Quantum dynamics under coherent and incoherent effects of a spin bath in   the Keldysh formalism: application to a spin swapping operation
  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. 
Ernesto P. Danieli, Horacio M. Pastawski and Gonzalo A. Alvarez