Fig. 2: Optimal clock stability.

a Allan deviation σ_y of an optical atomic clock at averaging time τ = 1 s as a function of Ramsey dark time T_R assuming a dead time T_D = 0.5 s and laser noise corresponding to the currently best ultra-stable clock lasers (cL) as characterized in Table 1. Solid lines are instabilities from the full noise model, Eq. (2), with N = 10 (blue) and N = 2000 (red) uncorrelated clock atoms. Dashed lines show the three contributing noise processes: QPN (blue and red), CTL (green), and the Dick noise (black). Symbols are numerical simulations of the closed feedback loop in agreement with the analytic model until the onset of fringe-hops leads to a sudden, strong increase in instability. b Bound on the minimal instability \({\sigma }_{\min }\) as a function of dead time for four different types of clock lasers (tL, cL, pL1, pL2) as defined in Table 1. Inset: Normalizing by the laser coherence time Z (as defined in the main text and given in Table 1) reveals an almost universal scaling with \(\sqrt{Z}{\sigma }_{\min }=3.0\times 1{0}^{-16}\ {({T}_{{\rm{D}}}/Z)}^{0.7}\) at longer dead times. We use the transition frequency ν₀ ≈ 429.228 THz of ⁸⁷Sr for calculations.