Spin entanglement in antiferromagnetic spin-1 Bose-Einstein condensates

We study the spin entanglement in a spin-1 Bose-Einstein condensate with antiferromagnetic atomic interactions using the Hartree-Fock approach. Based on the isomorphism between symmetric $N$-qubit states and spin-$j=N/2$ states, we analyze the negativity of the spin-1 ground state of the condensate viewed as a two spins $1/2$, and explore its dependence with the temperature and external Zeeman fields. The scope of this type of entanglement is highlighted and contrasted with other types of entanglement, as the mode and particle entanglement. It is shown that, at finite temperatures, there is a strong dependence of the negativity with respect to the strengths of quadratic Zeeman fields and the spin-spin interactions of the condensate. Interestingly, in the antiferromagnetic ground-state phase, the negativity and the linear Zeeman field are connected quadratically through the equation of a simple circle, in which its radius depends on the temperature. On the other hand, for the polar phase, being the phase that exhibits the highest degree of entanglement, we were able to identify a clear dependency on the spin-spin interactions and the Zeeman fields that can be expressed analytically in a closed form as a function of the temperature. This results might be relevant for applications in quantum information and metrology.