One-step achievement of robust multipartite Greenberger-Horne-Zeilinger state and controlled-phase gate via Rydberg interaction

We present a proposal for generation of a robust tripartite Greenberger-Horne-Zeilinger state among three-individual neutral Rydberg atoms. By modulating the relation between two-photon detuning and Rydberg interaction strength $U_{ij}(r)$, an effective Raman coupling is obtained between the hyperfine ground states $|F=2,M=2\rangle$ of three $^{87}$Rb atoms and the Rydberg states $|rrr\rangle$ via the third-order perturbation theory. This method is also capable of implementing a three-qubit controlled-phase gate with each qubit encoded into the hyperfine ground states $|F=1,M=1\rangle$ and $|F=2,M=2\rangle$. As an extension, we generalize our scheme to the case of multipartite GHZ state and quantum gate in virtue of high-order perturbation theory.