Non-equilibrium dynamics in two-dimensional quantum systems

The understanding and precise prediction of non-equilibrium quantum many-body dynamics, in particular across a critical point, remains a difficult task due to the relevance of all length scales near the critical point. Furthermore, the number of parameters required to characterise the state of the system increases exponentially with the number of particles, making the numerical investigation of such a system extremely difficult. In this thesis, we use ultracold 87Rb atoms prepared in a bilayer two-dimensional (2D) trap to probe the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in detail, both in and out of equilibrium. These experiments use a multiple-radiofrequency dressed trap, which allows dynamical control of the trapped atoms as well as the precise determination of the many-body wavefunction. For the characterisation of the 2D Bose gases using matter-wave interferometry, a novel technique was developed to obtain high contrast fringes by selective imaging of slices of the atomic cloud. This allows the observation of local fluctuations, such as phase correlation function, local vortex density and coherence full counting statistics. Utilising these observables, we have identified the BKT critical point and characterised microscopic features of harmonically-trapped 2D Bose gases in equilibrium. With this information about the system, we probe the non-equilibrium dynamics of 2D Bose gases following a quench across the BKT critical point. The system is quenched by a coherent splitting, which introduces a sudden reduction of density resulting in the quench from the superfluid to the thermal phase. We monitor the dynamics towards the vortex-proliferated state and find that the vortex-unbinding dynamics is well described by the real-time renormalisation group theory. Finally, we show preliminary results for a tunnel-coupled bilayer 2D gas, in which we probe the oscillations of the relative phase of the two layers of the superfluid. The results presented in this thesis demonstrate that the multiple-RF dressing technique is a very powerful tool for investigating quantum many-body phenomena. This paves the way for future studies of non-equilibrium critical dynamics and their description with renormalisation-group theory.