Time reversal of cross-polarization in solid-state NMR

Cross-polarization at the Hartmann-Hahn condition in solid-state NMR frequently is described in terms of thermodynamics. Spin temperatures characterizing the canonical density operator are assigned to the Zeeman reservoirs of the two spins and the cross-polarization process brings about a state of equilibrium of the two reservoirs with a common temperature. In such a model, cross-polarization from an initially polarized spin species (I spins) to another spin species (S spins) is inherently an irreversible process accompanied by an increase in the entropy of the system. However, a cross-polarization echo can be generated whereby the polarization transferred to the S spins returns to the I spins, restoring the initial density operator. Therefore a thermodynamic description should be applied with care even in samples where the build-up and the decay of the magnetization can be approximated well by multiexponential processes. Such cross-polarization echoes are formed by the consecutive application of two pulse trains that produce effective Hamiltonians differing in sign. The 'time reversal' of cross-polarization is consistent with both the increase in Zeeman entropy during the approach to equilibrium and with the constraint of unitary quantum evolution.