An explicit Schr\"odinger picture for Aharonov's Modular Variable concept

We propose to address in a natural manner, the modular variable concept explicitly in a Schr\"odinger picture. The idea of Modular Variables was introduced in 1969 by Aharonov, Pendleton and Petersen to explain certain non-local properties of quantum mechanics. Our approach to this subject is based on Schwinger's finite quantum kinematics and it's continuous limit.Comment: 16 pages, 9 figure

Berry phase in magnetic systems with point perturbations

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the Wilczek-Zee phase for an infinitely degenerated eigenvalue.Comment: LaTeX, 26 page

Theory of a magnetic microscope with nanometer resolution

We propose a theory for a type of apertureless scanning near field microscopy that is intended to allow the measurement of magnetism on a nanometer length scale. A scanning probe, for example a scanning tunneling microscope (STM) tip, is used to scan a magnetic substrate while a laser is focused on it. The electric field between the tip and substrate is enhanced in such a way that the circular polarization due to the Kerr effect, which is normally of order 0.1% is increased by up to two orders of magnitude for the case of a Ag or W tip and an Fe sample. Apart from this there is a large background of circular polarization which is non-magnetic in origin. This circular polarization is produced by light scattered from the STM tip and substrate. A detailed retarded calculation for this light-in-light-out experiment is presented.Comment: 17 pages, 8 figure

Weak values of a quantum observable and the cross-Wigner distribution

We study the weak values of a quantum observable from the point of view of the Wigner formalism. The main actor is here the cross-Wigner transform of two functions, which is in disguise the cross-ambiguity function familiar from radar theory and time-frequency analysis. It allows us to express weak values using a complex probability distribution. We suggest that our approach seems to confirm that the weak value of an observable is, as conjectured by several authors, due to the interference of two wavefunctions, one coming from the past, and the other from the future.Comment: Submitted for publicatio

A remark on interacting anyons in magnetic field

In this remark, we note that the anyons, interacting with each other through pairwise potential in external magnetic field, exhibit a simple quantum group symmetry.Comment: IPT-EPFL preprint, typos fixed, minor corrections, references updated, submitted to Physics Letter A

Secant varieties of toric varieties

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in \PP^r using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety \Sec X_P. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties $X_A$ embedded using a set of lattice points A \subset P\cap\ZZ^n containing the vertices of $P$ and their nearest neighbors.Comment: v1, AMS LaTex, 5 figures, 25 pages; v2, reference added; v3, This is a major rewrite. We have strengthened our main results to include a classification of smooth lattice polytopes P such that Sec X_P does not have the expected dimension. (See Theorems 1.4 and 1.5.) There was also a considerable amount of reorganization, and some expository material was eliminated; v4, 28 pages, minor corrections, additional and updated reference

Forest productivity under elevated CO 2 and O 3 : positive feedbacks to soil N cycling sustain decade‐long net primary productivity enhancement by CO 2

The accumulation of anthropogenic CO 2 in the Earth’s atmosphere, and hence the rate of climate warming, is sensitive to stimulation of plant growth by higher concentrations of atmospheric CO 2 . Here, we synthesise data from a field experiment in which three developing northern forest communities have been exposed to factorial combinations of elevated CO 2 and O 3 . Enhanced net primary productivity (NPP) ( c. 26% increase) under elevated CO 2 was sustained by greater root exploration of soil for growth‐limiting N, as well as more rapid rates of litter decomposition and microbial N release during decay. Despite initial declines in forest productivity under elevated O 3 , compensatory growth of O 3 ‐tolerant individuals resulted in equivalent NPP under ambient and elevated O 3 . After a decade, NPP has remained enhanced under elevated CO 2 and has recovered under elevated O 3 by mechanisms that remain un‐calibrated or not considered in coupled climate–biogeochemical models simulating interactions between the global C cycle and climate warming.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/88002/1/j.1461-0248.2011.01692.x.pd

Multi-Resolution Analysis and Fractional Quantum Hall Effect: an Equivalence Result

In this paper we prove that any multi-resolution analysis of \Lc^2(\R) produces, for some values of the filling factor, a single-electron wave function of the lowest Landau level (LLL) which, together with its (magnetic) translated, gives rise to an orthonormal set in the LLL. We also give the inverse construction. Moreover, we extend this procedure to the higher Landau levels and we discuss the analogies and the differences between this procedure and the one previously proposed by J.-P. Antoine and the author.Comment: Submitted to Journal Mathematical Physisc

Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem

We present a new approach to the problem of Bloch electrons in magnetic ( sometimes called Azbel-Hofstadter problem) field, by making explicit a natural relation between the group of magnetic translations and the quantum group $U_{q}(sl_2)$. The approach allows us to express the "mid" band spectrum of the model and the Bloch wave function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems. The zero mode wave functions are found explicitly in terms of $q$-deformed classical orthogonal polynomials.In this paper we present solution for the isotropic problem. We also present a class of solvable quasiperiodic equations related to $U_{q}(sl_2)$.Comment: 19 pages, Revte

Light scattering from disordered overlayers of metallic nanoparticles

We develop a theory for light scattering from a disordered layer of metal nanoparticles resting on a sample. Averaging over different disorder realizations is done by a coherent potential approximation. The calculational scheme takes into account effects of retardation, multipole excitations, and interactions with the sample. We apply the theory to a system similar to the one studied experimentally by Stuart and Hall [Phys. Rev. Lett. {\bf 80}, 5663 (1998)] who used a layered Si/SiO$_2$/Si sample. The calculated results agree rather well with the experimental ones. In particular we find conspicuous maxima in the scattering intensity at long wavelengths (much longer than those corresponding to plasmon resonances in the particles). We show that these maxima have their origin in interference phenomena in the layered sample.Comment: 19 pages, 12 figure