  Due to the potential application in quantum information process, geometric
phase of interacting system arouse many interests. Some physicists concentrate
on the system in pure classical envi- ronment, while others study the system in
pure quantized environment. So a natural question is asked: how about an
interacting system in composite environments made up of both classical and
quantized field. In this letter, we analyze a quantum system composed of two
interacting spins, of which one is in classical magnetic field and the other is
in quantized field. First, classical magnetic field driven Berry phases for the
whole system and subsystem are studied. The effect of couplings between
particles and photon on these phases are analyzed. In comparison with the
dynamical quantized field, We find that even a static quantized field in its
vacuum state can also have an effect on Berry phase. Second, quantized field
driven Berry phases for the whole system and sub- system are formulated,
including both one and two mode of this field. The vacuum induced effects are
elaborated, moreover compared with the constant vacuum induced phases in former
papers, the counterpart in this letter varies according to classical magnetic
field, couplings and other pa- rameters. For the two mode quantized field, the
rigorous relationship between the concurrence and Berry phase for subsystem are
built up.
