH^+. Five of the 25 sight lines are found to be fractionated toward lower 12-to-13 values, while three sight lines in the sample are fractionated toward higher ratios, signaling the predominance of either isotopic charge exchange or selective photodissociation, respectively. There are no obvious trends of the ^12CO-to-^13CO ratio with physical conditions such as gas temperature or density, yet ^12CO/^13CO does vary in a complicated manner with the column density of either CO isotopologue, owing to varying levels of competition between isotopic charge exchange and selective photodissociation in the fractionation of CO. Finally, rotational temperatures of H_2 show that all sight lines with detected amounts of ^13CO pass through gas that is on average colder by 20 K than the gas without ^13CO. This colder gas is also sampled by CN and C_2 molecules, the latter indicating gas kinetic temperatures of only 28 K, enough to facilitate an efficient charge exchange reaction that lowers the value of ^12CO/^13CO.Comment: 1-column emulateapj, 23 pages, 9 figure

Abundances and Isotope Ratios in the Magellanic Clouds: The Star Forming Environment of N113

With the goal of deriving the physical and chemical conditions of star forming regions in the Large Magellanic Cloud (LMC), a spectral line survey of the prominent star forming region N113 is presented. The observations cover parts of the frequency range from 85 GHz to 357 GHz and include 63 molecular transitions from a total of 16 species, among them spectra of rare isotopologues. Maps of selected molecular lines as well as the 1.2 mm continuum distribution are also presented. Molecular abundances in the core of the complex are found to be consistent with a photon dominated region (PDR) that is nitrogen deficient, with the potential exception of N2H+. Densities range from 5x10^3 cm-3 for CO to almost 10^6 for CS and HCN, indicating that only the densest regions provide sufficient shielding even for some of the most common species. An ortho- to para-H_2CO ratio of ~3 hints at H_2CO formation in a warm (>=40 K) environment. Isotope ratios are 12C/13C ~ 49+-5, 16O/18O ~ 2000+-250, 18O/17O ~ 1.7+-0.2 and 32S/34S ~ 15. Agreement with data from other star forming clouds shows that the gas is well mixed in the LMC . The isotope ratios do not only differ from those seen in the Galaxy. They also do not form a continuation of the trends observed with decreasing metallicity from the inner to the outer Galaxy. This implies that the outer Galaxy, is not providing a transition zone between the inner Galaxy and the metal poor environment of the Magellanic Clouds. A part of this discrepancy is likely caused by differences in the age of the stellar populations in the outer Galaxy and the LMC.Comment: 50 pages, 13 figures, accepted for publication in Ap

Spectrum and diffusion for a class of tight-binding models on hypercubes

We propose a class of exactly solvable anisotropic tight-binding models on an infinite-dimensional hypercube. The energy spectrum is analytically computed and is shown to be fractal and/or absolutely continuous according to the value hopping parameters. In both cases, the spectral and diffusion exponents are derived. The main result is that, even if the spectrum is absolutely continuous, the diffusion exponent for the wave packet may be anything between 0 and 1 depending upon the class of models.Comment: 5 pages Late

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Self-similar sets: projections, sections and percolation

We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.Postprin

The dynamical evolution of the circumstellar gas around low-and intermediate-mass stars I: the AGB

We have investigated the dynamical interaction of low- and-intermediate mass stars (from 1 to 5 Msun) with their interstellar medium (ISM). In this first paper, we examine the structures generated by the stellar winds during the Asymptotic Giant Branch (AGB) phase, using a numerical code and the wind history predicted by stellar evolution. The influence of the external ISM is also taken into account. We find that the wind variations associated with the thermal pulses lead to the formation of transient shells with an average lifetime of 20,000 yr, and consequently do not remain recorded in the density or velocity structure of the gas. The formation of shells that survive at the end of the AGB occurs via two main processes: shocks between the shells formed by two consecutive enhancements of the mass-loss or via continuous accumulation of the material ejected by the star in the interaction region with the ISM. Our models show that the mass of the circumstellar envelope increases appreciably due to the ISM material swept up by the wind (up to 70 % for the 1 Msun stellar model). We also point out the importance of the ISM on the deceleration and compression of the external shells. According to our simulations, large regions (up to 2.5 pc) of neutral gas surrounding the molecular envelopes of AGB stars are expected. These large regions of gas are formed from the mass-loss experienced by the star during the AGB evolution.Comment: 43 pages, 15 figures. Accepted for publication in the Astrophysical Journa

Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws

We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our construction is based on some ``continuous stochastic multiplication'' from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.Comment: 24 pages, 4 figure