Quasi-adiabatic preparation of antiferromagnetic-like state of Rydberg excitations of atoms in a lattice*

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. We find that 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 involving the minimal energy gap, extrapolation of which allows us to estimate the preparation fidelity for much larger systems.