  We first consider stimulated Raman adibatic passages (STIRAP) in a closed
four-level tripod system. In this case, the adiabatic eigenstates of the system
acquire real geometric phases. When the system is open and subject to
decoherence they acquire complex geometric phases that we determine by a Monte
Carlo wave function approach. We calculate the geometric phases and the state
evolution in the closed as well as in the open system cases and describe the
deviation between these in terms of the phases acquired. When the system is
closed, the adiabatic evolution implements a Hadamard gate. The open system
implements an imperfect gate and hence has a fidelity below unity. We express
this fidelity in terms of the acquired geometric phases.
