The dimensionality of a system strongly affects its physical properties ; the phase transitions that take place and the type of order that arises depend on the dimension. In low dimensional systems phase coherence proves more difficult to achieve as both thermal and quantum fluctuations play a stronger role. The two-dimensional Bose fluid is of particular interest as even if full order is precluded, a residual "quasi-long" range order arises at low temperatures. Then two ingredients have a significant effect on the state of the system : (i) the finite size of a real system enables one to recover of a macroscopic occupation of a single-particle state ; (ii) the interactions between particles lead to the emergence of a non-conventional type of phase transition toward a superfluid state.
In this thesis, we present an experimental study of the two-dimensional (2D) Bose gas using two different energy landscapes to trap our atoms. In the first part, we use the spatial dependence of some local properties of an inhomogeneous gas to characterize the state of the equivalent homogeneous system. We extract its equation of state with a high accuracy from the gas density profiles and test its superfluid behavior by measuring the heating induced by a moving local perturbation. In the second part, we observe and characterize the emergence of an extended phase coherence in a 2D homogeneous gas in particular via a 3D-to-2D dimensional crossover. We investigate the dynamical establishment of the coherence via a rapid crossing of the dimensional crossover and observe topological defects in the final superfluid state. We compare our findings with the predictions for the Kibble—Zurek mechanism.