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The best possible improvement from entangling the atoms, given this jitter, is not known.","posed_since":"","precise":"For a Ramsey-type optical clock with $N$ atoms and local-oscillator frequency noise with power spectrum $S(f)\\sim h_\\alpha f^{-\\alpha} (\\alpha=0, 1, 2)$, determine the optimal asymptotic scaling with $N$ of the long-term Allan deviation $\\sigma_y(\\tau)$, optimizing over entangled states, adaptive measurements and feedback, and whether any protocol reaches $1/N$ scaling up to logarithmic factors.","problem_ref":null,"references":"","settled_by":"A matching lower bound and explicit protocol for the Allan deviation exponent as a function of N for each noise type.","status_note":"Protocols with near-Heisenberg scaling up to logarithmic factors are known for white phase noise; matching bounds for 1/f and random-walk laser noise are lacking.","title":"Optimal scaling of atomic-clock stability with atom number under laser 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