Many-Particle Entanglement Criterion for Superradiantlike States

We derive a many-particle entanglement criterion for mixed states using a relation between single-mode and many-particle nonclassicalities. The criterion relies on the measurement of collective spin observables. It works very well not only in the vicinity of the Dicke states, but also for the superpositions of Dicke states: superradiant ground states of finite or infinite number of particles and time evolution of single-photon superradiance from an extended sample where random phases appear. We also obtain a criterion for ensemble-field entanglement, which is successful for such kinds of states. We also observe an interesting phenomenon: even though the collective excitation of this many-particle system has a sub-Poissonian character, which results in entanglement, the wave function displays bunching.

Figure 1

Linear entropy $Q$ [63, 64], many-particle entanglement criterion by Duan [17], and the new many-particle inseparability criterion ${\xi }_{\mathrm{new}}$ successfully predict the entanglement in Dicke states $|S,m⟩$. $Q,{\xi }_{\text{Duan}}>0$, and ${\xi }_{\mathrm{new}}<0$ imply entanglement.

Figure 2

(a) The number of photons and the excitation of the ensemble in the ground state of the Dicke Hamiltonian (3). Above the critical atom-photon coupling strength, $g>g_c$, superradiant phase transition occurs. (b) Many-particle entanglement. Linear entropy $Q$ and the new criterion ${\xi }_{\mathrm{new}}$ (squeezing in $⟨(\mathrm{\Delta }\widehat{\mathcal{R}}{)}^2⟩$) accompanies the order parameters of the phase transition. $Q>0$ and ${\xi }_{\mathrm{spin}}$, ${\xi }_{\mathrm{new}}<0$ imply entanglement. ${\xi }_{\text{Duan}}$ (not plotted) cannot witness the entanglement.

Figure 3

Ensemble-field entanglement in the ground state of the Dicke Hamiltonian (3) for finite or infinite number of particles. $\mu <0$ witnesses the entanglement. ${\mu }_{\mathrm{new}}^{\mathrm{HZ}}$ is obtained from the HZ [30] criterion. ${\mu }_{\mathrm{new}}^{\mathrm{SR}}$ is obtained from the stronger form of the HZ criterion [76].

Figure 4

Temporal behavior of many-particle entanglement (${\xi }_{\mathrm{new}}$) and ensemble-field entanglement (${\mu }_{\mathrm{new}}$) for single-photon superradiance of $N=2000$ atoms placed randomly in a sphere larger than wavelength. (${\mu }_{\mathrm{new}}$ is scaled for visual purposes.)