Multiple quantum (MQ) NMR spectroscopy is a powerful tool to study nuclear spin distributions in a great range of materials: liquid crystals, simple organic substances, amorphous hydrogenated silicon and many others [1]. It brings information about the dynamics of spin clusters and sheds light on quantum decoherence mechanisms in highly correlated spin states.
The free induction decay (FID), the most common observable in NMR, is determined by a small fraction of the density matrix elements due to a limited number of non-zero matrix elements of the spin projection operators. At the same time, the state of a spin system in the MQ NMR experiment is determined by all elements of the density matrix. Thus, MQ NMR experiments provide more information than the usual NMR [2]. In particular, this information resource can be used for the transfer of quantum information.
The dependence of the intensities of MQ NMR coherences on their orders (the profile of MQ NMR coherences) is an important characteristics of the above classes of systems because it contains a wealth of physical-chemical information. A simple combinatorial model [1] predicts a Gaussian profile of MQ NMR coherences. This conclusion was confirmed qualitatively in some experiments [1]. Since the model is based on very rough assumptions, one can expect discrepancies with experimental data in many cases. Numerous experimental data were reviewed in Ref.[3] and the authors found that the profiles of the MQ NMR intensities are rather exponential than Gaussian.
Our main goal is to create a full quantum-mechanical theory which would explain the observed profiles of the intensities of MQ NMR coherences. At this time, the problem cannot be solved for arbitrary three-dimensional spin systems. Thus, as a first step, we consider one-dimensional systems. We introduce a model of an isolated spin chain and show that the MQ NMR Hamiltonian of such a system is an XY Hamiltonian which can be diagonalized exactly in the approximation of the nearest neighbor interactions [4].
Eventually, we have found [4] that only the intensities of MQ NMR coherences of the zeroth and +/- second orders do not vanish. Such result means that the model does not yield a full solution to the problem. The situation does not change when we consider MQ NMR dynamics of one-dimensional systems at low temperatures and even in alternating spin chains in the approximation of the nearest neighbor interactions.
Next, we consider nanopore materials whose nanopores are filled with spin-carrying atoms (molecules). Since molecular spin diffusion is faster than spin flip-flop processes, one can average the dipole-dipole interactions (DDI) over molecular diffusion. The DDI coupling constants are the same for all spin pairs and we can investigate the MQ NMR dynamics in systems with several hundred spins [5]. Our numerical analysis showed that the profile of the intensities of the MQ NMR coherences is exponential in agreement with the experimental data [3].
In conclusion we note that modern NMR spectrometers have allowed us to perform experimental investigations of dynamics and relaxation in low-dimensional systems [6] which are in a good agreement with the developed theory [2,4].

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