Since the early days of quantum theory and even more since development of the field of quantum computation, it is well established that correlation and entanglement are fundamental in quantum mechanics and a remarkable effort is on the way to better understand these concepts, both from the theoretical and the experimental point of view. In this proposal a series of new experiments devoted to the study of quantum entanglement in a mesoscopic system are presented. All the proposed experiments will be carried out on a sample of sodium atoms cooled down to quantum degeneracy. In such a system, the spin dependent interaction gives rise to full many-body entanglement whose observation is the central theme of this proposal. Due to the antiferromagnetic character of this spin interaction in sodium, several quantum states can be engineered going from Shroedinger cats to fragmented Bose-Einstein condensates. Beside the intrinsic interest of observing these many-body states, the ability to access and manipulate them could shine new light on the ubiquitous phenomenon of decoherence. To be able to access such highly entangled states, which have never been observed in any domain of physics for a number of particle higher than ten, several technical obstacles have to be overcome. A first crucial point will be the reliable production of such mesoscopic samples and a matching ability to precisely count the number of particle. An important part of this proposal is the implementation of the required experimental techniques, many of them already tested in different environments, into a new experimental apparatus. In a wider perspective, highly entangled mesoscopic states can lead to groundbreaking developments in the field of quantum metrology since these states can overcome the so called quantum projection noise limit, a fundamental limit of conventional atomic interferometers.
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