Direct-product formalism for calculating magnetic resonance signals in many-body systems of interacting spins

Abstract

Modern applications of nuclear magnetic resonance and electron spin resonance, and especially quantum computing problems call for more effective formalisms to describe relaxation and evolution of various orders of coherence in the presence of motions of the interacting spins. Here we suggest a formulation for the description of multiquantum spin states based on direct-product structures that take into account the inherent permutation symmetries and quantum coherences of a multispin system. Convenient recursion relations are obtained for the matrix representations of the N-body Hamiltonian superoperator and pulse propagators. This allows one to obtain compact expressions for the evolution of magnetic resonance signals when the dipolar interactions amongst spin-bearing molecules are modulated by their motions. These expressions include the free-induction decay and solid echoes, as well as the decay of higher-order coherences under the assumption of statistical independence of the motions of the spin-bearing molecules. Exact results, not requiring this assumption, are obtained for the case of the truncated dipolar Hamiltonian. Important phenomena that arise in multispin systems, such as instantaneous diffusion and spectral diffusion arising from motions, are studied more rigorously by solving the equation for the time evolution of the spin-density states. The many-body magnetic-resonance signals in the presence of motions are obtained by solving the appropriate stochastic Liouville equations. These solutions may be compared to solid-echo experiments to extract translational diffusion coefficients even in the slow motional regime.