Dissipative preparation of steady Greenberger-Horne-Zeilinger states for Rydberg atoms with quantum Zeno dynamics

Inspired by a recent work [Reiter, Reeb, and S{\o}rensen, Phys. Rev. Lett. {\bf117}, 040501 (2016)], we present a simplified proposal for dissipatively preparing a Greenberger-Horne-Zeilinger (GHZ) state of three Rydberg atoms in a cavity. The $Z$ pumping is implemented under the action of the spontaneous emission of $\mathrm{\Lambda }$-type atoms and the quantum Zeno dynamics induced by strong continuous coupling. In the meantime, a dissipative Rydberg pumping breaks up the stability of the state $\mathrm{\mid }{\mathrm{G}\mathrm{H}\mathrm{Z}}_{+}⟩$ in the process of $Z$ pumping, making $\mathrm{\mid }{\mathrm{G}\mathrm{H}\mathrm{Z}}_{-}⟩$ be the unique steady state of system. Compared with the former scheme, the number of driving fields acting on atoms is greatly reduced and only a single-mode cavity is required. The numerical simulation of the full master equation reveals that a high fidelity $\sim 98\mathrm{\%}$ can be obtained with the currently achievable parameters in the Rydberg-atom-cavity system.