Karabanov, A., G. Kwiatkowski, and W. Köckenberger, Quantum Mechanical Simulation of Cross Effect DNP Using Krylov–Bogolyubov Averaging. Appl. Magn. Reson., 2012. 43(1): p. 43-58.
http://dx.doi.org/10.1007/s00723-012-0367-0
Krylov–Bogolyubov averaging is applied to reduce the dimension in quantum mechanical simulations of the cross effect dynamic nuclear polarization experiment. The exact form of the averaged master equation, describing the polarization dynamics of individual nuclear spins, is provided. The relevant relaxation superoperator is derived in the conventional product basis avoiding the diagonalization of the Hamiltonian for the purpose of finding its eigenbasis. Furthermore, we show that it is sufficient to retain the relaxation terms arising from the full set of electron Zeeman states and the terms arising from paramagnetic relaxation of the nuclear spins for deriving the relaxation superoperator. The subspace of the Liouville space to which the spin dynamics can be confined by the averaging procedure is identified. In addition, it is demonstrated that the state space can be truncated at a low spin correlation order, thus reducing the dimensions even further. Numerical results, illustrating the theory, are presented.