5th Meeting of the GDR Quantum Dynamics

Invited Speakers

Titles and abstracts

Jean-François Arnoldi: Ruelle resonances at the semiclassical limit

Since the work of Ruelle in the late '70 it is known that quantitative properties (e.g. rates of mixing and so forth) of certain chaotic dynamical systems can be deduced through the spectral study of a linear operator associated to the dynamics called the transfer operator (In adapted functional spaces). For a certain class of so-called partially-expanding systems (more precisely SU(2) extensions of expanding maps) the transfer operator decomposes into a direct sum. The terms of this sum can be interpreted as a semiclassical family of operators and exhibits a spectral behaviour similar to the one of quantum propagators studied in the context of quantum chaos by Sjostrand, Zworski and Nonnenmacher. In this talk we shall expose some results obtained concerning the spectrum of these operators when restricted to adapted spaces of distributions : an asymptotic spectral gap and and a fractal Weyl upper bound.

Stéphane Attal: Classical actions of quantum environments and obtuse random variables

We are interested in the action of a quantum bath on a quantum system, under the model of repeated quantum interactions. We focus on those dynamics where the quantum bath acts as a classical noise. This case appears to be intimately related to particular complex random variables, the obtuse random variables, with a very rich underlying algebraic structure connected to the behavior of the associated noises.

Nils Berglund: Irreversible diffusions and non-selfadjoint operators: Results and open problems

The spectral theory of reversible diffusions in the small-noise regime is by now well understood. The generator of their Markov semigroup is equivalent to a Schrödinger operator, and its exponentially small eigenvalues have been characterised by a number of different methods, including large deviations, semiclassical analysis and potential theory. The study of the irreversible case, which involves a non-selfadjoint generator, is substantially harder. I will discuss an approach based on Laplace transforms of hitting times for discrete-time, continuous-space Markov chains, describing random Poincaré maps. This approach yields information on exponentially small eigenvalues of the generator, and on mean transition times between attractors. As an illustration, I will discuss in detail what happens when a two-dimensional diffusion leaves a region surrounded by an unstable periodic orbit.

Based on joint work with Barbara Gentz (Bielefeld).

Jean-Marc Bouclet: Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

We consider the Laplace-Beltrami operator on a non compact manifold with asymptotically hyperbolic ends and show that the bottom of its continuous spectrum cannot be an eigenvalue. We will present in detail the geometric framework and the motivation of this problem for scattering theory. We will also explain the main steps of the proof and in particular how it follows from Carleman estimates proved by R. Mazzeo for such geometries.

Sébastien Breteaux: Minimisation of the energy of an electron interacting with a photon field for quasifree states

Dominique Delande: Many-body Anderson localization in cold atomic gases

A quantum particle placed in a disordered potential may be localized thanks to quantum interferences between the various the multiply scattered paths. This phenomenon - known as strong or Anderson localization - is a pure one-body effect. How particle-particle interaction affects Anderson localization is a difficult problem, far from being solved, with lots of open issues and contradictory predictions. Ultra-cold atomic gases can be prepared experimentally in well controlled disordered configurations and/or with tunable atom-atom interaction, opening the way to experimental answers. I will discuss, using quasi-exact numerical simulations, how typical few/many-body one-dimensional excitations, bright and dark solitons, behave in the presence of disorder.

Maxime Gazeau: Analysis of light propagation in birefringent optical fibers

The study of light propagation in monomode optical fibers requires to take care of various complex phenomena such as the polarization mode dispersion (PMD) and the Kerr effect. It has been proved that the slowly varying envelope of the electric field is well described by a coupled non linear schrödinger equation with random coefficients called the Manakov PMD equation. The particularity of this equation is the presence of various length scales whose ratio is given by a small parameter. In this talk I will introduce the asymptotic dynamic as this parameter goes to zero and numerical schemes for this limit equation. I will also present some numerical simulations of the PMD and in particular variance reduction methods to estimate the probability density function of the Differential Group Delay (DGD).

Nicolas Godet: Blow-up for the nonlinear Schrodinger equation on manifolds

We study the quintic nonlinear Schrodinger equation posed on a manifold M. If M is the real line, and near the ground state, the blow up theory for this equation is well-known. In this setting, essentially only one regime is possible; it is characterized by an almost self-similar blow-up rate and is known to be stable by perturbation of the initial data. In this talk, we show a property of geometric stability of this regime in the sense that we prove that it persists in noneuclidean geometries but with a radial symmetry assumption and for a supercritical equation.

Mathieu Lewin: The largest number of electrons that a nucleus can keep for a long time

Christian Maes: Dynamical activity in action

Dynamical ensembles for driven systems are determined by both thermodynamic (entropic) and kinetic factors. The kinetic factor is associated to dynamical activity in the action Lagrangian on path-space, and starts to play a major role when well away from equilibrium. We show how dynamical fluctuation theory gives rise to a natural extension of the Clausius heat theorem.

The dynamical activity is also responsible for the frenetic contribution in nonequilibrium response theory. Finally, we indicate how the dynamical activity complements the entropy production in determining the direction of the ratchet current. The mathematical framework is mostly that of Markov jump processes, so that at least some regime of quantum dynamics is involved.

Jens Marklof: Free path lengths in quasicrystals

I will report on current joint work with A. Strömbergsson (Uppsala), in which we study the distribution of the free path length in the Lorentz gas, where the spherical scatterers are located at the vertices of a quasicrystal. In the limit of small scatterer density (Boltzmann-Grad limit), we establish the existence of a limit distribution, and show that it is universal within certain families of quasicrystals. Previous results on kinetic transport in the Lorentz gas were limited to random or periodic scatterer configurations.

Grégoire Misguich: Numerical simulations of the non-equilibrium dynamics in the XXZ spin chain

We investigate the dynamics of an interacting spin-1/2 chain ("Ising-Heisenberg" Hamiltonian, also called "XXZ" model) which is prepared in an spatially inhomogeneous initial state.

This initial state is constructed to have different values of the magnetization on the left and right halves of the chain. We study this problem numerically using the Time-Evolving Block Decimation (TEBD) algorithm, which is based on a matrix-product representation of the wave-function. The dynamics shows two magnetization fronts which propagate to the left and to the right and we focus on the steady-state region which develops at long times in the center of the chain and which carries some spin current.

Nicolas Popoff: Spectrum of the magnetic Laplacian in 3D-domains with edges

In
application, we get an asymptotics in the semi-classical limit for the
first eigenvalue of the magnetic laplacian on a bounded 3D-domain with a
curved edge. We compare this asymptotics with known results for the
regular case. We also discuss asymptotic behaviour of the first
eigenvalue for more general corner domains.

Christoph Schenke: Superfluidity and macroscopic superpositions of Tonks-Girardeau bosons stirred on a 1D ring

Abstract

Benjamin Schlein: Mean field and Gross-Pitaevskii limits of quantum dynamics (Participation cancelled)

The derivation of effective evolution equations is a central question in non-equilibrium statistical mechanics. It turns out that, in the mean field limit, the many body quantum evolution can be approximated by the nonlinear Hartree equation. We will describe how optimal estimates on the rate of the convergence and a central limit theorem for the fluctuations around the Hartree dynamics can be obtained making use of so called coherent states. The Gross-Pitaevskii regime is a very singular mean field limit, relevant for the description of the dynamics of Bose-Einstein condensates. In this case, we will show how the convergence towards the limiting dynamics can be established by correcting the coherent states with appropriate Bogoliubov transformations, describing the correlation structure developed by the many body evolution.

Nikolaj Veniaminov: Thermodynamic limit for a system of interacting fermions in a random medium. Pieces one-dimensional model

In this talk, we study a particular one-dimensional random model for a system of interacting fermions. One is interested in the ground state wavefunction as well as in the associated ground state energy of the system in the thermodynamic limit for a small particle density. We quantify the influence of interactions on the ground state by estimating the difference between the ground state energy with and without interactions. We describe next the difference between the wavefunctions in interacting and noninteracting cases in terms of their functional structure. Finally, we describe the one- and two-particle density matrices of the ground state.

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- Jean-François Arnoldi (Institut Fourier, Grenoble)
- Stéphane Attal (ICJ, Lyon)
- Nils Berglund (MAPMO, Orléans)

- Jean-Marc Bouclet (IMT, Toulouse)

- Sébastien Breteaux (Technical University of Braunschweig)
- Dominique Delande (Laboratoire Kastler Brossel, Paris 6)

- Maxime Gazeau (Laboratoire Paul Painlevé, Lille 1)
- Nicolas Godet (Université de Nice)

- Mathieu Lewin (CNRS, Université de Cergy Pontoise)

- Christian Maes (KU Leuven)

- Jens Marklof (School of mathematics, Bristol)
- Grégoire Misguich (IPHT, CEA Saclay)

- Nicolas Popoff (IRMAR, Rennes)
- Christoph Schenke (Genève)

- Benjamin Schlein (Hausdorff Center for Mathematics, Bonn)
- Nikolaj Veniaminov (CEREMADE, Paris Dauphine)

Titles and abstracts

Jean-François Arnoldi: Ruelle resonances at the semiclassical limit

Since the work of Ruelle in the late '70 it is known that quantitative properties (e.g. rates of mixing and so forth) of certain chaotic dynamical systems can be deduced through the spectral study of a linear operator associated to the dynamics called the transfer operator (In adapted functional spaces). For a certain class of so-called partially-expanding systems (more precisely SU(2) extensions of expanding maps) the transfer operator decomposes into a direct sum. The terms of this sum can be interpreted as a semiclassical family of operators and exhibits a spectral behaviour similar to the one of quantum propagators studied in the context of quantum chaos by Sjostrand, Zworski and Nonnenmacher. In this talk we shall expose some results obtained concerning the spectrum of these operators when restricted to adapted spaces of distributions : an asymptotic spectral gap and and a fractal Weyl upper bound.

Stéphane Attal: Classical actions of quantum environments and obtuse random variables

We are interested in the action of a quantum bath on a quantum system, under the model of repeated quantum interactions. We focus on those dynamics where the quantum bath acts as a classical noise. This case appears to be intimately related to particular complex random variables, the obtuse random variables, with a very rich underlying algebraic structure connected to the behavior of the associated noises.

Nils Berglund: Irreversible diffusions and non-selfadjoint operators: Results and open problems

The spectral theory of reversible diffusions in the small-noise regime is by now well understood. The generator of their Markov semigroup is equivalent to a Schrödinger operator, and its exponentially small eigenvalues have been characterised by a number of different methods, including large deviations, semiclassical analysis and potential theory. The study of the irreversible case, which involves a non-selfadjoint generator, is substantially harder. I will discuss an approach based on Laplace transforms of hitting times for discrete-time, continuous-space Markov chains, describing random Poincaré maps. This approach yields information on exponentially small eigenvalues of the generator, and on mean transition times between attractors. As an illustration, I will discuss in detail what happens when a two-dimensional diffusion leaves a region surrounded by an unstable periodic orbit.

Based on joint work with Barbara Gentz (Bielefeld).

Jean-Marc Bouclet: Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

We consider the Laplace-Beltrami operator on a non compact manifold with asymptotically hyperbolic ends and show that the bottom of its continuous spectrum cannot be an eigenvalue. We will present in detail the geometric framework and the motivation of this problem for scattering theory. We will also explain the main steps of the proof and in particular how it follows from Carleman estimates proved by R. Mazzeo for such geometries.

Sébastien Breteaux: Minimisation of the energy of an electron interacting with a photon field for quasifree states

We
introduce the non-relativistic Pauli-Fierz model for an electron
interacting with a photon field. To obtain an upper bound on the ground
state energy a good strategy is to compute the minimum of the
energy on a subclass of states where the computations are simpler.
We introduce two such classes of states, coherent states and quasifree
states. We present the energy functionals in terms of parameters
describing the coherent/quasifree states. We prove under some hypotheses:

- the existence and uniqueness of a minimizer for these functionals,

- how to get an expansion of the energy at this minimizer in terms of the coupling constant appearing in the Pauli-Fierz model and of the total momentum.

- the existence and uniqueness of a minimizer for these functionals,

- how to get an expansion of the energy at this minimizer in terms of the coupling constant appearing in the Pauli-Fierz model and of the total momentum.

Dominique Delande: Many-body Anderson localization in cold atomic gases

A quantum particle placed in a disordered potential may be localized thanks to quantum interferences between the various the multiply scattered paths. This phenomenon - known as strong or Anderson localization - is a pure one-body effect. How particle-particle interaction affects Anderson localization is a difficult problem, far from being solved, with lots of open issues and contradictory predictions. Ultra-cold atomic gases can be prepared experimentally in well controlled disordered configurations and/or with tunable atom-atom interaction, opening the way to experimental answers. I will discuss, using quasi-exact numerical simulations, how typical few/many-body one-dimensional excitations, bright and dark solitons, behave in the presence of disorder.

Maxime Gazeau: Analysis of light propagation in birefringent optical fibers

The study of light propagation in monomode optical fibers requires to take care of various complex phenomena such as the polarization mode dispersion (PMD) and the Kerr effect. It has been proved that the slowly varying envelope of the electric field is well described by a coupled non linear schrödinger equation with random coefficients called the Manakov PMD equation. The particularity of this equation is the presence of various length scales whose ratio is given by a small parameter. In this talk I will introduce the asymptotic dynamic as this parameter goes to zero and numerical schemes for this limit equation. I will also present some numerical simulations of the PMD and in particular variance reduction methods to estimate the probability density function of the Differential Group Delay (DGD).

Nicolas Godet: Blow-up for the nonlinear Schrodinger equation on manifolds

We study the quintic nonlinear Schrodinger equation posed on a manifold M. If M is the real line, and near the ground state, the blow up theory for this equation is well-known. In this setting, essentially only one regime is possible; it is characterized by an almost self-similar blow-up rate and is known to be stable by perturbation of the initial data. In this talk, we show a property of geometric stability of this regime in the sense that we prove that it persists in noneuclidean geometries but with a radial symmetry assumption and for a supercritical equation.

Mathieu Lewin: The largest number of electrons that a nucleus can keep for a long time

Christian Maes: Dynamical activity in action

Dynamical ensembles for driven systems are determined by both thermodynamic (entropic) and kinetic factors. The kinetic factor is associated to dynamical activity in the action Lagrangian on path-space, and starts to play a major role when well away from equilibrium. We show how dynamical fluctuation theory gives rise to a natural extension of the Clausius heat theorem.

The dynamical activity is also responsible for the frenetic contribution in nonequilibrium response theory. Finally, we indicate how the dynamical activity complements the entropy production in determining the direction of the ratchet current. The mathematical framework is mostly that of Markov jump processes, so that at least some regime of quantum dynamics is involved.

Jens Marklof: Free path lengths in quasicrystals

I will report on current joint work with A. Strömbergsson (Uppsala), in which we study the distribution of the free path length in the Lorentz gas, where the spherical scatterers are located at the vertices of a quasicrystal. In the limit of small scatterer density (Boltzmann-Grad limit), we establish the existence of a limit distribution, and show that it is universal within certain families of quasicrystals. Previous results on kinetic transport in the Lorentz gas were limited to random or periodic scatterer configurations.

Grégoire Misguich: Numerical simulations of the non-equilibrium dynamics in the XXZ spin chain

We investigate the dynamics of an interacting spin-1/2 chain ("Ising-Heisenberg" Hamiltonian, also called "XXZ" model) which is prepared in an spatially inhomogeneous initial state.

This initial state is constructed to have different values of the magnetization on the left and right halves of the chain. We study this problem numerically using the Time-Evolving Block Decimation (TEBD) algorithm, which is based on a matrix-product representation of the wave-function. The dynamics shows two magnetization fronts which propagate to the left and to the right and we focus on the steady-state region which develops at long times in the center of the chain and which carries some spin current.

Nicolas Popoff: Spectrum of the magnetic Laplacian in 3D-domains with edges

In this talk we are interested in the spectrum of the Schrödinger operator with magnetic field on a domain
with edges in the semi-classical limit. We are led to study a model
magnetic Laplacian on infinite wedges. The problem reduces to the
analysis of a one-parameter family of 2D-magnetic Schrödinger operators
with an electric potential on an infinite sector. The behavior of this
family depends on the orientation of the magnetic field. We study what
happens when the opening angle of the sector is small.

Christoph Schenke: Superfluidity and macroscopic superpositions of Tonks-Girardeau bosons stirred on a 1D ring

Abstract

Benjamin Schlein: Mean field and Gross-Pitaevskii limits of quantum dynamics (Participation cancelled)

The derivation of effective evolution equations is a central question in non-equilibrium statistical mechanics. It turns out that, in the mean field limit, the many body quantum evolution can be approximated by the nonlinear Hartree equation. We will describe how optimal estimates on the rate of the convergence and a central limit theorem for the fluctuations around the Hartree dynamics can be obtained making use of so called coherent states. The Gross-Pitaevskii regime is a very singular mean field limit, relevant for the description of the dynamics of Bose-Einstein condensates. In this case, we will show how the convergence towards the limiting dynamics can be established by correcting the coherent states with appropriate Bogoliubov transformations, describing the correlation structure developed by the many body evolution.

Nikolaj Veniaminov: Thermodynamic limit for a system of interacting fermions in a random medium. Pieces one-dimensional model

In this talk, we study a particular one-dimensional random model for a system of interacting fermions. One is interested in the ground state wavefunction as well as in the associated ground state energy of the system in the thermodynamic limit for a small particle density. We quantify the influence of interactions on the ground state by estimating the difference between the ground state energy with and without interactions. We describe next the difference between the wavefunctions in interacting and noninteracting cases in terms of their functional structure. Finally, we describe the one- and two-particle density matrices of the ground state.

Back to the conference homepage