  We perform Ramsey spectroscopy on the ground state of ultra-cold 87Rb atoms
magnetically trapped on a chip in the Knudsen regime. Field inhomogeneities
over the sample should limit the 1/e contrast decay time to about 3 s, while
decay times of 58 s are actually observed. We explain this surprising result by
a spin self-rephasing mechanism induced by the identical spin rotation effect
originating from particle indistinguishability. We propose a theory of this
synchronization mechanism and obtain good agreement with the experimental
observations. The effect is general and susceptible to appear in other physical
systems.
