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Quantum dynamics generated by the two-axis counter-twisting Hamiltonian

We study the quantum dynamics generated by a two-axis counter-twisting Hamiltonian from an initial spin coherent state in a spin-$1/2$ ensemble. A characteristic feature of the two-axis counter-twisting Hamiltonian is the existence of four neutrally stable and two saddle unstable fixed points. The presence of the last one is responsible for a high level of squeezing. The squeezing is accompanied by the appearance of several quantum states of interest in quantum metrology with Heisenberg-limited sensitivity, and we show fidelity functions for some of them. We present exact results for the quantum Fisher information and the squeezing parameter. Although, the overall time evolution of both changes strongly with the number of particles, we find that they have regular dynamics for short times. We explain scaling with the system size by using a Gaussian approach.