Quantum mechanical transitions in a dissipative environment

We have constructed an algorithm to treat quantum mechanical dynamics, such as the nonadiabatic transitions in a dissipative environment, by making us e of the formula of reduced density matrix, i.e., Quantum Master Equation ( QME). We applied the method to investigate dynamics of magnetization of nan oscale magnets, such as Mn-12 and Fe-8, ina time dependent field. Due to th e discreteness of the energy levels in small systems the resonating tunneli ng phenomena play an important role. At relatively high temperatures the ex cited levels contribute to the transition and the transition rate depends s ignificantly on the temperature. On the other hand, at very lour temperatur es the magnetization process does not depend on the temperature. Even at su ch a low temperature, the effect of environments is still relevant. We demo nstrate such a process with the method of QME. We also analyze temperature dependent phenomena of hysteresis of the nanoscale magnets.