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We first consider stimulated Raman adibatic passages (STIRAP) in a closedfour-level tripod system. In this case, the adiabatic eigenstates of the systemacquire real geometric phases. When the system is open and subject todecoherence they acquire complex geometric phases that we determine by a MonteCarlo wave function approach. We calculate the geometric phases and the stateevolution in the closed as well as in the open system cases and describe thedeviation between these in terms of the phases acquired. When the system isclosed, the adiabatic evolution implements a Hadamard gate. The open systemimplements an imperfect gate and hence has a fidelity below unity. We expressthis fidelity in terms of the acquired geometric phases.

Geometric phases in open tripod systems