Stationary three-dimensional entanglement via dissipative Rydberg pumping

We extend the recent result of a bipartite Bell singlet [A. W. Carr and M. Saffman, Phys. Rev. Lett. 111, 033607 (2013)] to a stationary three-dimensional entanglement between two-individual neutral Rydberg atoms. This proposal makes full use of the coherent dynamics provided by a Rydberg-mediated interaction and the dissipative factor originating from the spontaneous emission of a Rydberg state. The numerical simulation of the master equation reveals that both the target state negativity $\mathcal{N}\left({\widehat{\rho }}_{\infty }\right)$ and fidelity $\mathcal{F}\left({\widehat{\rho }}_{\infty }\right)$ can exceed 99.90%. Furthermore, a steady three-atom singlet state $|S_3\rangle$ is also achievable based on the same mechanism.

Figure 1

Level diagram of two identical Rydberg atoms. The transition from ground state $|g_{L(0,R)}\rangle$ to the Rydberg state $|e_{L(0,R)}\rangle$ is driven by a classical field with Rabi frequency ${\Omega }_{L(0,R)}$, detuned by $-\Delta$. The resonant couplings between ground states are realized by two microwave fields with Rabi frequencies ${\omega }_{L0}$ and ${\omega }_{0R}$, respectively. For convenience, we assume that the Rydberg interaction strengths satisfy ${\mathcal{U}}_{LL}={\mathcal{U}}_{00}={\mathcal{U}}_{RR}=2\mathcal{U}$ and ${\mathcal{U}}_{0L}={\mathcal{U}}_{L0}={\mathcal{U}}_{0R}={\mathcal{U}}_{R0}={\mathcal{U}}_{LR}={\mathcal{U}}_{RL}=\mathcal{U}$. The excited state can spontaneously decay into three ground states with branching ratio $\gamma /3$.

Figure 2

Negativity $\mathcal{N}\left({\widehat{\rho }}_{\infty }\right)$ versus single-photon detuning parameter $\Delta$ and spontaneously decay rate $\gamma$. The other parameters are $\Omega /2\pi =0.02$ MHz, $\mathcal{U}=2\Delta$, and $\omega =3{\Omega }^2/4\Delta$.

Figure 3

Dependence of steady-state fidelity $\mathcal{F}\left({\widehat{\rho }}_{\infty }\right)=\langle \Psi |{\widehat{\rho }}_{\infty }|\Psi \rangle$ on the single-photon detuning parameter $\Delta$. The other parameters are $\Omega /2\pi =0.02$ MHz, $\gamma =1$ kHz, $\mathcal{U}=2\Delta$, and $\omega =3{\Omega }^2/4\Delta$. The inset shows the time evolutions of populations for an initially mixed state corresponding to $\Delta /2\pi =0.5$ MHz.

Figure 4

Fourth-order Runge-Kutta numerical simulation of fidelity for creation of the three-atom singlet state $|S_3\rangle$, corresponding to the parameters $\Omega /2\pi =0.02$ MHz, $\gamma =1$ kHz, $\mathcal{U}=0.2\Delta$, $\omega =3{\Omega }^2/4\Delta$, and $\Delta /2\pi =0.5$ MHz. The initial state is selected as $|g_L{g}_L{g}_L\rangle$ and the fidelity approaches around 80% after 300 ms.