As occurs with the XY (flip-flop) dynamics, the double-quantum (flip flip-flop flop) dynamics in a linear chain with nearest neighbor interaction can be mapped to non-interacting fermions. A manifestation of this property is that only zero and second order coherences are excited under typical nuclear magnetic resonance (NMR) conditions. We test the conditions required to observe one-body dynamics and the rate of decoherence in a network of coupled spins 1/2 in hydroxyapatite (HAp) using solid state NMR. HAp behaves as a one-dimensional spin chain, as the minimum distance between hydrogen spins in different chains is about three times larger than the distance between adjacent protons within the chain. Due to the distance dependence of the dipolar interaction the ratio between intra and inter-chain couplings becomes larger than twenty. Moreover, the rapid dynamics within the chain tends to freeze the interchain dynamics, quantum Zeno effect, leading to a ratio between the intra and inter-chain time scales of more than two orders of magnitude.
Two different implementations of an effective double quantum Hamiltonian, H_{DQ}, that mixes subspaces with different spin projections, are used to create many-body superposition states: the multiple quantum coherences. Our experimental results confirm the theoretical predictions showing that, starting in equilibrium condition, all the coherences orders above two cancel out. We go out of this one-body to a many-body condition by changing the effective spin-spin interaction or alternatively, the dimensionality of the coupling network. In the first case, the dynamics of HAp under a rotated dipolar Hamiltonian, H_{XX}, that does not present the mapping property, shows higher orders of coherences. In the second case, H_{DQ} is applied to adamantane (a standard 3d system). Decoherence is tested in both systems through experimental implementations of Loschmidt echoes where the time reversal is executed on H_{DQ}. Remarkably, the results in HAp presents an exponential decay, in contrast with a more inertial decay in the 3-d system.
