Near-Heisenberg-Limited Atomic Clocks in the Presence of Decoherence

The ultimate stability of atomic clocks is limited by the quantum noise of the atoms. To reduce this noise it has been suggested to use entangled atomic ensembles with reduced atomic noise. Potentially this can push the stability all the way to the limit allowed by the Heisenberg uncertainty relation, which is denoted the Heisenberg limit. In practice, however, entangled states are often more prone to decoherence, which may prevent reaching this performance. Here we present an adaptive measurement protocol that in the presence of a realistic source of decoherence enables us to get near-Heisenberg-limited stability of atomic clocks using entangled atoms. The protocol may thus realize the full potential of entanglement for quantum metrology despite the detrimental influence of decoherence.

Figure 1

(a) The atomic state just before the measurement of $J_z$ for (A) uncorrelated atoms, (B) moderately squeezed atoms, and (C) highly squeezed atoms. (b) Stability as a function of the Ramsey time ($\gamma T$) for $N={10}^5$. ▪, (▾) is the conventional protocol of Ref. 5 for optimal squeezing (uncorrelated) atoms while ●, (▴) is the adaptive protocol for optimal squeezing (uncorrelated) atoms. The adaptive protocol allows for $\gamma T\sim 0.3$ while the conventional protocol only allows for $\gamma T\sim 0.1$ (see the Supplemental Material [32]).

Figure 2

Operation of an atomic clock. A clock cycle of duration $T_c$ starts with initializing the atoms and ends with the measurements and feedback on the LO. The bottom part of the figure shows the adaptive protocol consisting of a series of weak measurements with intermediate feedback. The feedback seeks to rotate the atomic state to have mean spin almost along the $y$ axis before the final projective measurement and subsequent feedback on the LO.

Figure 3

Optimized stability of an atomic clock for a LO subject to (a) white noise and (b) $1/f$ noise. ○, □, ▵, ▿ are the full quantum simulation while ●, ▪, ▴, ▾ are the Gaussian simulation. The Gaussian simulation can be extended down to $N=100$, which gives more or less identical results to the full quantum simulation. ○, ● (▴, ▵) are the adaptive scheme and □, ▪ (▿, ▾) are the conventional protocol with (without) entanglement. The dotted lines are the analytical results and the solid line is the Heisenberg limit for the maximal Ramsey time $\gamma T=0.3$ (a) and $\gamma T=0.2$ (b).