Magnus expansion paradoxes in the study of equilibrium magnetization and entanglement in multi-pulse spin locking

Divergence of the Magnus expansion leads to paradoxes in the spin dynamics of solid-state NMR and in quantum informatics. This review presents results on quasi-equilibrium magnetization in a system of dipole–dipole (DD) coupled spins at times $T_2\ll t \ll T_{1\rho}$ in multiple-pulse spin locking ($T_2$ is the transverse spin relaxation time and $T_{1\rho}$ is the rotating-frame spin–lattice relaxation time). It is shown how contradictions between the results obtained with the Magnus expansion and experimental data can be removed. Systems of two and three DD coupled spins in multi-pulse spin locking are considered, and the entanglement evolution is investigated using both the Magnus expansion and the exact solution. The critical temperature for an entangled state is also found.