The multipartite Greenberger-Horne-Zeilinger (GHZ) states are indispensable elements for various quantum information processing tasks. Here we put forward two deterministic proposals to dissipatively prepare tripartite GHZ states in a neutral atom system. The first scheme can be considered as an extension of a recent work [T. M. Wintermantel, Y. Wang, G. Lochead, textit{et al}, {Phys. Rev. Lett. textbf{124}, 070503 (2020)}]. By virtue of the polychromatic driving fields and the engineered spontaneous emission, a multipartite GHZ state with odd numbers of atoms are generated with a high efficiency. This scheme effectively overcomes the problem of dependence on the initial state but sensitive to the decay of Rydberg state. In the second scenario, we exploit the spontaneous emission of the Rydberg states as a resource, thence a steady tripartite GHZ state with fidelity around $98%$ can be obtained by simultaneously integrating the switching driving of unconventional Rydberg pumping and the Rydberg antiblockade effect.
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