  We develop the Keldysh formalism for the polarization dynamics of an open
spin system. We apply it to the swapping between two qubit states in a model
describing an NMR cross-polarization experiment. The environment is a set of
interacting spins. For fast fluctuations in the environment, the analytical
solution shows effects missed by the secular approximation of the Quantum
Master Equation for the density matrix: a frequency decrease depending on the
system-environment escape rate and the quantum quadratic short time behavior.
Considering full memory of the bath correlations yields a progressive change of
the swapping frequency.
