Environment-induced corrections to the spin Hamiltonian as dynamic frequency shifts in nuclear magnetic resonance

Abstract

We derive an expression for the correction to the spin-system Hamiltonian that arises due to the system–bath interaction, starting both from the standard master equation for the spin density matrix and a perturbative diagonalization of the system–bath Hamiltonian to the second order in the interaction. We show that the dynamic frequency shifts observed in the evolution of the nuclear spin coherences are a result of these Hamiltonian corrections. We present a systematic decomposition of the relaxation superoperator into Hermitian and anti-Hermitian parts as opposed to the usual practice of partitioning it into real and imaginary parts. We point out that the relaxation-induced corrections to the coherent motion arise exclusively from the anti-Hermitian part and the dissipative effects, from the Hermitian part, both, in general, being complex. However, the secular terms of this correction are found to depend only on the imaginary and the real parts, respectively.