Scaling of Fluctuations in a Trapped Binary Condensate

We demonstrate that measurements of number fluctuations within finite cells provide a direct means to study fluctuation scaling in a trapped two-component condensate. This quantum system supports a second-order phase transition between miscible (co-spatial) and immiscible (symmetry-broken) states that is driven by a diverging susceptibility to magnetic fluctuations. As the transition is approached from the miscible side the magnetic susceptibility is found to depend strongly on the geometry and orientation of the observation cell. However, a scaling exponent consistent with that for the homogenous gas ($\gamma = 1$) can be recovered, for all cells considered, as long as the fit excludes the region in the immediate vicinity of the critical point. As the transition is approached from the immiscible side, the magnetic fluctuations exhibit a non-trivial scaling exponent $\gamma \simeq 1.30$. Experimentally, the observation cells may be formed either by considering individual imaging pixels or by combining pixels to form larger cells, and fluctuation statistics can be obtained by repeated \emph{in situ} images. Interestingly, on both sides of the transition, we find it best to extract the exponents using an observation cell that covers half of the trapped system. This implies that relatively low-resolution \emph{in situ} imaging will be adequate for the investigation of these exponents. We also investigate the gap energy and find exponents $\nu z$ = 0.505 on the miscible side and, unexpectedly, $\nu z$ = 0.60(3) for the immiscible phase.