Geometric phase gate without dynamical phases

A general scheme for an adiabatic geometric phase gate is proposed which is maximally robust against parameter fluctuations. While in systems with SU(2) symmetry geometric phases are accompanied by dynamical phases and are thus not robust, we show that in the more general case of an SU(2)⊗SU(2) symmetry it is possible to obtain a nonvanishing geometric phase without dynamical contributions. The scheme is illustrated for a phase gate using two systems with dipole-dipole interactions in external laser fields which form an effective four-level system.