Fourier analysis of Ramsey fringes observed in a continuous atomic fountain for in situ magnetometry

Ramsey fringes observed in an atomic fountain are formed by the superposition of the individual atomic signals. Due to the atomic beam residual temperature, the atoms have slightly different trajectories and thus are exposed to a different average magnetic field, and a velocity-dependent Ramsey interaction time. As a consequence, both the velocity distribution and magnetic field profile are imprinted in the Ramsey fringes observed on Zeeman sensitive microwave transitions. In this work, we perform a Fourier analysis of the measured Ramsey signals to retrieve both the time-averaged magnetic field associated with different trajectories and the velocity distribution of the atomic beam. We use this information to reconstruct Ramsey fringes and establish an analytical expression for the value of the microwave frequency for which individual Ramsey fringes add most constructively and are thus visible in the microwave spectrum.