Two-component Bose-Einstein condensate (${\psi }_1,{\psi }_2$)

Two separately conserved bosonic species have two condensate fields and independent chemical potentials. A contact interaction energy density $U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ gives ${\mu }_1=U_{11}{n}_1+U_{12}{n}_2$ and ${\mu }_2=U_{12}{n}_1+U_{22}{n}_2$ in a uniform mixture. Coherent interconversion would change the conservation constraints and cannot be included while retaining independent particle-number constraints unchanged.

In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.