Spin-1/2 geometric phase driven by decohering quantum fields

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We calculate the geometric phase of a spin-1/2 system driven by a one and two mode quantum field subject to decoherence. Using the quantum jump approach, we show that the corrections to the phase in the no-jump trajectory are different when considering an adiabatic and non-adiabatic evolution. We discuss the implications of our results from both the fundamental as well as quantum computational perspective.