Evolution of entanglement under an Ising-like Hamiltonian with particle losses

We present analytical compact solution for the density matrix and all correlation functions of two collective-macroscopic spins evolving via Ising-like Hamiltonian in the presence of particle losses. The losses introduce non-local phase noise which destroys highly entangled states arising in the evolution. On the other hand, the states appearing at relatively short timescales, possessing EPRlike entanglement will survive. Applying our solutions to the recently proposed scheme to entangle two Bose-Einstein condensates, we estimate the optimal number of atoms for EPR correlations.