Optimally squeezed spin states

We consider optimally spin-squeezed states that maximize the sensitivity of the Ramsey spectroscopy, and for which the signal to noise ratio scales as the number of particles $N$. Using the variational principle we prove that these states are eigensolutions of the Hamiltonian $H(\lambda)=\lambda S_z^2-S_x,$ and that, for large $N$, the states become equivalent to the quadrature squeezed states of the harmonic oscillator. We present numerical results that illustrate the validity of the equivalence.