Séminaire Christian Bonhomme "The evolution operator in magnetic resonance: from Floquet, Magnus, Feynman ... to Path-Sum." (LCMCP)

The evolution operator in magnetic resonance: from Floquet, Magnus, Feynman ... to Path-Sum.

Pr. Christian Bonhomme

Campus Sorbonne Université Tour 32-42  salle 101

Abstract: Spin dynamics in Magnetic Resonance (NMR, EPR, DNP...) follows the Liouville-von Neumann equation, or in other words, the time-dependent Schrödinger equation. In particular, the evolution operator U(t',t) is in the heart of this equation and must be calculated at the best level. In the past, several perturbative (and approximate approaches) have been proposed such as the Magnus and Floquet expansions (when the hamiltonian is periodic), leading to spectacular new concepts in solid state NMR.

In this contribution, an exact, unconditional and superexponential convergent solution for U(t',t) is obtained in all cases using graph theory and the Path-Sum theorem (or PS, arithmetics on graphs) demonstrated by P.-L. Giscard in 2015. It corresponds to the infinite resummation of Feynman diagrams, here in a discrete case. Several examples including MAS spin diffusion in François Me/BuSn12, and the exact expression of the Bloch-Siegert effect at all orders will be presented. I will propose most demonstrations related to graph theory and PS at the blackboard in order to demystify the mathematical background.