Multirate characterization of relaxation mechanisms for two nonequivalent nuclear spins 1/2 in a liquid using maximally entangled pseudo-pure quantum states

Multirate characterization of spin responses in nuclear magnetic resonance (NMR) is a promising approach to fingerprinting complex molecules in the presence of multiple relaxation mechanisms. Here we present experimental and theoretical investigations simultaneously accessing 8 relaxation rates describing the density matrix of two adjacent non-equivalent nuclear spins 1/2 $¹$H and $^{\ 13}$C belonging to a molecule in a liquid. The selected nuclear pair is stable with respect to chemical exchange. Some of the rates are obtained from conventional measurements of inversion recovery and nuclear Overhauser effect, while other, less conventional ones, are extracted from the relaxation initialized by the maximally entangled pseudo-pure Bell states (Bell PPSs) of the spin pair. The Bell PPSs are created using a hereby introduced method based on a detuned Hartmann-Hahn double resonance condition. Microscopic theory behind the measured relaxation rates is presented, and its consistency is demonstrated by several parameter-free tests. In particular, it is shown both theoretically and experimentally, that the eigenmodes of the off-diagonal relaxation of the two-spin density matrix can be selectively initialized using Bell PPSs. Our multirate analysis suggests that the measured off-diagonal relaxation is partly due to an unconventional mechanism arising from very weak J-couplings of the spin pair with fluctuating distant nuclear spins. Furthermore, we identify a dimensionless ratio of diagonal relaxation rates, which is determined exclusively by intra-pair magnetic dipolar interaction and hence possesses a universal value for a broad class of nuclear spin pairs. This value is consistent with both our experiments and other experiments reported in the literature.