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Which component shows spin squeezing under twisting Hamiltonian?

Given a many body spin system under the action of the twisting Hamiltonian: $$H = A J_{z}^{2}$$ where $N$ is the number of spins, $$\mathbf{J} = \sum_{n=1}^{N} \mathbf{S}_{n}$$ is the collective spin operator, $\mathbf{S}_{n}$ is the spin operator for the $n$th spin, and $A$ characterizes the strength.

Which component of the spin system will display reduced variance/will be squeezed? Can you answer this without assumptions on the initial state? Can it be answered for the case $\langle J_{z} \rangle=0$?

The context of the question is the notion of spin squeezed states, as originally put forward by Wineland et al. and Kitagawa & Ueda