General solutions for quantum dynamical systems driven by time-varying Hamiltonians: applications to NMR

P.-L Giscard (1, 2) , Christian Bonhomme (3)

1 ULCO - Université du Littoral Côte d'Opale
2 Department of Computer Sciences, York University
3 LCMCP-SMiLES - Spectroscopie, Modélisation, Interfaces pour L'Environnement et la Santé

Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood. Since the inception of Magnus’ expansion in 1954, no fundamentally novel mathematical method for solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems and is invariant under scale transformations. We solve for the dynamics of all two-level systems as well as of many-body Hamiltonians with a particular emphasis on NMR applications (Bloch-Siegert effect and N-spin systems involving the dipolar Hamiltonian and spin diffusion).

Mots clés: Bloch-Siegert effect; Analytical and numerical methods; Path-sum; Time-varying Hamiltonian; N spins systems and dipolar couplings; Nuclear Magnetic Resonance

Domaines: Physique mathématique [math-ph]; Chimie-Physique [physics.chem-ph]; Physique Quantique [quant-ph]; Systèmes dynamiques [math.DS]; Physique [physics]

P.-L Giscard, Christian Bonhomme. General solutions for quantum dynamical systems driven by time-varying Hamiltonians: applications to NMR. 2019. ⟨hal-02125160v1⟩