Two-axis spin squeezing in two cavities

Ultracold atoms in an ultrahigh-finesse optical cavity are a powerful platform to produce spin squeezing since photon of cavity mode can induce nonlinear spin-spin interaction and thus generate a one-axis twisting Hamiltonian $H_{\text{OAT}}=qJ_{x}^{2}$, whose corresponding maximal squeezing factor scales as $N^{-2/3}$, where $N$ is the atomic number. On the contrary, for the other two-axis twisting Hamiltonian $H_{\text{TAT}}=q(J_{x}^{2}-J_{y}^{2})$, the maximal squeezing factor scales as $N^{-1}$, approaching the Heisenberg limit. In this paper, inspired by recent experiments of cavity-assisted Raman transitions, we propose a scheme, in which an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers, to realize a tunable two-axis spin Hamiltonian $$. For proper parameters, the above one- and two- axis twisting Hamiltonians are recovered, and the scaling of $N^{-1}$ of the maximal squeezing factor can occur naturally. On the other hand, in the two-axis twisting Hamiltonian, spin squeezing is usually reduced when increasing the effective atomic resonant frequency $\omega_{0}$. Surprisingly, we find that by combined with the dimensionless parameter $\chi(>-1)$, the effective atomic resonant frequency $\omega_{0}$ can enhance spin squeezing largely. These results are benefit for achieving the required spin squeezing in experiments.