Robust generation of entangled state via ground-state antiblockade of Rydberg atoms

We propose a mechanism of ground-state antiblockade of Rydberg atoms, which is then exploited to prepare two-atom entangled state via three different kinds of pulses. First we use the pulses in the form of sin2 and cos2 functions and obtain a maximally entangled state at an accurate interaction time. Then the method of stimulated Raman adiabatic passage (STIRAP) is adopted for the entanglement generation, which is immune to the fluctuations of revelent parameters but requires a long time. Finally we capitalize the advantages of the former two methods and employ shortcuts to adiabatic passage (STAP) to generate the maximal entanglement. The strictly numerical simulation reveals that the current scheme is robust against spontaneous emission of atoms due to the virtual excitation of Rydberg states, and all of the above methods favor a high fidelity with the present experimental technology.

However, it should be noted that the populations of the excited Rydberg states will decrease the fidelity of entangled state due to the spontaneous emission of atoms since the lifetime of Rydberg state is finite. Very recently, Shao et al. put forward an efficient scheme of ground-state blockade for N-type Rydberg atoms by virtue of Rydberg antiblockade effect and Raman transition 32 , which averts the spontaneous emission of the excited Rydberg state, and keep the nonlinear Rydberg-Rydberg interaction (RRI) at the same time. Inspired by this scheme, in this paper, we propose a mechanism of ground-state antiblockade for Rydberg atoms, i.e., the effectively coherent Rabi oscillation between two ground states gg and ee can be achieved. As its application, we will explore three ways to implement the two-atom maximally entangled state. First, we adopt the pulses in the form of sin 2 and cos 2 functions and obtain a high-fidelity maximally entangled state at an accurate interaction time. The second method takes advantage of STIRAP which is insensitive to parameter fluctuations but needs a relatively long time. Finally, we use the shortcuts to adiabatic passage which combines the former two methods' advantages to generate entangled state. The prominent advantage of our scheme is that the quantum information is encoded into the ground states of Rydberg atoms, and the evolution process of system is robust against atomic decay for two-atom entangled state preparation.

Theoretical Model
As shown in Fig. 1, we consider two identical Rydberg atoms trapped in two separate microscopic dipole traps. The states | 〉 g and | 〉 e are the hyperfine states in the ground-state manifold, respectively, and state | 〉 r is the excited Rydberg state. One atomic transition | 〉 | 〉 ↔ | 〉 g e r ( ) is driven by a classical laser field with Rabi frequency Ω Ω′ ( ) 1 2 , detuned by ∆ 1 (−∆ 2 ), the other atomic transition | 〉 | 〉 ↔ | 〉 g e r ( ) is driven by a classical laser field with Rabi frequency Ω′ Ω ( ) 1 2 and the corresponding detuning is −∆ 2 (∆ 1 ). The Hamiltonian of the whole system can be written as  and We can deem Hamiltonian of Eq. (5) an effective Λ-type three-level system with an excited state | 〉 rr and two ground states | 〉 gg and | 〉 ee as shown in Fig. 2  where the Stark-shift terms originating from the two-photon transition are disregarded. It should be noted that in order to obtain this kind of spin squeezing-like Hamiltonian, six lasers were applied by Bouchoule et al. 33 , however, four lasers are enough in our proposal. In governed by the original Hamiltonian Ĥ I . It shows that the ground state | 〉 gg resonantly interacts with the ground state | 〉 ee under the condition of large detuning and there is nearly no population for the states | 〉 ge or | 〉 eg . In addition, from the Hamiltonian of Eq. (5), we can readily find the dark state is Here we adopted four pulses Ω 1 , Ω 1 , Ω′ 1 , Ω′ 2 .
b a where Ω = Ω + Ω a b 2 2 . Therefore, we can manipulate the evolution of quantum states with various adiabatic passages.
/ 01 0 . It is easy to find that we can obtain a high population for the state ϕ | 〉 at the time t = T/8 (T is pulse period).
Stimulated Raman adiabatic passage. We choose parameters for the laser pulses suitably to fulfill the boundary condition of the STIRAP Thus, the Rabi frequencies Ω t ( ) 1 and Ω t ( ) 2 in the original Hamiltonian Ĥ I are chosen as where Ω 0 is the peak Rabi frequency, t c is the pulse duration, and τ is the delay between the pulses. The shapes of pulses are shown in Fig. 5(a), where the parameters have been chosen as = Ω t 3000 / c 0 , = .
T t 0 2 c , and τ = . t 0 04 c . Figure 5  time is required, i.e. = Ω t 2100 / 0 for achieving the target state, and the population of the target state φ | 〉 remains unit when ≥ Ω t 2100 / 0 . Compared with the former method, the STIRAP is not restricted to an accurate interaction time but requires a relatively long time.
2 . If the detuning ν δ δ = + ′ ≠ ( 0 ) is considered as shown in Eq. in order to cancel the first two terms, and the final Hamiltonian becomes 2 . We will show below the numerical analysis of the creating the two-atom Bell state governed by the STAP. Here the Rabi frequencies Ω′ t ( ) where Ω 0 is the pulse amplitude. The forms of above pulses just correspond to Ω = Ω = − Ω ∼ t i t ( ) ( ) 28 1 2 for the original Hamiltonian Ĥ I of Eq. (1). In Fig. 6(a)

Discussion
We have illustrated how to prepare the maximally entangled state | 〉 − | 〉 gg ee ( ) / 2 in the ideal situation by manipulating pulses in different ways. However, the actual system will interact with the environment inevitably, which affects the availability of these methods. Thus it is necessary to investigate the influence of spontaneous emission of atoms on our proposal. When the dissipation is considered, the evolution of the system can be modeled by a master equation in Lindblad form 40,41 In experiments, the ground-state antiblockade model can be realized in 87 Rb atoms which are trapped in two tightly focused dipole traps 21,42  MHz. The spontaneous emission rate from the Rydberg state is γ π = × . 2 48 kHz. By substituting these values into the master equation, we find the fidelities of generating two-atom entanglement with the above three methods are all beyond 99%.
In summary, we have put forward an efficient scheme for the ground-state antiblockade of Rydberg atoms and prepare two-atom entangled state. Three kinds of pulses are exploited to obtain the maximally entangled state, and a high fidelity is achievable with the current experimental parameters. Most interestingly, this process is robust against the decoherence induced by spontaneous emission of atoms. We hope that our scheme could find some applications in the near future.   . Schematic view of atomic-level configuration for the generation of antisymmetric Bell state. | 〉 r is the Rydberg state, while | 〉 g and | 〉 e are two ground states. ∆ rr denotes the RRI strength. For atom 1, the transition | 〉 ↔ | 〉 g r is driven by a classical laser field with Rabi frequency Ω 1 and the transition | 〉 ↔ | 〉 e r is driven by a classical laser field with Rabi frequency Ω′ 2 . For atom 2, the transition | 〉 ↔ | 〉 g r is driven by a classical laser field with Rabi frequency Ω 2 and the transition | 〉 ↔ | 〉 e r is driven by a classical laser field with Rabi frequency Ω′ 1 . ∆ 1 (2) represents the corresponding detuning parameter.