MAGnetic resonance techniques and Innovative Combinatorial Algebra – MAGICA

Magnetic resonance techniques have a deep impact as fundamental characterization methods in materials science. The key to further progress lies in mastering the associated spin dynamics. Solving the quantum equations of motion is especially difficult for systems submitted to time-dependent forces. In such cases the only available mathematical techniques are numerical or perturbative in nature. MAGICA relies on recently developed mathematics (Path-Sum) for exactly solving the dynamics of spin systems. Path-Sum stems from algebraic and combinatorial properties of walks on graphs which lead to an analytical, non-perturbative and convergent expression for the solution of the quantum equations of motion in the presence of time-dependent driving. Very recently (Phy. Rev. Res., 2020), C. Bonhomme and P.-L. Giscard have demonstrated the power of this approach by providing an exact analytical and non-perturbative description of the Bloch-Siegert effect at all orders and for all excitation regimes as well as a mathematical characterization of Coherent Destruction of Tunneling with an unprecedented level of accuracy. Illustrating further the capabilities of Path-Sum for quantum chemistry, spin diffusion has been quantitatively analyzed in the case of a system involving 42 coupled protons.

MAGICA proposes the full blown development of these recent breakthroughs with cutting-edge mathematical techniques to get novel theoretical, numerical and experimental insights into spin dynamics for NMR and DNP. MAGICA is built on three intentionally interconnected pillars. The first theme of research is devoted to the extension of Path-Sum to systems of coupled partial and fractional differential equations. These developments will lead to the analytical resolution of Bloch, Bloch-McConnell et Bloch-Torrey equations, either classical of fractional. Direct applications to Rabi frequency modulated continuous wave spectroscopy, chemical exchange theory, anomal diffusion in heterogeneous materials and CEST (Chemical Exchange Saturation Transfer) imaging will be implemented. Although ambitious mathematically, a clear roadmap for achieving these objectives is laid out from past research. The second theme of research is focused on extending of Path-Sum to ultra-large systems of spins via the newly introduced Lanczos Path-Sum method. This numerical method has just been developed by P.-L. Giscard in close collaboration with S. Pozza, member of the MAGICA consortium but not explicitly funded by ANR. The mathematical foundations of Lanczos Path-Sum are therefore established but its first applications will appear in MAGICA: we expect major improvement of simulation capabilities in numerical platforms such as SIMPSON, explicit calculations involving ultra-large systems such as those encountered in infinite spin chains and DNP (thousands of spins). The third and last topic of research pertains to experimental chemistry, yet is closely connected with the first two work packages. Primarily, we will study two classes of materials: apatite and calcium oxalates. The very first Rabi frequency modulated continuous wave excitation experiments will be implemented in solid state NMR on these materials. Relaxation times in heterogeneous materials will be analyzed using fractional Bloch equations, understood analytically through the first work package. In parallel, we propose to revisit spin diffusion in 1, 2 and 3D H-bond networks using the inherent scale-invariance of Path-Sum, which constitutes a highly innovative approach. Finally, chemically tunable materials will be synthesized and deeply studied by DNP in order to validate and feedback into the Lanczos Path-Sum simulations. We aim at establishing a strong dialog between theory and experiment on this point. This is a high risk/high gain MAGICA task. MAGICA is therefore an interdisciplinary project, at the crossroads of NMR spectroscopy, pure and numerical mathematics.