  Non-equilibrium quantum dynamics represents an emerging paradigm for
condensed matter physics, quantum information science, and statistical
mechanics. Strongly interacting Rydberg atoms offer an attractive platform to
study driven-dissipative dynamics of quantum spin models with long-range order.
Here, we explore the conditions under which stationary many-body entanglement
persists with near-unit fidelity and high scalability. In our approach,
coherent many-body dynamics is driven by Rydberg-mediated laser transitions,
while atoms at the lattice boundary reduce the entropy of the many-body state.
Surprisingly, the many-body entanglement is established by continuously
evolving a locally dissipative Rydberg system towards the steady-state, as with
optical pumping. We characterize the dynamics of multipartite entanglement in a
1D lattice by way of quantum uncertainty relations, and demonstrate the
long-range behavior of the stationary entanglement with finite-size scaling,
reaching "hectapartite" entanglement under experimental conditions. Our work
opens a route towards dissipative preparation of many-body entanglement with
unprecedented scaling behavior.
