From Quantum to Classical
Du quantique au classique
22 – 26 April 2019

Scientific Committee
Comité scientifique

Frank Boyer (Université Toulouse III – Paul Sabatier)
Stéphane Nonnenmacher (Université Paris-Sud)
Julien Sabin (Université Paris-Sud)

Organizing Committee
Comité d’organisation

Stéphane Nonnenmacher (Université Paris-Sud)
Julien Sabin (Université Paris-Sud)

This topical school will be part of the “Etats de la Recherche” sessions. Its aim will be to introduce several aspects of the crossover between quantum mechanics (which describes microscopic particles) and effective dynamics closer to “classical physics” (Newton’s mechanics, statistical physics).

​This school will be composed of 6 lectures of 3 hours each, which we split into the following 3 themes :

– The semiclassical limit of quantum mechanics, and its applications to the spectral theory of Schrödinger type operators. One lecture will address the question of the phase space localization of quantum eigenmodes, in relation with the corresponding classical dynamical. A second lecture will instead apply semiclassical methods to understand fine properties of chaotic dynamics.

– The effective description of N-particle quantum systems, in the thermodynamic limit. The aim is to replace the microscopic description (N-body Hamiltonian operator) by an effective macroscopic theory, which takes the form of a nonlinear field theory describing a collective observable. The two lectures will study fermionic systems, among which an electron gas submitted to a strong magnetic field (Quantum Hall Effect).     
– A real quantum system always interacts with some “environment”, which is difficult to describe in detail. This interaction can be taken into account by an effective dynamics of the “small” system, different from the Schrödinger equation. This effective dynamics leads to interesting effects, like the decoherence of the quantum system, which brings the latter from a “pure” quantum state to a statistical mixture. Considering the interaction between two quantum systems leads to interesting concepts and methods, making up the quantum information theory ; for instance the phenomenon of entanglement between the two systems. One lecture will introduce these concepts and explain some of their applications.

Cette école thématique s’inscrira dans le cadre des “Etats de la Recherche” organisés par la SMF. Elle aura pour objectif d’introduire plusieurs aspects du passage entre la mécanique quantique (décrivant des particules au niveau microscopique), et une dynamique effective proche de la physique “classique” (équations de Newton, physique statistique).

Cette école sera composée de 6 cours de 3h, qui s’articuleront en trois thèmes.

— La limite semiclassique de la mécanique quantique, et ses applications à la théorie spectrale des opérateurs de type Schrödinger. Un des cours portera sur l’étude de la localisation dans l’espace des phases de modes propres quantiques, en lien avec la dynamique classique sous-jacente. Un second cours portera, inversement, sur les applications de méthodes semiclassiques à l’étude de systèmes dynamiques chaotiques.

— Les descriptions effectives de N particules quantiques en interaction, dans la limite thermodynamique (N grand).
On veut alors remplacer la description microscopique (opérateur de Schrödinger à N particules) par une théorie effective macroscopique, qui prend souvent la forme d’une équation de champ non-linéaire décrivant une observable collective de ces particules. Les deux cours porteront sur des systèmes de fermions, en particulier les gaz d’électrons placés dans un champ magnétique fort, qui donnent lieu à l’effet Hall quantique.

​— La théorie de l’information quantique permet de décrire l’interaction entre deux systèmes quantiques, par exemple le phénomène d’intrication entre les deux soussystèmes. Un cours fournira les concepts de base de l’information quantique. Si l’un des sous-systèmes est un gigantesque “environnement”, on s’attache plutôt à décrire l’évolution effective du “petit” système, qui diffère de l’évolution de Schrödinger. Cette dynamique effective permet de rendre compte du phénomène de décohérence du système quantique, qui fait passer d’un état pur à un mélange statistique.

Theme Quantum information

  • Guillaume Aubrun (Université Lyon 1) – Entanglement as a resource in quantum information theory

Guillaume Aubrun (Université Lyon 1) – Entanglement as a resource in quantum information theory​

We introduce the concept of quantum entanglement, fundamental in quantum physics. We review some basic properties of entangled states (pure and mixed), and explain how entanglement can be used as a resource in quantum information theory. This leads to various measures of entanglement (e.g. entanglement cost, distillable entanglement) to quantify how much entanglement a given system contains.

  • Yan Pautrat (Université Paris-Sud) – Effective dynamics of open quantum systems, decoherence and indirect measurements

Yan Pautrat (Université Paris-Sud) – Effective dynamics of open quantum systems, decoherence and indirect measurements

The lecture will describe the effective dynamics of small systems in interaction with an environment and their properties, in particular their asymptotic (in time) behavior. A connection will be given to the theory of indirect measurements and recent results on their behavior

Theme​ Semiclassics

  • Frédéric Faure (Université Grenoble Alpes) – Microlocal methods in chaotic dynamics: from classical to quantum

Frédéric Faure (Université Grenoble Alpes– Microlocal methods in chaotic dynamics: from classical to quantum

In the 1980’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (“Ruelle resonances”). Following this approach and use of microlocal analysis, we will consider the geodesic flow on a smooth strictly negative curvature Riemannian manifold M. 1) the flow is Anosov (i.e. uniformly hyperbolic, with sensitivity to initial conditions) 2) the generator of the geodesic flow, namely the vector field on S*M, has an intrinsic discrete spectrum called Ruelle resonances (for this we need to consider specific Anisotropic Sobolev spaces). 3) The Ruelle resonances are structured in vertical bands. 4) The first (right most) band that governs long time fluctuations of classical probabilities is the spectrum of an effective quantum wave equation (this may be surprising because there is no added quantization procedure). 5) We will discuss consequences for the zeros of dynamical zeta functions. Relations between the Ruelle spectrum and periodic orbits.
This shows that the problematic of classical chaos and quantum chaos are closely related. Part 2) is from a joint work with J. Sjöstrand. Parts 3) and 4) are from a joint work with Masato Tsujii.

  • Gabriel Rivière (Université de Nantes) – Semiclassical behaviour of quantum eigenstates

Gabriel Rivière (Université de Nantes) – Semiclassical behaviour of quantum eigenstates

Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.

Theme N-body quantum mechanics

  • Marcello Porta (University of Tübingen) – Mean field dynamics of many-body fermionic systems

Marcello Porta (University of Tübingen) – Mean field dynamics of many-body fermionic systems

Systems of interest in physics are typically formed by a huge number of particles, way too large to extract quantitative information from the fundamental evolution equation, the Schrödinger equation. However, in some scaling regimes, the many-body evolution of physical observables can be approximated by nonlinear effective equations, depending on much less degrees of freedom than the full Schrödinger dynamics. In this talk, I will consider the evolution of many-body fermionic systems, in the mean-field regime. I will discuss mathematical methods for the derivation of the time-dependent Hartree-Fock equation, a well-known example of nonlinear effective evolution equation for interacting fermions. The methods combine ideas from semiclassical analysis together with tools from second quantization. If time permits, I will report about the extension of these techniques to the derivation of the Vlasov equation, or to the computation of the ground state energy for trapped fermionic systems beyond the quasi-free approximation.

  • Nicolas Rougerie (CNRS/ Université Grenoble Alpes) – Mathematical topics around fractional quantization​

Nicolas Rougerie (CNRS/ Université Grenoble Alpes– Mathematical topics around fractional quantization

​The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles.
These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof. Some progress has been made on some of these, but lots remains to be done, and open problems will be mentioned.