Controlling phase separation of a two-component Bose-Einstein condensate by confinement

We point out that the widely accepted condition ${g}_{11}{g}_{22}<{g}_{12}^{2}$ for phase separation of a two-component Bose-Einstein condensate is insufficient if kinetic energy is taken into account, which competes against the intercomponent interaction and favors phase mixing. Here ${g}_{11}$, ${g}_{22}$, and ${g}_{12}$ are the intra- and intercomponent interaction strengths, respectively. Taking a $d$-dimensional infinitely deep square well potential of width $L$ as an example, a simple scaling analysis shows that if $d=1$ ($d=3$), phase separation will be suppressed as $L\ensuremath{\rightarrow}0$ ($L\ensuremath{\rightarrow}\ensuremath{\infty}$) whether the condition ${g}_{11}{g}_{22}<{g}_{12}^{2}$ is satisfied or not. In the intermediate case of $d=2$, the width $L$ is irrelevant but again phase separation can be partially or even completely suppressed even if ${g}_{11}{g}_{22}<{g}_{12}^{2}$. Moreover, the miscibility-immiscibility transition is turned from a first-order one into a second-order one by the kinetic energy. All these results carry over to $d$-dimensional harmonic potentials, where the harmonic oscillator length ${\ensuremath{\xi}}_{\mathrm{ho}}$ plays the role of $L$. Our finding provides a scenario of controlling the miscibility-immiscibility transition of a two-component condensate by changing the confinement, instead of the conventional approach of changing the values of the $g$'s.

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