at $T=0$ they do not [@Bisset2015; @Bienaime2016]. In fact, they instead approach the noninteracting, quantum shot-noise limit [@Astrakharchik2007; @Klawunn2011]. This occurs because at the transition $a_{12}^{\rm c,TD}$ the interactions have no preference for miscibility over immiscibility, and so the spin fluctuations behave as if the BEC was noninteracting.

We demonstrate that the situation changes dramatically, however, if one enters the regime in which quantum pressure becomes important. Here the critical point $a_{12}^{\rm c}$ is shifted higher than the thermodynamic-limit prediction, i.e. $a_{12}^{\rm c}>a_{12}^{\rm c,TD}$, thus opening a gap [@Navarro2009; @Wen2012]. For such a system, if the intercomponent scattering length lies within this gap, i.e. $a_{12}^{\rm c,TD}<a_{12}<a_{12}^{\rm c}$, then quantum spin excitations experience an effective attractive interaction, in the sense that they act to decrease the interaction energy. We show that the resulting spin fluctuations are then greatly enhanced. An intuitive picture for the physics at play: the trap pins the two components together, forcing miscibility of the ground state solution within the regime where the interactions would otherwise prefer immiscibility. Quantum spin fluctuations are then enhanced as a reward for lowering the interaction energy.

By numerically solving the Gross-Pitaevskii equations (GPE) and Bogoliubov de Gennes (BdG) equations, we demonstrate that 