Stability of Atomic Clocks Based on Entangled Atoms

We analyze the effect of realistic noise sources for an atomic clock consisting of a local oscillator that is actively locked to a spin-squeezed (entangled) ensemble of $N$ atoms. We show that the use of entangled states can lead to an improvement of the long-term stability of the clock when the measurement is limited by decoherence associated with instability of the local oscillator combined with fluctuations in the atomic ensemble’s Bloch vector. Atomic states with a moderate degree of entanglement yield the maximal clock stability, resulting in an improvement that scales as $N^{1/6}$ compared to the atomic shot noise level.

Figure 1

(a) Representation of the probability distribution on the Bloch sphere for a spin squeezed state $|\psi (\kappa )⟩$, with $\kappa =N^{1/4}$ corresponding to the squeezing parameter $\xi =N^{-1/4}$ ($N=10$, both the initial state and the state just before detection are shown for clarity). Thick lines indicate initial uncertainties in $\overrightarrow{J}$. (b) Noise spectra due to LO frequency fluctuations when free running, when stabilized to uncorrelated atoms ($\xi =1$), and when stabilized to spin squeezed atoms ($\xi =N^{-1/6}$), $N={10}^3$, and $\gamma T={10}^{-2}$. (c) Inverse fractional frequency stability $1/{\sigma }_y$ (arbitrary units) vs Ramsey time for white LO noise, $N={10}^5$. Points: numerical simulations; lines: analytical results. Uncorrelated atoms (○, $\xi =1$) and spin squeezed atoms (◇, $\xi =N^{-1/4}$), both for linear feedback (full lines, empty symbols) and nonlinear feedback (dashed lines, filled symbols).

Figure 2

Inverse fractional frequency stability $1/{\sigma }_y$ (arbitrary units) vs number of atoms $N$, with Ramsey time optimized for (a) white noise and (b) $1/f$ noise. Points: numerical simulations; lines: analytical results. Uncorrelated atoms (○) and optimal spin squeezed atoms (◇), both for linear feedback (full lines, empty symbols) and nonlinear feedback (dashed lines, filled symbols).