Phase diffusion in trapped-atom interferometers

We evaluate the performance and phase diffusion of trapped $^{87}$Rb atoms in an atom-chip sensor with Ramsey interferometry and Hahn's spin echo in the time and phase domains. We trace out how the phase uncertainty of interference fringes grows with time. The phase-domain spin echo enables us to attain many-second-long phase diffusion with a low-cost local oscillator that otherwise seems unrealistic to obtain with such an oscillator. In the Ramsey experiment we record interference fringes with contrast decaying in $12$ s, and with a frequency uncertainty of $80$ mHz corresponding to the dephasing time of $2.8$ s. A clear distinction is drawn between the decoherence of the atomic ensemble, and the dephasing originating from the local oscillator. Spin echo cancels most of the perturbations affecting the Ramsey experiments, and leaves the residual phase noise of only $19$ mHz mostly attributed to the local oscillator frequency instability, yielding the increased coherence time of $11.9$ s which coincides with the contrast decay time in the Ramsey sequence. A number of perturbation sources leading to homogeneous and inhomogeneous dephasing is discussed. Our atom-chip sensor is useful in probing fundamental interactions, atomtronics, microcantilever, and resonant cavity profiling in situ.