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We consider the decoherence of a pseudo-spin ensemble under collective randomrotations, and study, both theoretically and experimentally, how anondestructive measurement combined with real-time feedback correction canprotect the state against such a decoherence process. We theoreticallycharacterize the feedback efficiency with different parameters --- coherence,entropy, fidelity --- and show that a maximum efficiency is reached in the weakmeasurement regime, when the projection of the state induced by the measurementis negligible. This article presents in detail the experimental resultspublished in [Phys. Rev. Lett. \textbf{110}, 210503 (2013)], where the feedbackscheme stabilizes coherent spin states of trapped ultra-cold atoms, andnondestructively probed with a dispersive optical detection. In addition, westudy the influence of several parameters, such as atom number and rotationangle, on the performance of the method. We analyze the various decoherencesources limiting the feedback efficiency and propose how to mitigate theireffect. The results demonstrate the potential of the method for the real-timecoherent control of atom interferometers.

Feedback control of coherent spin states using weak nondestructive measurements