QUROPE Quantum Information Processing and Communication in Europe

In atomic clocks, the frequency of a local oscillator is stabilized based on the feedback signal obtained by periodically interrogating an atomic reference system. The instability of the clock is characterized by the Allan variance, a measure widely used to describe the noise of frequency standards. We provide an explicit method to find the ultimate bound on the Allan variance of an atomic clock in the most general scenario where N atoms are prepared in an arbitrarily entangled state and arbitrary measurement and feedback schemes are allowed, including those that exploit coherences between succeeding interrogation steps. While the method is rigorous and completely general, it becomes numerically inefficient for large N and long averaging times. This could be remedied by incorporating numerical methods based on e.g. a matrix product states approximation.