  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.
