  We examine the adiabatic preparation of spatially-ordered Rydberg excitations
of atoms in finite one-dimensional lattices by frequency-chirped laser pulses,
as realized in a number of recent experiments simulating quantum Ising model.
Our aims are to unravel the microscopic mechanism of the phase transition from
the unexcited state of atoms to the antiferromagnetic-like state of Rydberg
excitations by traversing an extended gapless phase, and to estimate the
preparation fidelity of the target state in a moderately sized system amenable
to detailed numerical analysis. We find that, in the basis of the bare atomic
states, the system climbs the ladder of Rydberg excitations predominantly along
the strongest-amplitude paths towards the final ordered state. We show that,
despite its complexity, the interacting many-body system can be described as an
effective two-level system involving a pair of lowest-energy instantaneous
collective eigenstates of the time-dependent Hamiltonian. The final preparation
fidelity of the target state can then be well approximated by the Landau-Zener
formula, while the nonadiabatic population leakage during the passage can be
estimated using a perturbative approach applied to the instantaneous collective
eigenstates.
