  The geometric (Berry) phase of a two-level system in a dissipative
environment is analyzed by using the second-quantized formulation, which
provides a unified and gauge-invariant treatment of adiabatic and nonadiabatic
phases and is thus applicable to a quantitative analysis of transitional
regions away from ideal adiabaticity. In view of the recent experimental
observation of the Berry phase in a superconducting qubit, we illustrate our
formulation for a concrete adiabatic case in the Ohmic dissipation. The
correction to the total phase together with the geometry-dependent dephasing
time is given in a transparent way. The behavior of the geometric phase away
from ideal adiabaticity is also analyzed in some detail.
