Coherence properties are central to quantum systems and are at the heart of phenomena such as superconductivity. In this work, we studied coherence properties of an ultracold Bose gas in a two-dimensional optical lattice across the thermal phase transition. To infer the phase coherence and phase fluctuation profiles, we used direct matter-wave imaging of higher Talbot revivals and introduced a phase microscope based on a site-resolved mapping of phase fluctuations to density fluctuations during matter-wave imaging. We observed the algebraic decay of the phase correlations in the superfluid phase and a linear temperature increase of the exponent. These techniques may enable studying coherence properties in strongly correlated quantum systems with full spatial resolution.
Brüggenjürgen et al. (Thu,) studied this question.