Symplectic integrators for the multilevel Redfield equation

We develop a method for solving the Redfield equation using symplectic integrators (SI). We show how the Redfield equation for a reduced density matrix of a multi-level system interacting with a thermal bath can be rewritten in a symplectic form so that numerical algorithms used in classical mechanics can readily be applied. The symplectic structure of the Redfield equation is established for a superoperator which is complex, non-Hermitian, and explicitly time-dependent in some cases. The efficiency of several SI schemes are studied and compared for a model dissipative system: electron transfer coupled to a vibrational mode and a thermal bath.