Instability and entanglement of the ground state of the Dicke model

Abstract

Using tools of quantum information theory we show that the ground state of the Dicke model exhibits an infinite sequence of instabilities (quantum-phase-like transitions). These transitions are characterized by abrupt changes of the bi-partite entanglement between atoms at critical values $\kappa_j$ of the atom-field coupling parameter $\kappa$ and are accompanied by discontinuities of the first derivative of the energy of the ground state. We show that in a weak-coupling limit ($\kappa_1\leq \kappa \leq \kappa_2$) the Coffman-Kundu-Wootters (CKW) inequalities are saturated which proves that for these values of the coupling no intrinsic multipartite entanglement (neither among the atoms nor between the atoms and the field) is generated by the atom-field interaction. We analyze also the atom-field entanglement and we show that in the strong-coupling limit the field is entangled with the atoms so that the von Neumann entropy of the atomic sample (that serves as a measure of the atom-field entanglement) takes the value $S_A={1/2}\ln (N+1)$. The entangling interaction with atoms leads to a highly sub-Poissonian photon statistics of the field mode.