While ultra-cold gases have been traditionally confined in harmonic potentials, trapping atoms in spatially uniform potentials has recently received growing interest as it opens up new possibilities to measure long-range correlations or study the dynamics of phase transitions by looking at the establishment of phase coherence over the entire system.

I will first describe our experimental setup which consists of a Bose-gas in an in-plane "box-potential" plus a harmonic confinement out of plane (frequency nu_z). Whereas most 2D experiments rely on completely freezing out one spatial degree of freedom (k_B*T << h*nu_z), I will discuss how dimensionality can be reduced at much higher temperatures (k_B*T > h*nu_z) by Bose condensing to lower dimensions. I will present evidence of this transverse condensation by looking at the in-plane coherence of the gas.

I will then report on a series of experiments where we study the dynamics of the coherence establishment by imposing a temperature quench on the system across the transverse condensation transition. I will show that it leads to the stochastic nucleation of topological defects (vortices) and I will compare measurements of their distribution for different cooling rate with predictions based on the Kibble-Zurek mechanism.