Magic density in a self-rephasing ensemble of trapped ultracold atoms

We investigate the collective spin dynamics of a self-rephasing bosonic ensemble of Rb-87 trapped in a one-dimensional vertical optical lattice. We show that the combination of the frequency shifts induced by atomic interactions and inhomogeneous dephasing, together with the spin self-rephasing mechanism, leads to the existence of a "magic density": i.e., a singular operating point where the clock transition is first-order insensitive to density fluctuations. This feature is very appealing for improving the stability of quantum sensors based on trapped pseudo-spin-1/2 ensembles. Ramsey spectroscopy of the vertical bar 5s(2) S-1/2, F = 1, m(F) = 0 > -> vertical bar 5s(2) S-1/2, F = 2, F = 2,m(F )=0 > hyperfine transition is in qualitative agreement with a numerical model based on coupled Bloch equations of motion for energy-dependent spin vectors.