  The ability to control and tune interactions in ultracold atomic gases has
paved the way towards the realization of new phases of matter. Whereas
experiments have so far achieved a high degree of control over short-ranged
interactions, the realization of long-range interactions would open up a whole
new realm of many-body physics and has become a central focus of research.
Rydberg atoms are very well-suited to achieve this goal, as the van der Waals
forces between them are many orders of magnitude larger than for ground state
atoms. Consequently, the mere laser excitation of ultracold gases can cause
strongly correlated many-body states to emerge directly when atoms are
transferred to Rydberg states. A key example are quantum crystals, composed of
coherent superpositions of different spatially ordered configurations of
collective excitations. Here we report on the direct measurement of strong
correlations in a laser excited two-dimensional atomic Mott insulator using
high-resolution, in-situ Rydberg atom imaging. The observations reveal the
emergence of spatially ordered excitation patterns in the high-density
components of the prepared many-body state. They have random orientation, but
well defined geometry, forming mesoscopic crystals of collective excitations
delocalised throughout the gas. Our experiment demonstrates the potential of
Rydberg gases to realise exotic phases of matter, thereby laying the basis for
quantum simulations of long-range interacting quantum magnets.
