Observing Properties of an Interacting Homogeneous Bose--Einstein Condensate: Heisenberg-Limited Momentum Spread, Interaction Energy and Free-Expansion Dynamics

We study the properties of an atomic Bose--Einstein condensate produced in an optical-box potential, using high-resolution Bragg spectroscopy. For a range of box sizes, up to $70~\mu$m, we directly observe Heisenberg-limited momentum uncertainty of the condensed atoms. We measure the condensate interaction energy with a precision of $k_B \times 100$ pK and study, both experimentally and numerically, the dynamics of its free expansion upon release from the box potential. All our measurements are in good agreement with theoretical expectations for a perfectly homogeneous condensate of spatial extent equal to the size of the box, which also establishes the uniformity of our optical-box system on a sub-nK energy scale.