This thesis describes experimental studies regarding superfluidity in a two-dimensional gas of ultracold atoms trapped in a uniform potential. It is conceptually divided into three parts.

In the first part we offer a theoretical discussion of a superfluid, first from a general perspective and then concentrating on its distinct features when restricted to two-dimensions. One of the hallmarks of superfluidity in all dimensions, predicted by the highly successful hydrodynamic two-fluid model and observed in both liquid helium and ultracold atomic gases, is the existence of two kinds of sound excitations, the first and second sound. Unlike its three dimensional counter-part, however, superfluidity in two dimensions is associated with the pairing of vortices of opposite circulation as described by the Berezinskii-Kosterlitz-Thouless (BKT) theory, rather than the emergence of true long-range order. One of the most well-known features of BKT superfluidity is the universal jump in its superfluid density at a critical temperature without any discontinuities in the fluid’s thermodynamic properties.

In the second part we describe the experimental realisation and the characterisation of a versatile two-dimensional box trap for the confinement of a 39K atomic gas. Our apparatus is the outcome of merging a previously existing experimental setup with a series of modifications and extensive additions. Importantly for this thesis, we are able to tune the interactions of the gas with the aid of a magnetic Feshbach resonance, reaching the hydrodynamic regime where the predictions of the two-fluid model for a two-dimensional superfluid are expected to be valid.

With a homogeneous and tunable two-dimensional gas at hand, in the third part of this thesis we describe our experimental method to observe both first and second sound; the latter is seen for the first time in any two-dimensional fluid. From the two temperature-dependent measured sound speeds we deduce its superfluid density, a central quantity for a superfluid that had so far remained elusive in ultracold gases. Our results agree with BKT theory, including the prediction for the universal superfluid-density jump.