Abstract
We develop an abstract model of atomic clocks that fully describes the dynamics of repeated synchronization between a classical oscillator and a quantum reference. We prove existence of a stationary state of the model and study its dependence on the control scheme, the interrogation time and the stability of the oscillator. For unbiased atomic clocks, we derive a fundamental bound on atomic clocks long time stability for a given local oscillator noise. In particular, we show that for a local oscillator noise with integrated frequency variance scaling as $${T^\alpha}$$ for short times T, the optimal clock time variance scales as $${F^{-(\alpha +1)/(\alpha +2)}}$$ with respect to the quantum Fisher information, F, associated to the quantum reference. In an attempt to prove the bounds without the unbiasedness assumption, we derive a new Cramer-Rao type inequality.
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