  Any state of matter is classified according to its order, and the kind of
order a physical system can posses is profoundly affected by its
dimensionality. Conventional long-range order, like in a ferromagnet or a
crystal, is common in three-dimensional (3D) systems at low temperature.
However, in two-dimensional (2D) systems with a continuous symmetry, true
long-range order is destroyed by thermal fluctuations at any finite
temperature. Consequently, in contrast to the 3D case, a uniform 2D fluid of
identical bosons cannot undergo Bose-Einstein condensation. Nevertheless, it
can form a "quasi-condensate" and become superfluid below a finite critical
temperature. The Berezinskii-Kosterlitz-Thouless (BKT) theory associates this
phase transition with the emergence of a topological order, resulting from the
pairing of vortices with opposite circulations. Above the critical temperature,
proliferation of unbound vortices is expected. Here we report the observation
of a BKT-type crossover in a trapped quantum degenerate gas of rubidium atoms.
Using a matter wave heterodyning technique, we observe both the long-wavelength
fluctuations of the quasi-condensate phase and the free vortices. At low
temperatures, the gas is quasi-coherent on the length scale set by the system
size. As the temperature is increased, the loss of long-range coherence
coincides with the onset of proliferation of free vortices. Our results provide
direct experimental evidence for the microscopic mechanism underlying the BKT
theory, and raise new questions regarding coherence and superfluidity in
mesoscopic systems.
