## Section: Partnerships and Cooperations

### National Initiatives

#### ANR Programme blanc international (BLAN) LODIQUAS 2012-2015

Participants : Philippe Chartier, Florian Méhats, François Castella, Mohammed Lemou.

The project, entitled "LODIQUAS" (for: Low DImensional QUANtum Systems), received fundings for 4 postdocs (48 months) and one pre-doc (36 months). The whole project involves the following researchers : Norbert Mauser (Vienna), Erich Gornik (Vienna), Mechthild Thalhammer (Innsbruck), Christoph Naegerl (Innsbruck), Jörg Schmiedmayer (Vienna), Hans-Peter Stimming (Vienna),Francis Nier (Rennes), Raymond El Hajj (Rennes), Claudia Negulescu (Toulouse), Fanny Delebecque (Toulouse), Stéphane Descombes (Nice), Christophe Besse (Lille).

The expected scientific and technological progress brought by the present project are as follows.

Quantum technology as the application of quantum effects in macroscopic devices has an increasing importance, not only for far future goals like the quantum computer, but already now or in the near future. The present project is mainly concerned with the mathematical and numerical analysis of these objects, in conjunction with experimental physicists. On the side of fermions quantum electronic structures like resonant tunnelling diodes show well studied non classical effects like a negative differential resistance that are exploited for novel devices. On the side of bosons the creation and manipulation of Bose Einstein Condensates (the first creation of BECs by Ketterle et al merited a Nobel prize) has become a standard technique that allows to study fundamental quantum concepts like matter-wave duality with increasingly large objects and advanced quantum effects like decoherence, thermalization, quantum chaos. In state-of-the-art experiments e.g. with ultracold atoms in optical lattices the bosonic or fermionic nature of quantum objects can change and it makes a lot of sense to treat the models in parallel in the development of mathematical methods. The experimental progress in these fields is spectacular, but the mathematical modelling and analysis as well as the numerical simulation are lagging behind. Low dimensional models are mostly introduced in a heuristic way and there is also a need for systematic derivations and comparison with the 3-d models. To close the gap is a main goal of this project that aims to deliver reliable tools and programme packages for the numerical simulation of different classes of quantum systems modelled by partial differential equation of NLS type. Virtually all participants have a strong track record of international collaboration, they grew up with the concept of the European Research Area where science knows no boundaries and scientists used to work in different countries, as it was the case in a pronounced way in mathematics and in quantum physics in the thirties of the last century. The Pre- and Post-Docs to be funded by this project will be trained in this spirit of mobility between scientific fields and between places.

This project gave rise to the following scientific achievements

**PhD students**

*Boris Pawilowski*, has been hired as a PhD student,
under the supervision of F. Nier and N. Mauser. His contract started
october 2012, and the PhD thesis was defended on December 2015.
His PhD subject is
"Mean field limit for discrete models and nonlinear discrete
Schrödinger equations".

**Postdocs**

*Loïc Le Treust* has been hired as a Postdoc,
under the supervision of F. Méhats (main) and N. Mauser. His contract
started October 2013, and it did last
two years, in Rennes and Vienna.

*Yong Zhang*, under contract in Vienna, has been invited for several one month periods in Rennes.
There are works in progress with F. Méhats and P. Chartier.

*Kristelle Roidot*, had a six months contract in Vienna, and this gave rise to
works with N. Mauser, C. Klein, J.-C. Saut, S. Descombes.

**Workshops**

**July 2012**, kick-off meeting of the LODIQUAS project, WPI, Vienna (one week, approx. 40 people, amongst which most of the participants of the project).

**February 2013**, WPI, Vienna, with a similar organization as the kick-off meeting.

**July 2013**, WPI, Vienna.
At the WPI for one week.
"Quantized Vortices in Superfluidity and
Superconductivity and Related Problems", organisers W. Bao, C. Bardos, Q.
Du, N. Mauser.

**September 2013**, WPI Vienna, "Modified dispersion for dispersive equations and
systems ", organisers R. Carles, Mauser, J.C. Saut.

**September 2013**, WPI Vienna, "Modified dispersion for dispersive equations and
systems ", organisers R. Carles, Mauser, J.C. Saut.

**October 2014**, WPI Vienna, "Blow-up and Dispersion in nonlinear Schrödinger and Wave equations",
organizers G. Lebeau, A. Jüngel,
O. Ivanovici, J.-C. Saut, H.-P. Stimming.

**December 2014**, Saint-Malo, "Lodiquas Meeting",
organisers F. Castella and P. Chartier.

**December 2015**, Dinard, "Joint Lodiquas and Ipso Meeting",

#### ANR MOONRISE: 2015-2019

Participants : Nicolas Crouseilles, Philippe Chartier, Florian Méhats, François Castella, Mohammed Lemou.

The project *Moonrise* submitted by F. Méhats has been funded by the ANR for 4 years, for the period 2015-2019. This project aims at exploring modeling, mathematical and numerical issues originating from the presence of high-oscillations in nonlinear PDEs from the physics of nanotechnologies (quantum transport) and from the physics of plasmas (magnetized transport in tokamaks). The partners of the project are the IRMAR (Rennes), the IMT (Toulouse) and the CEA Cadarache. In the IPSO team, F. Castella, P. Chartier, N. Crouseilles and M. Lemou are members of the project Moonrise.

#### IPL (FRATRES)

IPSO is associated to IPL FRATRES which started in june 2015. The aim of this project is to organize Inria teams activities which develop mathematical and numerical tools in magnetically confined nuclear fusion. The ambition is to prepare the next generation of numerical modeling methodologies able to use in an optimal way the processing capabilities of modern massively parallel architectures. This objective requires close collaboration between a) applied mathematicians and physicists that develop and study mathematical models of PDE; b) numerical analysts developing approximation schemes; c) specialists of algorithmics proposing solvers and libraries using the many levels of parallelism offered by the modern architecture and d) computer scientists. The project road map ambitions to contribute in close connection with National and European initiatives devoted to nuclear Fusion to the improvement and design of numerical simulation technologies applied to plasma physics and in particular to the ITER project for magnetic confinement fusion.