Ultimate bounds for precision of atomic clock frequency measurement: Rabi, Ramsey, and coherent-population-trapping techniques

We investigate the ultimate quantum limits to the achievable uncertainty in the estimation of the transition frequency between two atomic levels. We focus on Rabi, Ramsey, and coherent-population-trapping (CPT) techniques, which are widely employed in experiments. We prove that in the Rabi and Ramsey schemes, measuring the atomic population allows one to reach the minimum uncertainty, but for the CPT setup, a measurement involving the coherences between the levels results in a further improvement of the estimation. As a paradigmatic example of possible noises, the effect of cavity coupling fluctuations is considered in the Rabi and Ramsey scenarios. As a figure of merit, we assess the Fisher information of the population measurement and compare its value with the maximum one given by the quantum Fisher information, which is achieved by optimizing over all the possible measurements allowed by quantum mechanics.