EMAQS

Manipulating the evolution of atoms and molecules at the quantum level has been a goal from the very beginnings of the laser technology. However, designing laser pulses based on intuition alone did not succeed until the researchers understood that this problem should be attacked with the tools of control theory which greatly contributed to the first positive experimental results. Different applications followed with electromagnetic radiations used for High Harmonic Generation of high frequency lasers, Nuclear Magnetic Resonance applications in medicine, quantum gates design in next generation computers, etc. The field of manipulating molecular evolution is now an emerging technology that requires active input from researchers from different perspectives. In the mathematical formalization the coherent and conservative evolution is determined by the Schrödinger equation that involves the Hamiltonian of the system; in a more general formalism including decoherence and irreversible effects, the state is described by a density matrix operator and evolves according to the Lindblad-Kossakowski master equation. A first and crucial question concerning the manipulation of the quantum evolution by an external field is whether this is possible at all i.e. whether any possible state can be reached with a well-chosen control field; in technical terms this question is called the `controllability of the equation'. One may further refine this question depending on whether the controllability is investigated exactly or approximately, in finite time or asymptotically, in open loop or in closed loop (feedback stabilization)...The study of the control of equations modeling quantum systems is one of the goals of this project. The mere positive conclusion that a control exists does not indicate how to find it in practice. Therefore one needs to formulate numerical algorithms that find the control and are compatible with experimental constraints. Consequently an additional goal of the project is to design such algorithms for specific situations relevant for applications. To be efficient in practice these algorithms have also to take into account the uncertainties and errors and find ways to deal with them. Finally, an estimation (also called `inverse') problem can be formulated: suppose that one does not know the Hamiltonian but can instead measure, for several control fields some aggregate quantity depending on the wave function. How much information on the Hamiltonian can be recovered from such measurements and which are the most efficient procedures to realize this operation? It is well known that these 3 subjects (open loop control, feedback stabilization and estimation) are closely related; which gives additional consistency to our research project. A strong specificity of this proposal is the direct collaboration between physicists, chemists and applied mathematicians. This interaction aims at providing us challenging mathematical problems directly linked to applications. Moreover, the theoretical and numerical advances obtained could be experimentally tested, because the members of the project have direct access to two distinct experimental settings, one at the LKB laboratory (one of the leading physicists at LKB is in the ARMINES-CAS team) and one at Princeton University (group of Prof. H. Rabitz with whom the CEREMADE team has longstanding collaborations). Our goal is to produce mathematical results with an important experimental applicability; which justifies the study of infinite dimensional control systems, numerics, uncertainties, estimation and inversion. Finally, our team presents complementary skills for this purpose: classical and numerical analysis of PDEs, mathematical system theory, quantum physics. Our goal is to achieve the synthesis of these skills.