  We examine the adiabatic preparation of crystalline phases of Rydberg
excitations in a one-dimensional lattice gas by frequency sweep of the
excitation laser, as proposed by Pohl et al. [Phys. Rev. Lett. 104, 043002
(2010)] and recently realized experimentally by Schau{\ss} et al. [Science 347,
1455 (2015)]. We find that the preparation of crystals of a few Rydberg
excitations in a unitary system of several tens of atoms requires exceedingly
long times for the adiabatic following of the ground state of the system
Hamiltonian. Using quantum stochastic (Monte-Carlo) wavefunction simulations,
we show that realistic decay and dephasing processes affecting the atoms during
the preparation lead to a final state of the system that has only a small
overlap with the target crystalline state. Yet, the final number and highly
sub-Poissonian statistics of Rydberg excitations and their spatial order are
little affected by the relaxations.
