Phase variable and phase relaxation processes in the Liouville space

The quantum dissipative dynamics of a phase variable is studied. A phase variable is treated as a quantum mechanical operator in Liouville space. The time translation properties of a phase operator and its eigenstate are investigated in terms of a time-evolution generator describing the phase relaxation process. The model corresponds to a quantum mechanical generalization of the random frequency modulation. An arbitrary state of a system can be expressed in terms of superpositions of the eigenstates of a phase operator. It is found that the width of a wave packet expressed by the phase eigenstates spreads monotonously with time, and that any stationary state is a completely random phase state. Such a phase randomizing process makes the entropy of a system increase. This process is also discussed with regard to stochastic processes. The Langevin equation for the phase variable and the Fokker-Planck equation for the probability distribution function in the phase eigenspace are obtained.