1,628 research outputs found
Inversion of Toeplitz operators, Levinson equations, and Gohberg-Krein factorization—A simple and unified approach for the rational case
AbstractThe problem of the inversion of the Toeplitz operator TΦ, associated with the operator-valued function Φ defined on the unit circle, is known to involve the associated Levinson system of equations and the Gohberg-Krein factorization of Φ. A simplified and self-contained approach, making clear the connections between these three problems, is presented in the case where Φ is matrix-valued and rational. The key idea consists in looking at the Levinson system of equations associated with Φ−1(z−1), rather than that associated with Φ(z). As a consequence, a new invertibility criterion for Toeplitz operators with rational matrix-valued symbols is derived
An electronic Mach-Zehnder interferometer in the Fractional Quantum Hall effect
We compute the interference pattern of a Mach-Zehnder interferometer
operating in the fractional quantum Hall effect. Our theoretical proposal is
inspired by a remarkable experiment on edge states in the Integer Quantum Hall
effect (IQHE). The Luttinger liquid model is solved via two independent
methods: refermionization at nu=1/2 and the Bethe Ansatz solution available for
Laughlin fractions. The current differs strongly from that of single electrons
in the strong backscattering regime. The Fano factor is periodic in the flux,
and it exhibits a sharp transition from sub-Poissonian (charge e/2) to
Poissonian (charge e) in the neighborhood of destructive interferences
Euclidean versus hyperbolic congestion in idealized versus experimental networks
This paper proposes a mathematical justification of the phenomenon of extreme
congestion at a very limited number of nodes in very large networks. It is
argued that this phenomenon occurs as a combination of the negative curvature
property of the network together with minimum length routing. More
specifically, it is shown that, in a large n-dimensional hyperbolic ball B of
radius R viewed as a roughly similar model of a Gromov hyperbolic network, the
proportion of traffic paths transiting through a small ball near the center is
independent of the radius R whereas, in a Euclidean ball, the same proportion
scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at
the center of the hyperbolic ball scales as the square of the volume, whereas
the same traffic load scales as the volume to the power (n+1)/n in the
Euclidean ball. This provides a theoretical justification of the experimental
exponent discrepancy observed by Narayan and Saniee between traffic loads in
Gromov-hyperbolic networks from the Rocketfuel data base and synthetic
Euclidean lattice networks. It is further conjectured that for networks that do
not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of
maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure
The Large Scale Curvature of Networks
Understanding key structural properties of large scale networks are crucial
for analyzing and optimizing their performance, and improving their reliability
and security. Here we show that these networks possess a previously unnoticed
feature, global curvature, which we argue has a major impact on core
congestion: the load at the core of a network with N nodes scales as N^2 as
compared to N^1.5 for a flat network. We substantiate this claim through
analysis of a collection of real data networks across the globe as measured and
documented by previous researchers.Comment: 4 pages, 5 figure
Localization and diffusion in Ising-type quantum networks
We investigate the effect of phase randomness in Ising-type quantum networks.
These networks model a large class of physical systems. They describe micro-
and nanostructures or arrays of optical elements such as beam splitters
(interferometers) or parameteric amplifiers. Most of these stuctures are
promising candidates for quantum information processing networks. We
demonstrate that such systems exhibit two very distinct types of behaviour. For
certain network configurations (parameters), they show quantum localization
similar to Anderson localization whereas classical stochastic behaviour is
observed in other cases. We relate these findings to the standard theory of
quantum localization.Comment: 12 page
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
Molecules in external fields: a semiclassical analysis
We undertake a semiclassical analysis of the spectral properties (modulations
of photoabsorption spectra, energy level statistics) of a simple Rydberg
molecule in static fields within the framework of Closed-Orbit/Periodic-Orbit
theories. We conclude that in addition to the usual classically allowed orbits
one must consider classically forbidden diffractive paths. Further, the
molecule brings in a new type of 'inelastic' diffractive trajectory, different
from the usual 'elastic' diffractive orbits encountered in previous studies of
atomic and analogous systems such as billiards with point-scatterers. The
relative importance of inelastic versus elastic diffraction is quantified by
merging the usual Closed Orbit theory framework with molecular quantum defect
theory.Comment: 4 pages, 3 figure
Nearest-neighbor distribution for singular billiards
The exact computation of the nearest-neighbor spacing distribution P(s) is
performed for a rectangular billiard with point-like scatterer inside for
periodic and Dirichlet boundary conditions and it is demonstrated that for
large s this function decreases exponentially. Together with the results of
[Bogomolny et al., Phys. Rev. E 63, 036206 (2001)] it proves that spectral
statistics of such systems is of intermediate type characterized by level
repulsion at small distances and exponential fall-off of the nearest-neighbor
distribution at large distances. The calculation of the n-th nearest-neighbor
spacing distribution and its asymptotics is performed as well for any boundary
conditions.Comment: 38 pages, 10 figure
Weak localization of light by cold atoms: the impact of quantum internal structure
Since the work of Anderson on localization, interference effects for the
propagation of a wave in the presence of disorder have been extensively
studied, as exemplified in coherent backscattering (CBS) of light. In the
multiple scattering of light by a disordered sample of thermal atoms,
interference effects are usually washed out by the fast atomic motion. This is
no longer true for cold atoms where CBS has recently been observed. However,
the internal structure of the atoms strongly influences the interference
properties. In this paper, we consider light scattering by an atomic dipole
transition with arbitrary degeneracy and study its impact on coherent
backscattering. We show that the interference contrast is strongly reduced.
Assuming a uniform statistical distribution over internal degrees of freedom,
we compute analytically the single and double scattering contributions to the
intensity in the weak localization regime. The so-called ladder and crossed
diagrams are generalized to the case of atoms and permit to calculate
enhancement factors and backscattering intensity profiles for polarized light
and any closed atomic dipole transition.Comment: 22 pages Revtex, 9 figures, to appear in PR
Robustness of energy landscape control to dephasing
As shown in previous work, in some cases closed quantum systems exhibit a non-conventional absence of trade-off between performance and robustness in the sense that controllers with the highest fidelity can also provide the best robustness to parameter uncertainty. As the dephasing induced by the interaction of the system with the environment guides the evolution to a more classically mixed state, it is worth investigating what effect the introduction of dephasing has on the relationship between performance and robustness. In this paper we analyze the robustness of the fidelity error, as measured by the logarithmic sensitivity function, to dephasing processes. We show that introduction of dephasing as a perturbation to the nominal unitary dynamics requires a modification of the log-sensitivity formulation used to measure robustness about an uncertain parameter with nonzero nominal value used in previous work. We consider controllers optimized for a number of target objectives ranging from fidelity under coherent evolution to fidelity under dephasing dynamics to determine the extent to which optimizing for a specific regime has desirable effects in terms of robustness. Our analysis is based on two independent computations of the log-sensitivity: a statistical Monte Carlo approach and an analytic calculation. We show that despite the different log-sensitivity calculations employed in this study, both demonstrate that the log-sensitivity of the fidelity error to dephasing results in a conventional trade-off between performance and robustness
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