An analysis of the stationary operation of atomic clocks

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.