Instability and Entanglement of the Ground State of the Dicke Model

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 (${\kappa }_1\le \kappa \le {\kappa }_2$) the Coffman-Kundu-Wootters 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 show that in the strong-coupling limit the entangling interaction with atoms leads to a highly sub-Poissonian photon statistics of the field mode.

Figure 1

The energy $E$ (measured in units of $\hslash \omega$) of the ground state of the DM with $N=12$ atoms as a function of the scaled coupling parameter $\kappa /\omega$. We see a sequence of instabilities (quantum-phase-like transitions) that are marked by vertical lines corresponding to different values of the coupling ${\kappa }_j$. The first interval of values of the coupling parameter $0\le \kappa \le {\kappa }_1$ is characterized by the excitation number equal to zero. The second interval ${\kappa }_1\le \kappa \le {\kappa }_2$ is characterized by one excitation, etc. The first derivative of the ground-state energy (black line) is not a continuous function of $\kappa$ (see the inset that illustrates the behavior of the energy of the ground state around the critical point ${\kappa }_j$). In the figure we plot also a bi-partite concurrence as a function of $\kappa$. The concurrence exhibits abrupt changes at “critical” values of ${\kappa }_j$. The maximal value of bi-partite concurrence in the Dicke system appears for ${\kappa }_1\le \kappa \le {\kappa }_2$ when the ground state of the system contains just one excitation ($p=1$). Then, the value of the concurrence decreases with the increase of $p$. Nevertheless, even for $p\gg 12$ the concurrence is nonzero.

Figure 2

The bi-partite concurrence of the ground state of the DM as a function of number of atoms $N$ and the number of excitations $p$. Since the bi-partite concurrence decreases as $1/N$, we plot the “total atomic bi-partite entanglement” ${\tau }_A=C^{2}N(N-1)/2$, where the factor $N(N-1)/2$ corresponds to the number of possible pairs in the atomic sample. We see that ${\tau }_A$ as a function of $N$ takes the largest value for $p=1$ and in the large $N$ limit tends to the value $1/2$. We see that even in the strong-coupling case when a $p=N$ arbitrary pair of atoms in the sample is entangled. This is reflected by a nonzero value of ${\tau }_A$ [see the diagonal line in the $(p,N)$ plane of the figure]. In this case, for $N$ large enough the field exhibits sub-Poissonian photon statistics with the mean-photon number $\overline{n}=2N/3$ and the dispersion $\Delta =\sqrt{5N/26}$. We conclude that the larger the number of excitations $p$ the smaller is the amount of bi-partite entanglement between atoms. In the strong-coupling limit a strong atom-field entanglement is established that is quantified by the von Neumann entropy $S_A=S_F=\frac{1}{2}\ln (N+1)$.