Squeezing of Spin Waves in Atomic Ensembles

Squeezing the collective spin of an atomic ensemble via QND measurement is based on the lighhift interaction between a cloud of atoms and a laser probe. When the shot noise resolution of the laser probe is below the projection noise of the atoms, the resulting backaction can reduce the uncertainty for a collective atomic observable. Most current models of this process rely on idealized one-dimensional plane wave approximations of the underlying light-matter interaction, which are not appropriate for describing a real system consisting of an atomic cloud in dipole trap interacting with a paraxial probe laser. We derive from first principles a model for three-dimensional QND spin squeezing of an ensemble of alkali atoms. The model includes spin waves, diffraction, propagation phase, paraxial modes, and optical pumping, based on a full master equation description. Our model easily generalizes to atoms with hyperfine spin f \textgreater 1/2, for which initial state preparation of the ensemble using internal hyperfine control can enhance the entangling power of the Faraday interaction [Norris et al., PRL 109, 173603 (2012)]. Including dissipative dynamics, we find optimal geometries to maximize spin squeezing for a variety of state preparations and spin sizes.