while in the latter case, there is genuine multi-atomic correlation. In this section, we investigate how the two ways of squeezing are related to each other.

Internal spin squeezing {#sec:internal-squeezing}
-----------------------

We start our analysis with the situation in which the quantum uncertainty in the transverse component of each atomic spin, $\hat F_z^{(j)}$, is reduced independently. No entanglement is created among the atoms in this way. To exemplify our analysis, we will consider an ensemble of cesium atoms in the hyperfine level $F=4$ of the atomic ground state $6S_{1/2}$ that is illuminated by an intense laser field. The light-atom interaction is off-resonantly tuned to the $6S_{1/2}$–$6P_{3/2}$ transition, hence photons are not absorbed, and transitions among the hyperfine sublevels as well as the ground-state energy shifts are mediated only by transfer of photons between different polarization components. Namely, for the $j$th atom interacting with a light field propagating in the $x$ direction, the tensor light shift reads [@PhysRevA.71.032348] $$\begin{gathered}
 \label{eq:AL-tensorial}
 \hat H^{(j)}_{AL} = g_0 \hat \phi(0) + g_1 \hat s_z(0) \hat F_x^{(j)}
\\
 + g_2 \Bigl(
 - \hat \phi(0) \hat F_x^{(j)2}
 + \hat s_-(0) \hat F_{x+}^{(j)2}
 + \hat s_+(0) \hat F_{x-}^{(j)2}
 \Bigr),\end{gathered}$$ where the effective coupling constants $g_i$ sum up all 