Inertial sensors using ultracold neutral atoms and atom interferometry techniques have demonstrated performances competing or even beating the more conventional light based interferometers. To further improve their sensitivity, quantum-non demolition (QND) detection schemes are presently investigated in order to engineer the initial state creating correlations between the atoms. Spin squeezed samples have been recently obtained, and it allowed to surpass the shot-noise detection limit. In this new regime the limit is set by the Heisenberg uncertainty principle, which states a 1/N ultimate signal-to-noise ratio. Implementing quantum limited atom interferometry will be a groundbreaking achievement, opening the door to many new advances in terms of both scientific discoveries and technological applications. The proposal merges atom interferometry and QND measurements with cavity cooling quantum electrodynamics (QED), in order to exploit the high degree of control of the atom-radiation interaction. A high finesse cavity will serve both to confine the atomic ensemble with an optical dipole trap, eventually reaching Bose-Einstein Condensation (BEC), and to develop new schemes of QND detection taking advantage of the gain factor provided by the cavity. The ultracold atomic sample will then be levitated against gravity with a sequence of coherent vertical momentum transfers, obtained by periodically shining the freely-falling atoms with phase-locked Raman beams. Keeping the resonance condition for the number of atoms versus time results in the determination of the gravity acceleration, and the measurement will be non-destructive. Finally the gravimeter will be loaded with a squeezed atomic sample, obtained through a QND measurement. Sub--shot--noise sensitivity approaching the Heisenberg limit should then be achieved for the interferometer, whereas a QND detection scheme will allow continuous readout interferometry.
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