An N-atom Collective State Atomic Clock with Root-N Fold Increase in Effective Frequency and Root-N Fold Reduction in Fringe Width

We describe a collective state atomic clock with Ramsey fringes narrowed by a factor of $\sqrt{N}$ compared to a conventional clock, N being the number of non-interacting atoms, without violating the uncertainty relation. This narrowing is explained as being due to interferences among the collective states, representing an effective $\sqrt{N}$ fold increase in the clock frequency, without entanglement. The detection process, which measures a collective state, can be used to increase the quantum efficiency of detection significantly, yielding a net improvement in stability by as much as a factor of 10.