QUAntum COntrol: PDE systems and MRI applications – QUACO

The goal of quantum control is to design efficient population transfers between quantum states. This task is crucial in atomic and molecular physics, with applications ranging from photochemistry to quantum information, and has attracted increasing attention among quantum physicists, chemists, computer scientists and control theorists alike. The research challenges of quantum control are extremely varied, being of theoretical, algorithmic, and experimental nature. Our project is part of the already broad task of exploring the dynamical properties of controlled quantum systems, with a particular attention to control developments of Magnetic Resonance Imaging (MRI). A quantum system may be controlled by exciting it with one or several external fields, such as magnetic or electric fields. The goal of quantum control theory is to adapt the tools originally developed by control theory to the specific features of the quantum realm. Some of these features are the conservation of the norm (coming from the probabilistic nature of wave functions) for closed systems, the notion of density matrix, the effects of measure and decoherence on quantum evolution. These features affect the design of controls : both feedback laws and control signals that are too large or last for too long, for instance, can lead to a falsification of the model describing the system. Many control strategies have been proposed and implemented, both on numerical simulations and on physical systems, but there is still a large gap to fill before getting a complete picture of the control properties of quantum systems. Our project aims chiefly at contributing to quantum control theory in two directions :
-) improving the comprehension of the dynamical properties of controlled quantum systems evolving in infinite-dimensional state spaces, and in particular characterize their controllability as finely as possible ;
-) improve the efficiency of control algorithms for MRI and develop qualitative optimal control tools for the simultaneous control of spin systems.

In the last few years the members of the project developed geometric control techniques both for the qualitative characterization of optimal control laws in quantum control and for the controllability analysis of a wide range of quantum models characterized by discrete spectrum Hamiltonians defined on infinite-dimensional spaces. These original contributions will be the starting point for further extensions towards physically relevant new challenges such as minimal – time estimation, controllability of open systems, optimal control in parallel transmission… The scientific project brings together mathematicians, automatic control specialists and physicists. Its spirit is resolutely interdisciplinary.