Olivier Pinaud

Transient simulations of a resonant tunneling diode

Stationary and transient simulations of a resonant tunneling diode in the ballistic regime are presented. The simulated model consists in a set of Schrodinger equations for the wave functions coupled to the Poisson equation for the electrostatic interaction. The Schrodinger equations are applied with open boundary conditions that model continuous injection of electrons from reservoirs. Automatic resonance detection enables reduction of the number of Schrodinger equations to be solved. A Gummel type scheme is used to treat the Schrodinger–Poisson coupling in order to accelerate the convergence. Stationary I–V characteristics are computed and the transient regime between two stationary states is simulated.

Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation

A numerical scheme for the one-dimensional stationary Schrodinger-Poisson model is described. The scheme is used to simulate a resonant tunneling diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discretization of the energy variable, which is a parameter for the Schrodinger equation, is improved by approaching the wavefunctions in the double barrier region by its projection on the resonant states (following the work of Presilla-Sjostrand and Jona-Lasinio [On Schrodinger equations with concentrated non-linearities, Ann. Phys. 240 (1995) 1-21]).

Kinetic Models for Imaging in Random Media

We derive kinetic models for the correlations and the energy densities of wave fields propagating in random media. These models take the form of radiative transfer and diffusion equations. We use these macroscopic models to address the detection and imaging of small objects buried in highly heterogeneous media. More specifically, we quantify the influence of small objects on (i) the energy density measured at an array of detectors and (ii) the correlation between the wave field measured in the absence of the object and the wave field measured in the presence of the object. We analyze the advantages and disadvantages of such measurements as a function of the level of disorder in the random media. Numerical simulations verify the theoretical predictions.

IMAGING USING TRANSPORT MODELS FOR WAVE–WAVE CORRELATIONS

We consider the imaging of objects buried in unknown heterogeneous media. The medium is probed by using classical (e.g. acoustic or electromagnetic) waves. When heterogeneities in the medium become too strong, inversion methodologies based on a microscopic description of wave propagation (e.g. a wave equation or Maxwells equations) become strongly dependent on the unknown details of the heterogeneous medium. In some situations, it is preferable to use a macroscopic model for a quantity that is quadratic in the wave fields. Here, such macroscopic models take the form of radiative transfer equations also referred to as transport equations. They can model either the energy density of the propagating wave fields or more generally the correlation of two wave fields propagating in possibly different media. In particular, we consider the correlation of the two fields propagating in the heterogeneous medium when the inclusion is absent and present, respectively. We present theoretical and numerical results showing that reconstructions based on this correlation are more accurate than reconstructions based on measurements of the energy density.

Dynamics of Wave Scintillation in Random Media

This paper concerns the asymptotic structure of the scintillation function in the simplified setting of wave propagation modeled by an Itô–Schrödinger equation. We show that the size of the scintillation function crucially depends on the smoothness of the initial conditions for the wave equation and on the size of the “array of detectors” where the wave fields are measured. In many practical settings, we show that the estimates are optimal and devise an equation for the appropriately rescaled scintillation function. The estimates are based on a careful analysis of Wigner transforms and of linear kinetic equations involving oscillatory integrals.

Adiabatic approximation of the Schrödinger–Poisson system with a partial confinement: The stationary case

Asymptotic quantum transport models of a two-dimensional gas are presented. The models are the stationary versions of those introduced in a previous paper by Ben Abdallah, Mehats, Pinaud. The starting point is a singular perturbation of the three-dimensional stationary Schrodinger–Poisson system posed on bounded domain. The electron injection in the device is modeled thanks to open boundary conditions. Under a small density assumption, the asymptotics lead to a full two-dimensional first-order approximation of the initial model. An intermediate model, called the “2.5D adiabatic model” in Ben Abdallah, Mehats, Pinaud is then introduced. It shares the same structure as the limit but is shown to be a second-order approximation of the three-dimensional model.

Mean-field Theory and Synchronization in Random Recurrent Neural Networks

In this paper, we first present a new mathematical approach, based on large deviation techniques, for the study of a large random recurrent neural network with discrete time dynamics. In particular, we state a mean field property and a law of large numbers, in the most general case of random models with sparse connections and several populations. Our results are supported by rigorous proofs. Then, we focus our interest on large size dynamics, in the case of a model with excitatory and inhibitory populations. The study of the mean field system and of the divergence of individual trajectories allows to define different dynamical regimes in the macroscopic parameters space, which include chaos and collective synchronization phenomenons. At last, we look at the behavior of a particular finite-size system submitted to gaussian static inputs. The system adapts its dynamics to the input signal, and spontaneously produces dynamical transitions from asynchronous to synchronous behaviors, which correspond to the crossing of a bifurcation line in the macroscopic parameters space.

Absorbing layers for the Dirac equation

This work is devoted to the construction of perfectly matched layers (PML) for the Dirac equation, that not only arises in relativistic quantum mechanics but also in the dynamics of electrons in graphene or in topological insulators. While the resulting equations are stable at the continuous level, some care is necessary in order to obtain a stable scheme at the discrete level. This is related to the so-called fermion doubling problem. For this matter, we consider the numerical scheme introduced by Hammer et al. 19], and combine it with the discretized PML equations. We state some arguments for the stability of the resulting scheme, and perform simulations in two dimensions. The perfectly matched layers are shown to exhibit, in various configurations, superior absorption than the absorbing potential method and the so-called transport-like boundary conditions.

Single scattering estimates for the scintillation function of waves in random media

The energy density of high frequency waves propagating in highly oscillatory random media is well approximated by solutions of deterministic kinetic models. The scintillation function determines the statistical instability of the kinetic solution. This paper analyzes the single scattering term in the scintillation function. This is the term of the scintillation function that is linear in the power spectrum of the random fluctuations. We show that the structure of the scintillation function is already quite complicated in this simplified setting. It strongly depends on the singularity of the initial conditions for the wave field and on the correlation properties of the random medium. We obtain limiting expressions for the scintillation function as the correlation length of the random medium tends to zero.

Hypoelliptic estimates in radiative transfer

ABSTRACT We derive the hypoelliptic estimates for a kinetic equation of the form where d ≥ 1, β ≥0, 𝕊d is the unit sphere in ℝd+1 and Δd is the Laplace-Beltrami operator on 𝕊d. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable k ∈ 𝕊d, we obtain estimates for the fractional derivatives of f in the (t, x) variables. Our proof follows the method of Bouchut based on the regularization of the momentum variable and on averaging lemmas on the sphere.