Brigitte Bidégaray-Fesquet

French National Centre for Scientific Research | CNRS · Laboratoire Jean Kuntzmann, Grenoble

We are interested in numerically solving a transitional model derived from the Bloch model. The Bloch equation describes the time evolution of the density matrix of a quantum system forced by an electromagnetic wave. In a high frequency and low amplitude regime, it asymptotically reduces to a non-stiff rate equation. As a middle ground, the transit...

In this paper, we define a splitting scheme for the [Formula: see text]-level Bloch model which makes use of exact numerical solutions of sub-equations. These exact solutions involve matrix exponentials which we want to avoid to calculate at each time-step. The resulting scheme is nonstandard and preserves qualitative properties of the Bloch equati...

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) scheme to adapt them to systems of ODEs. This leads to exact schemes in the linear case, and also improve the accuracy in the nonlinear case. In the Hamiltonian nonlinear case, it consists in adding correction terms to schemes derived by Mickens.

In this work, we discuss how to take into account electron–phonon interactions in a Bloch type model for the description of quantum dots. The model consists in coupling an equation on the density matrix with a set of equations on quantities called phonon-assisted densities, one for each phonon mode. After a description of the model, we discuss how...

Self‐triggered control is an improvement on event‐triggered control methods. Unlike the latter, self‐triggered control does not require monitoring the behavior of the system constantly. Instead, self‐triggered algorithms predict the events at which the control law has to be updated before they happen, relying on system model and past information. I...

We introduce an event-triggered algorithm for the stabilization of switched linear systems. We define a pseudo-Lyapunov function common to all the subsystems. The pseudo-Lyapunov function is compared, at every time instant, to an exponentially decreasing upper threshold. An event is generated when the two functions intersect, or when a new subsyste...

We extend to the N-level Bloch model the splitting scheme which use exact numerical solutions of sub-equations. These exact solutions involve matrix exponentials which we want to avoid to calculate at each time step. We use Newton interpolation to reduce the computational cost. The resulting scheme is nonstandard and preserves all qualitative prope...

Self-triggered control is an improvement on event-triggered control methods. Unlike the latter, self-triggered control does not require monitoring the behavior of the system constantly. Instead, self-triggered algorithms predict the events at which the control law has to be updated before they happen, relying on system model and past information.\\...

In this paper, we present splitting schemes for the two-level Bloch model. After proposing two ways to split the Bloch equation, we show that it is possible in each case to generate exact numerical solutions of the obtained sub-equations. These exact solutions involve matrix exponentials which can be expensive to compute. Here, for (Formula present...

Event-based control techniques are investigated for output reference tracking in the case of linear time-invariant systems. In event-based control, the controller remains at rest if the system is behaving according to some predefined conditions, the feedback loop being closed only when the system states violate these conditions. In this work a refe...

This chapter targets the synthesis and the design of filters using nonuniformly sampled data for use in asynchronous systems. Data is sampled with a level crossing technique. It presents several filtering strategies, namely Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filters. The synthesis of filters in the frequency domain ba...

We investigate the impact of additive noise and uncertainty on levels (offset) on the filtering of level-crossing sampled data. We therefore suppose that either the signal of the levels are affected by an error following a normal distribution with zero mean. The errors are then analyzed in terms of the standard deviation of the normal distribution....

Based on non-uniform sampling techniques and event-driven logic, signal processing is evolving to integrate new demands such as power consumption. As power is mainly connected to the processing activity and data volume, the level-crossing sampling scheme offers a simple way to reduce data volume and consequently processing activity. Nevertheless, t...

We study the dynamics of a one-dimensional lattice of nonlinearly coupled oscillators. The class of problems addressed include Newton's cradle with Hertzian contact interactions. The Cauchy problem is studied yielding lower bounds for the existence time. We then derive an asymptotic model for small amplitude solutions over large times. This allows...

Today, our digital society exchanges data ows as never it has been the case in the past. The amount of data is incredibly large and the future promises that not only human will exchange digital data but also technological equipment, robots, etc. We are close to open the door of the internet of things. This data orgy wastes a lot of energy and contr...

In this paper we first derive a Coulomb Hamiltonian for electron-electron interaction in quantum dots in the Heisenberg picture. Then we use this Hamiltonian to enhance a Bloch model, which happens to be nonlinear in the density matrix. The coupling with Maxwell equations when interaction with an electromagnetic field is also considered from the Ca...

Thanks to averaging processes and Gamma-convergence techniques, we are able to link a microscopic description of ferromagnetic materials based on spin lattices and their mesoscopic description in the static framework for the three fundamental contributions: exchange, magnetostatic and external field. The results are in accordance with the classical...

The reduction of the number of samples is a key issue in signal processing for mobile applications. We investigate the link between the smoothness properties of a signal and the number of samples that can be obtained through a level crossing sampling procedure. The algorithm is analyzed and an upper bound of the number of samples is obtained in the...

We present a numerical study of electromagnetic reflection and cavity modes of 1D-sub-wavelength rectangular metallic gratings exposed to TM-polarized light. Computations are made using the modal development. In particular we study the influence of the choice of boundary conditions on the metallic surfaces on the determination of modes, on specular...

We investigate the dynamics of a chain of oscillators coupled by fully-nonlinear interaction potentials. This class of models includes Newton's cradle with Hertzian contact interactions between neighbors. By means of multiple-scale analysis, we give a rigorous asymptotic description of small amplitude solutions over large times. The envelope equati...

We address the problem of quantifying the number of samples that can be obtained through a level crossing sampling procedure for applications to mobile systems. We specially investigate the link between the smoothness properties of the signal and the number of samples, both from a theoretical and a numerical point of view.

Master course

We propose a filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function thanks to non-uniform sampling. Following some previous work where we sampled non-uniformly already existing filter transfer functions, we propose now to design directly new filters in...

We propose a filtering technique which takes advantage of a specific non-uniform sampling scheme which allows the capture of a very low number of samples for both the signal and the filter transfer function. This approach leads to a summation formula which plays the same role as the discrete convolution for usual FIR filters. Here the formula is mu...

This article describes a new kind of processing chain based on a non-uniform sampling scheme provided by a level-crossing ADC. The chain implements IIR filters which process directly the non-uniform samples without resampling in a regular scheme. The non-uniformity in the sample times leads to choose a state representation for the filters. The stab...

The aim of this paper is to derive a raw Bloch model for the interaction of light with quantum boxes in the framework of a two-electron-species (conduction and valence) description. This requires a good understanding of the one-species case and of the treatment of level degeneracy. In contrast with some existing literature, we obtain a Liouville eq...

Today signal processing systems uniformly sample analog signals without taking advantage of their intrinsic properties. For instance, temperature, pressure, electrocardiograms, speech signals significantly vary only during short moments. The digitizing system does not take into account this specificity and furthermore is highly constrained by the S...

We present a numerical study of electromagnetic reflection and cavity modes of 1D-sub-wavelength rectangular metallic gratings exposed to TM-polarized light. Computations are made using the modal development. In particular we study the influence of the choice of boundary conditions on the metallic surfaces on the determination of modes, on specular...

We propose a FIR filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function thanks to non-uniform sampling. This approach leads to a summation formula which plays the role of the discrete convolution for usual FIR filters. Here the formula is much more comp...

We propose a FIR filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function thanks to non-uniform sampling. This approach leads to a summation formula which plays the role of the discrete convolution for usual FIR filters. Here the formula is much more comp...

The stability of five finite difference-time domain (FD-TD) schemes coupling Maxwell equations to Debye or Lorentz models have been analyzed in [1] (P.G. Petropoulos, "Stability and phase error analysis of FD-TD in dispersive dielectrics", IEEE Transactions on Antennas and Propagation, vol. 42, no. 1, pp. 62--69, 1994), where numerical evidence for...

This technical report yields detailed calculations of the paper [1] (B. Bidégaray-Fesquet, "Stability of FD-TD schemes for Maxwell-Debye and Maxwell-Lorentz equations", Technical report, LMC-IMAG, 2005) which have been however automated since (see http://ljk.imag.fr/membres/Brigitte.Bidegaray/NAUtil/). It deals with the stability analysis of variou...

The aim of this paper is first to review the derivation of a model describing the propagation of an optical wave in a photorefractive medium and to present various mathematical results on this model: Cauchy problem, solitary waves.

We present various mathematical results (Cauchy problem, solitary waves) for the Zozulya--Anderson model which describes the propagation of an optical wave through a photorefractive medium. This is a joint work with Jean-Claude~Saut.

The stability analysis of Finite Difference-Time Difference (FD-TD) schemes can be reduced via the von Neumann approach to the study of a sequence of polynomials with coefficients depending on the physical parameters of the medium and the numerical parameters. Such computations are very tedious in 3D since the simplest medium involves 9th degree po...

Ce rapport technique donne les détails de l'article [1]. Il s'agit de l'analyse de stabilité de différents schémas aux différences finies pour les équations de Maxwell–Debye et de Maxwell–Lorentz. Ce travail donne un prolongement systématique et rigoureuse à des travaux antérieurs de Petropoulos [5].

Bloch equations give a quantum description of the coupling between atoms and a driving electric force. It is commonly used in optics to describe the interaction of a laser beam with a sample of atoms. In this paper, we address the asymptotics of these equations for a high frequency electric field, in a weak coupling regime. The electric forcing is...

We consider Bloch equations which govern the evolution of the density matrix of an atom (or: a quantum system) with a discrete set of energy levels. The system is forced by a time dependent electric potential which varies on a fast scale and we address the long time evolution of the system. We show that the diagonal part of the density matrix is as...

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a K...

Master Course

In this article we derive new time discretizations for the numerical simulation of Maxwell-Bloch equations. These discretizations decouple the equations, thus leading to improved efficiency. This approach may be combined with the fulfilment of physical properties, such as positiveness properties, which are not accounted for by classical schemes. Ou...

This paper presents an iterative method for solving bound-constrained systems of nonlinear equations. It combines ideas from the classical trust-region Newton method for unconstrained nonlinear equations and the recent interior affine scaling approach ...

In this paper, we consider the nonlinear Schrödinger equation $u_t+i\Delta u -F(u)=0$ in two dimensions. We show, by an operator-theoretic proof, that the well-known Lie and Strang formulae (which are splitting methods) are approximations of the exact solution of order 1 and 2 in time.

La description semi-classique de l'interaction radiation-matière dans un contexte résonnant utilise le modèle de Maxwell-Bloch. Nous décrivons dans cet article quelques avancées récentes en modélisation tant physique que numérique qui permettent d'aborder la simulation d'une plus grande variété de phénomènes physiques. La maîtrise de ce modèle perm...

The Bloch equation models the evolution of the state of electrons in matter described by a Hamiltonian. To model more physical phenomena we have to introduce phenomenological relaxation terms. The introduction of these terms has to conserve some positiveness properties. The aim of this paper is to review possible relaxation models and to provide in...

In this article we implement dierent numerical schemes to simulate the Schrodinger- Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing s...

We present the Maxwell-Bloch equations that are a model for the semi-classical description of laser-matter interactions. After having discussed both models and their numerical coupling in a general context regarding relaxation terms and the number of energy levels, we give examples of simulations that show the capacities of our approach.

In this paper we study the local-in-time Cauchy problem for the Schrödinger–Debye equations. This model occurs in nonlinear optics and describes the non-resonant delayed interaction of an electromagnetic wave with a medium. We extend the study to nonphysical cases such as the three-dimensional case or more general nonlinearities.

We study the Cauchy problem for two systems of equations (Maxwell-Debye and Maxwell-Bloch) describing laser-matter interaction phenomena. We show that these problems are locally in time well-posed for initial data in dierent Sobolev spaces. In the case of Maxwell-Debye system, which contains some delay term, we study the limit of the solutions when...

We investigate a nonlinear set of coupled-wave equations describing the inertial regime of the strong Langmuir turbulence, which differs from the usual Zakharov equations by the inclusion in the first equation for E of a second time-derivative, multiplied by the parameter 1/ω2 that vanishes under the so-called time-envelope approximation . From the...

This article deals with the Cauchy problem for a nonlocal Zakharov equation. We will first recall the physical motivation (due to V.E. Zakharov) of this system. Then we will study the local Cauchy problem for certain initial data, and will identify the limit of the solutions when the ion velocity tends to infinity.

We construct invariant measures for Hamiltonian systems such as the nonlinear Schrödinger equation or the wave equation in order to prove Poisson's recurrence. The particular case of schemes (finite dimensional spaces) is also treated in order to explain the recurrence phenomenon which is observed during numerical simulations.

Cette thèse comporte trois parties traitant différents aspects de l'étude des équations d'ondes dispersives. Les modèles étudiés ont tous une origine physique.La première partie est consacrée à l'étude du problème de Cauchy pour un système de Zakharov non local. On effectue également l'étude de la limite lorsque la vitesse de la lumière tend vers l...

On étudie le problème de Cauchy associé à deux systèmes d'équations (Maxwell-Debye et Maxwell-Bloch) décrivant des phénomènes d'interaction laser-matière. On montre que ces problèmes sont bien posés localement en temps pour des données initiales appartenant à différents espaces de Sobolev. Dans le cas du système de Maxwell-Debye, qui comporte un te...

This article deals with the Cauchy problem for a nonlocal Zakharov equation. We will first recall the physical motivation (due to V.E. Zakharov) of this system. Then we will study the local Cauchy problem for certain initial data, and will identify the limit of the solutions when the ion velocity tends to infinity.