J_x \rangle^2$ is the noise in spin angle  [@PhysRevA.46.R6797]. []{data-label="fig:internal-squeezing"}](squeezing){width=".9\hsize"}

Projection based spin squeezing {#sec:qnd-squeezing}
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In the next step, we consider a collective process in which the noise in $\hat J_z$ is reduced via quantum non-demolition (QND) measurement and feedback [@EurophysLett.42.481; @PhysRevLett.85.1594; @Science.304.270; @PhysRevLett.93.163602; @PhysRevLett.93.173002; @PhysRevA.65.053819; @PhysRevA.65.061801; @PhysRevA.70.052324; @PhysRevA.72.052313]. This kind of squeezing creates interatomic entanglement. To measure $\hat J_z$ in a non-destructive way, we shall couple it to a “meter” system, e.g., a light field propagating along the $z$-axis. This probe field is far detuned, so that we can neglect the second order light shift. We also neglect spontaneous emission, absorption of the probe beam, and other sources of imperfections that may actually limit spin squeezing [@PhysRevA.65.061801; @PhysRevA.65.053819; @PhysRevA.70.052324; @PhysRevLett.93.173002]. The interaction Hamiltonian in this configuration is $\hat H_1 = g_1 \hat s_z(0) \hat J_z$.

The Stokes vector component $\hat s_z$ is a QND variable in this interaction: it is not modified by the interaction itself while the light passes different segments of the sample. It is therefore conventional to treat the accumulated interaction as if the ensemble of atoms as a whole were interacting with a single light mode integrated along the pulse, $\hat S_i (t) \equiv 