Factorization of Dephasing Process in Quantum Open System

The fluctuation-dissipation relation is well known for the quantum open system with energy dissipation. In this paper a similar underlying relation is found between the bath fluctuation and the dephasing of the quantum open system, of which energy is conserved, but the information is leaking into the bath. To obtain this relation we revisit the universal, but simple dephasing model with quantum non-demolition interaction between the bath and the open system. Then we show that the decoherence factor describing the dephasing process is factorized into two parts, to indicate the two sources of dephasing, the vacuum quantum fluctuation and the thermal excitations defined in the initial state of finite temperature.