  Ultracold atomic gases have been used extensively in recent years to realize
textbook examples of condensed matter phenomena. Recently, phase transitions to
ordered structures have been predicted for gases of highly excited, 'frozen'
Rydberg atoms. Such Rydberg crystals are a model for dilute metallic solids
with tunable lattice parameters, and provide access to a wide variety of
fundamental phenomena. We investigate theoretically how such structures can be
created in four distinct cold atomic systems, by using tailored
laser-excitation in the presence of strong Rydberg-Rydberg interactions. We
study in detail the experimental requirements and limitations for these
systems, and characterize the basic properties of small crystalline Rydberg
structures in one, two and three dimensions.
