Evolution of Spin Entanglement and an Entanglement Witness in Multiple-Quantum NMR Experiments

We investigate the evolution of entanglement in multiple-quantum (MQ) NMR experiments in crystals with pairs of close nuclear 1/2-spins. The initial thermodynamic equilibrium state of the system in a strong external magnetic field evolves under the nonsecular part of the dipolar Hamiltonian. As a result, MQ coherences of the zeroth and plus/minus second orders appear. A simple condition for the emergence of entanglement is obtained. We show that the measure of the spin-pair entanglement-concurrence-coincides qualitatively with the intensity of MQ coherences of the plus/minus second order and, hence, the entanglement can be studied with MQ NMR methods. We introduce an entanglement witness using MQ NMR coherences of the plus/minus second order.