Quantum Zeno dynamics (QZD) were recently demonstrated with Bose-Einstein condensates (BEC)  and with Rydberg atoms . In our experiment we work with internal degrees of freedom of a BEC of 87 Rb atoms created on an Atom Chip. In the presence of a constant magnetic bias field, we drive the dynamics between magnetics sub levels on the same hyperfine ground states using a radio-frequency (RF) field. We exploited QZD with a Raman coupling between different hyperfine levels to constrain atoms to evolve in two Hilbert subspaces. We choose the 4 logical input states of a 2-qbit gate among the two subspaces; with D-CRAB optimization  we design a control RF pulse to perform a phase gating operation. We make use of a recently developed state reconstruction technique  to perform input and output state tomography in order to characterize the gate operation. Given that in the same system we can make use of Raman and microwave transitions between the different hyperfine manifolds in order to perform single q-bit operations, our use of QZD allows the demonstration of a complete set of logical gates within a single rubidium atom.
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