A simple feedback-loop for a frequency lock on a narrow atomic transition ⋆

In this paper we consider an atomic system with a narrow transition probed with a usercontrolled electromagnetic radiation, the basis of atomic clocks. We propose a simple feedback-loop in order to lock automatically the probe frequency on the atomic transition, allowing continuous clock operation without any preparation steps. The proof of the convergence of the feedback-loop is based on averaging arguments, with approximations compatible with realistic physical parameters. Numerical simulations illustrate the robustness of the proposed feedback-loop versus measurement noise and bias.