Theoretical study of the Dick effect in a continuously operated Ramsey resonator

It is well established that passive frequency standards operated in pulsed mode may suffer a degradation of their frequency stability due to the frequency (FM) noise of the Local Oscillator (LO). In continuously operated frequency standards, it has been shown that a similar degradation of the frequency stability may arise, depending on the used modulation-demodulation scheme. In this paper, we report a theoretical analysis on the possible degradations of the frequency stability of a continuous fountain due to the LO FM noise. A simple model is developed to evaluate whether or not aliasing persists. This model is based on a continuous frequency control loop of a frequency standard using a Ramsey resonator. From this model, we derive a general formula, valid for all usual modulation-demodulation schemes, for the LO frequency fluctuations due to aliasing in closed loop operation. We demonstrate that in an ideal situation and for all usual modulation waveforms, no aliasing occurs if the half-period of modulation equals the transit time of atoms in the Ramsey resonator. We also deduce that in the same conditions, square-wave phase modulation provides the strongest cancellation of the LO instabilities in closed loop operation. Finally, we show that the “Dick formula” for the specific case of the pulsed fountain can be recovered from the model by a sampling operation.