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We consider an ensemble of trapped atoms interacting with a continuous wavelaser field. For sufficiently polarized atoms and for a polarized light field,we may approximate the non-classical components of the collective spin angularmomentum operator for the atoms and the Stokes vectors of the field byeffective position and momentum variables for which we assume a gaussian state.Within this approximation, we present a theory for the squeezing of the atomicspin by polarization rotation measurements on the probe light. We deriveanalytical expressions for the squeezing with and without inclusion of thenoise effects introduced by atomic decay and by photon absorption. The theoryis readily adapted to the case of inhomogeneous light-atom coupling [A. Kuzmichand T.A.B. Kennedy, Phys. Rev. Lett. Vol. 92, 030407 (2004)]. As a specialcase, we show how to formulate the theory for an optically thick sample byslicing the gas into pieces each having only small photon absorptionprobability. Our analysis of a realistic probing and measurement scheme showsthat it is the maximally squeezed component of the atomic gas that determinesthe accuracy of the measurement.Comment: 12 pages, 5 figure

Spin squeezing and precision probing with light and samples of atoms in the Gaussian description