developed a variational model to corroborate the phase diagram of the shifted immiscibility phase transition. Such a shift of $g_{12}^{\rm c}$ is crucial for reaching the regime of enhanced quantum fluctuations. We find excellent qualitative and good quantitative agreement between the numerical findings and the variational model, and the latter should be useful when scouting for experimentally relevant regimes. We numerically performed BdG calculations to show that the enhanced quantum spin fluctuations should be observable using either spin or single-component Bragg spectroscopy for two-component BECs.

This work opens a number of possibilities for future research. An obvious direction is for experiments to quantitatively test beyond-mean-field theories. Since the size of the quantum fluctuations is tunable, by adjusting $g_{12}$ to approach the immiscibility transition, it would be interesting to see at which point, for example, BdG theory breaks down. It is also an open question as to how quantum fluctuations might shift the immiscibility transition. Moreover, the proposed regime could be useful for quantitative comparisons between beyond-mean-field theories with Monte Carlo simulations. Another intriguing direction would be to go beyond BdG theory to study the quantum critical physics on approach to the immiscibility phase transition.

As a final note, the enhanced 