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We study the quantum phase transition of a N two-level atomic ensembleinteracting with an optical degenerate parametric process, which can bedescribed by the finite size Dicke Hamiltonian plus counter-rotating andquadratic field terms. Analytical closed forms of the critical coupling valueand their corresponding separable ground states are derived in the weak andstrong coupling regimes. The existence of bipartite entanglement between thetwo-level-system ensemble and photon field as well as between ensemblecomponents for moderate coupling is shown through numerical analysis. Given afinite size, our results also indicate the co-existence of squeezed fields andsqueezed atomic ensembles.

Quantum phase transition of nonlinear light in the finite size Dicke Hamiltonian