Analysis of a trapped atom clock with losses

We present a number of analytical means for the analysis of the evolution of an atom clock, emphasizing the role of many-body processes in the population and coherence loss. A master equation with one-body and two-body losses included is presented for a thermal cloud ensemble, suitable for modelling atom clock evolution. Approximate solutions for the coupled rate equations with one-body and two-body losses are obtained for the case of Ramsey free evolution, and their validity is analyzed. An approach is suggested to combine the analytical trap-induced fringe envelope function with a numerically evaluated many-body envelope function. A set of Ramsey and spin echo models is offered with analytical solutions for the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. An interferometric fringe in long evolution experiments is typically measured with a visibility decay primarily due to phase diffusion, phase instability, trapping effects and many-body losses. It constitutes a problem to extract parameters from a fringe obtained in the time domain because of the growing uncertainty of the fringe amplitude and phase. This visibility loss can be compensated for systems with unequal state population losses, as the one we have, by tailoring the pulse durations. Methods are suggested for the one-body and many-body cases.A comprehensive set of experiments is conducted on the characterization of the stability of the trapped atom clock: with condensate and thermal cloud ensembles, with phase-domain Ramsey and spin echo interferometries, with synchronization from a rubidium frequency standard and from an oven-controlled quartz oscillator. The types of noise present in the system are identified by means of an Allan deviation analysis. Despite the 50 times different cloud densities of the condensate and the thermal clouds, an equality of the phase stability in the Ramsey interferometry with BEC and thermal clouds is demonstrated. The spin echo interferometry has shown a slightly superior stability with the thermal clouds as compared to the BEC.