  We theoretically analyze recent experiments [G. Semeghini et al., Science
374, 1242 (2021)] demonstrating the onset of a topological spin liquid using a
programmable quantum simulator based on Rydberg atom arrays. In the experiment,
robust signatures of topological order emerge in out-of-equilibrium states that
are prepared using a quasi-adiabatic state preparation protocol. We show
theoretically that the state preparation protocol can be optimized to target
the fixed point of the topological phase -- the resonating valence bond (RVB)
state of hard dimers -- in a time that scales linearly with the number of
atoms. Moreover, we provide a two-parameter variational manifold of tensor
network (TN) states that accurately describe the many-body dynamics of the
preparation process. Using this approach we analyze the nature of the
non-equilibrium state, establishing the emergence of topological order.
