When femto-second (fs) time-resolved experiments are used to study ultrafast processes, quantum beat phenomena are often observed. In this paper, to analyze the fs time-resolved spectra, we will present the density matrix method, a powerful theoretical technique, which describes the dynamics of population and coherence of the system. How to employ it to study the pump-probe experiments and fs ultrafast processes is described. The ππ*→nπ* transition of pyrazine is used as an example to demonstrate the application of the density matrix method. Recently, Suzuki’s group have employed the 22 fs time resolution laser to study the dynamics of the ππ* state of pyrazine. In this case, conical intersection is commonly believed to play an important role in this non-adiabatic process. How to treat the effect of conical intersection on non-adiabatic processes and fs time-resolved spectra is presented. Another important ultrafast process, vibrational relaxation, which takes place in sub-ps and ps range and has never been carefully studied, is treated in this paper. The vibrational relaxation in water dimer is chosen to demonstrate the calculation. It should be noted that the vibrational relaxation of (H2O)2 has not been experimentally studied but it can be accomplished by the pump-probe experiments. When this femto-second (fs) time-resolved experiments are used to study ultrafast processes, quantum beat phenomena are often observed. We this present the density matrix method, a powerful theoretical technique , which describes the dynamics of population and coherence of the system. How to employ it to study the pump-probe experiments and fs ultrafast processes is described. The ππ * → nπ * transition of pyrazine is used as an example to demonstrate the application of the density matrix method. Recently, Suzuki’s group have employed the 22 fs time resolution laser to study the dynamics of the ππ * state of pyrazine. In this case, conical intersection is commonly believed to play an important role in this non-adiabatic process. How to treat the effect of conical intersection on non-adiabatic processes and fs time-resolved spectra is presented. Another important ultrafast process, vibrational relaxation, which takes place in sub-ps an The vibrational relaxation in water dimer is chosen to demonstrate the calculation. It should be noted that the vibrational relaxation of (H2O) 2 has not been experimentally studied but it can be accomplished by the pump-probe experiments.