even masses, to be equal. A possible complication for the imbalanced case, though, is that inhomogeneous trapping potentials can cause the binary mixture to become immiscible even when $g_{12}<\sqrt{g_{11}g_{22}}$ [@Lee2016], hence shifting $g_{12}^{\rm c}$ in the wrong direction. Fortunately, this negative $g_{12}^{\rm c}$ shift can be reversed by imbalancing the atom numbers in favor of the component with the strongest interactions. In any case, the enhanced quantum pressure in the regime of interest will act to raise $g_{12}^{\rm c}$ as required.

Although the flattened trap geometry considered in this paper has some advantages, such as for [*in situ*]{} imaging of the fluctuations and numerical tractability, our results should be qualitatively extendible to more spherical geometries. In fact, loosening the trap along the tight direction should allow the regime of enhanced quantum fluctuations to extend to larger atom numbers.

Not only are quantum spin fluctuations enhanced on approach to the transition, but so too are thermal fluctuations [@Bisset2015]. Fortunately, though, Bragg spectroscopy provides a window through which one can selectively observe the quantum contribution, even at finite $T$. To appreciate this, we estimate the temperature that would otherwise be required for quantum fluctuations to dominate. The excitations primarily responsible for enhanced 