  We study the geometric phase of a two-level atom coupled to an environment
with Lorentzian spectral density. The non-Markovian effect on the geometric
phase is explored analytically and numerically. In the weak coupling limit the
lowest-order correction to the geometric phase is derived analytically and the
general case is calculated numerically. It is found that the correction to the
geometric phase is significantly large if the spectral width is small and in
this case the non-Markovian dynamics has a significant impact to the geometric
phase. When the spectral width increases, the correction to the geometric phase
becomes negligible, which shows the robustness of the geometric phase to the
environmental white noises. The result is significant to the quantum
information processing based on the geometric phase.
