Modeling and Numerical Simulation of Low Dimensional Quantum Systems

This project is dedicated to the mathematical modelling and numerical simulation of stationary and time dependent NonLinear Schrödinger equations (NLS) for quantum mechanical systems that are confined in one or more space dimensions. Such systems, like the 2DEG (2 dimensional electron gas) or 1DBEC (one dimensional Bose Einstein Condensates) are interesting both from the theoretical point of view and for applications like quantum semi- conductors or atom-chips. A recent exciting application is graphene, a two dimensional state of carbon with very interesting properties. The first experimental realization has recently been awarded the Nobel prize. This project is one of the first systematic mathematical approaches to graphene. The dimension reduction can result from a (e.g. spherical) symmetry or a translational invariance in one or two space dimensions or from a confinement of the quantum particles in one, two ("quantum wires") or even 3 space dimensions ("quantum dots"). The confinement can be described by adding in the Hamilton operator a confining potential with a small parameter, e.g. an anisotropic harmonic oscillator potential or homogeneous Dirichlet boundary conditions in some direction(s). The small parameter limit then yields the correct asymptotic model. Despite their widespread use, the mathematical derivation of Schrödinger equations describing such confined systems has been started only recently, with important contributions from the participants of this project. We bring together French and Austrian applied mathematicians working both on the rigorous justification and mathematical analysis of such low dimensional models and their numerical methods and simulations. We add some of the top Austrian physicists grouped by the WPI who indeed use such NLS models and simulations for state of the art experiments with such fermionic and bosonic systems. No experiments are funded by this project, we aim at recalibrating modelling and numerical methods in direct dialogue and comparison of experiment and computer simulation. The funding of this project will mainly finance Postdocs (2 x 2 years) and PhD student(s) (2 x 1.5 years) as well as "travel money" (research visits, visiting experts). The project coordinator and the WPI have an excellent experience in leading international interdisciplinary projects on PDEs with application in physics, like the large European network HYKE or the Marie Curie training multi-site DEASE. The WPI has very strong scientific links with France, it even carries an UMI of the CNRS in Vienna, the "Institut CNRS Pauli". The IRMAR is one of the larger French mathematics institutes, with a strong applied math section well experienced in grants like ANR projects. The more senior French participants, including the French coordinator, have been participating in the WPI coordinated projects like HYKE, the scientific collaboration is also documented by joint French-Austrian publications and PhD theses in co-tutelle and post-doctoral training which are one of the deliveries of this project. Also, this project will develop novel simulation tools, e.g. a "programme package" for the numerical simulation of general NLS in 1-, 2- and 3-d for the general use in the community. Such simulation tools are very valuable both for experiments in fundamental research in quantum physics (e.g. using BECs of ultracold atoms) and for the development of quantum electronic devices (e.g. resonant tunnelling diodes) - the computer simulation can also significantly reduce the cost for the build-up of the experimental apparatus or for building prototypes. The programme packages will be adapted to modern "super computers" (parallel machines), but it should be noted that only half of the progress in computer simulation stems from better computers and half comes from better modelling and better numerical algorithms. To pay a few additional young applied mathematicians is far more cost effective than to buy the most super computer.

The LODIQUAS project was dedicated to the mathematical modelling, analysis and numerical simulation of stationary and time dependent NonLinear Schrödinger equations (NLS) for quantum mechanical systems that are confined in one or more space dimensions. Such systems, like the 2DEG (2 dimensional electron gas) or 1DBEC (one dimensional Bose Einstein Condensates) are interesting both from the theoretical, mathematical point of view and for applications in quantum technology like quantum semi-conductors or atom-chips. The dimension reduction results from models where we have a (e.g. spherical) symmetry or a translational invariance in one or two space dimensions or from models where the quantum particles are confined in one or two (quantum wires) dimensions, described by a confining potential with a small parameter, e.g. a harmonic oscillator potential. The small parameter limit then yields the correct asymptotic model. French and Austrian applied mathematicians in Rennes, Paris and Vienna, Innsbruck worked together both on the rigorous justification and mathematical analysis of such low dimensional models and on numerical methods and simulations for fermionic and bosonic systems. The project yielded 31 publications in refereed journals, with 11 joint papers of the French and the Austrian partner. Among the key results we mention a joint paper in SIAM J. Appl. Math where the dimension reduction for the 3-d Schrödinger equation with Coulomb interaction is systematically done for two 2-d models and a 1-d model, with numerical simulations showing the difference of the 2-d model with translational invariance which gives the 2-d Poisson equation and the 2-d model with confinement which gives the square root Laplacian model of convolution with 1/x. In another joint work the Stroboscopic Averaging Method (SAM) was developed as a tool to solve the NLS in the highly oscillatory, i.e. semi-classical regime. Another joint work deals with the resolution of the highly oscillatory Schrödinger type equation by the multi-revolution composition for time splitting schemes. Also, in joint work new improved error estimates for time splitting methods for semi-classical Schrödinger equations were obtained. Among the numerical methods developed in the project we mention several joint papers on Non-uniform FFT (NUFFT) methods for numerics of low-dimensional NLS-type equations, in particular for Schrödinger-Poisson, nonlocal Gross-Pitaevskii equations, Davey-Stewartson and the case of systems with vector-valued potential and for a dipolar BEC models. The NUFFT was also combined with a Gaussian sum approximation for nonlocal potentials, thus increasing once more the accuracy/efficiency of the method. The project budget mainly (co-)funded 2 Postdocs and a PhD thesis in co-tutelle between Univ.Rennes and Univ. Wien and to (co-)fund 3 large project meetings and 4 workshops.