The realisation of strongly correlated quantum many-body systems using ultracold atoms has enabled the direct observation and control of fundamental quantum effects beyond the reach of traditional solid-state materials. A prominent example in this respect is the transition from a superfluid to a Mott insulator, occurring when interactions between particles on a lattice dominate over their kinetic energy.

Among the unique opportunities offered by ultracold atom systems, the fact that the typical distance between two particles is of the order of half a micrometer makes it in principle possible to optically resolve the position of each individual atoms. Recently, such a high-resolution in-situ imaging could be implemented in new experimental setups, in our group and in the group of Markus Greiner in Harvard. This new probing technique opens the way to radically new type of measurements. It allows one to resolve the spatial structure imposed by the confining potential as well as to directly observe non-local correlations, which lie at the heart of many quantum states of matter.

In this talk, I will first describe our imaging capabilities and illustrate them with observations of bosonic Mott insulators in the so-called « atomic limit ». I will then present some very recent observations of quantum fluctuations close to the superfluid-to-Mott insulator transition in one and two dimensional geometries. These fluctuations appear as correlated particle-hole pairs in the Mott insulating regime and can be directly detected in our experiment. Finally, I will show that we can also reveal the presence of non-local correlations, which can be used to classify classify many-body quantum phases that are not amenable to a description through a local order parameter, typically used in the Landau paradigm of phase transitions.