I am exploring the one axis twisting (OAT) hamiltonian $\hat{H}=\chi S_z^2$ with $S_z=\sum_{i=1}^N\frac{\sigma_z^i}{2}$ and considering the initial state to be $\left|\psi(0) \right>=\left|+x\right>^{\otimes N}$, where +x means that the state is pointing along the x direction in the Bloch sphere, i denotes the index of the ith spin and $\chi$ is the coupling strength. Define the spin-squeezing $\xi^2$ as https://en.wikipedia.org/wiki/Spin_squeezing. For a large number of particles $N$ we can find the scaling of the minimum spin-squeezing ($\xi^2_{min}$) with the number of particles, a simple model for such scaling is $\xi^2_{min}(N)=C/N^{\alpha}$ (this can be done in general but the fitting improves at the large number of particles). My question is, is there any reference where this calculation or a numerical simulation has been done such that we have an estimation of $\alpha$ in OAT?. How can we include possible corrections beyond $N^{\alpha}$? Thanks.
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