Control of tunneling reactions with an external held in a four-level system: A general Redfield approach

We investigate the possibility of controlling tunneling in a four-level system by the application of an external field. The four level system is defined by a symmetric double well potential energy surface that supports two sets of tunnel doubler states, as is characteristic of some proton transfer potential energy surfaces. The time-dependent external field couples to the dipole moment operator of the tunneling system. In a four-level system, there is the possibility of intrawell transitions via vibrational relaxation and external field induced transitions, in addition to the interwell tunneling transitions. We obtain a generalization of Redfield theory that describes the time evolution of the elements of the reduced density matrix for the system in the presence of the external field. The elements of the resulting Redfield tensor are constructed with the use of symbolic algebra. The theory is illustrated with a simple example that shows how the external field can control the rate of population transfer (tunneling) in the presence of the vibrational transitions.