Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 83 / 4

Since each species number $N_i=\int |{\psi }_i{|}^2{d}^{3}x$ is separately conserved, minimize $E-{\mu }_1{N}_1-{\mu }_2{N}_2$ with two independent Lagrange multipliers. Variation with respect to each complex-conjugate field, with boundary terms removed, givesThese are the two stationary Gross–Pitaevskii equations; the time-dependent equilibrium fields are $e^{-i{\mu }_{i}t/\hslash }{\psi }_i(\mathbf{x})$. The cross term contributes $U_{12}|{\psi }_j{|}^2{\psi }_i$ to each equation, with no factor two, while differentiating the self-interaction term cancels its factor $1/2$. Separate number conservation does not require the two chemical potentials to be equal.

For an occupied uniform two-component Bose-Einstein condensate, the gradient and trapping terms vanish. Dividing each stationary equation by its nonzero field givesThe same relations follow by differentiating the uniform interaction-energy density $e=U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ with respect to each number density. When $D=U_{11}{U}_{22}-U_{12}^2\ne 0$, the inverse relations areThese describe a genuinely overlapping mixture only when both densities are nonnegative. An unoccupied component cannot be divided out of its stationary field equation; it requires the appropriate constrained equilibrium condition instead.

At fixed particle numbers, the perturbations satisfy $\int \delta n_i{d}^{3}x=0$. Expanding the uniform interaction energy, its linear term is $\int ({\mu }_1\delta n_1+{\mu }_2\delta n_2)d^{3}x=0$, and its exact quadratic change in this contact-interaction approximation isCompleting the square givesHence the miscibility criterion for a two-component condensate isThese are precisely strict positive definiteness of the interaction matrix, and make every nonzero admissible long-wavelength density perturbation cost energy. With both equilibrium densities positive, a negative eigenvalue would instead allow a small sinusoidal perturbation along its eigenvector, with zero mean and negative energy change. The omitted positive gradient term cannot cure arbitrarily long wavelengths in an unbounded system.

If cross-repulsion exceeds the geometric mean of the self-repulsions, the unstable direction favors separation of species; sufficiently strong cross-attraction favors a joint density increase and collapse. Equality gives a zero-cost direction at this order, so it is marginal rather than strictly stable. Constraints near an absent component or finite-system wavelength restrictions require a separate boundary analysis.

Set $A_i=m_i{\omega }_i^2/2$, so $V_i=A_i{r}^2$, and $D=U_{11}{U}_{22}-U_{12}^2$. In the overlap region, the Thomas–Fermi approximation for a condensate neglects the gradient terms and requiresInverting this matrix proves the unambiguous profilesUsing $R_i^2={\mu }_i/A_i$ and introducingthese Thomas–Fermi profiles for unequal condensate trap curvatures becomeThere is a genuine printed error: the PDF puts $\lambda$ rather than $1/\lambda$ in the second bracket as well. In the first bracket its coefficient requires $\lambda =A_2/A_1$, while in the second it requires $\lambda =A_1/A_2$; for nonzero cross-coupling these cannot both hold unless the curvatures are equal. The TeX additionally drops both occurrences of $\lambda$. The matrix-inverse formulas above give the corrected general answer directly.

For example, $U_{11}=U_{22}=1$, $U_{12}=1/2$, ${\mu }_1={\mu }_2=2$, $A_1=1$, $A_2=2$ give the physical overlap profiles $n_1=4/3$, $n_2=4/3-2r^2$. Using $\lambda =2$ in both printed brackets would instead give $n_2=4/3$, which fails the second equilibrium equation at every $r\ne 0$ in the overlap region.

Only the region where both computed densities are positive belongs to this overlap solution. Beyond its edge, a remaining species has $n_i=({\mu }_i-A_i{r}^2)/U_{ii}$ where positive, and both vanish outside the support; simply retaining negative values from the coupled inverse is unphysical. The actual chemical potentials are determined by the two normalization integrals. If the reference uncoupled radii are defined at the coupled solution's chemical potentials, the identities $R_i^2={\mu }_i/A_i$ are algebraic scales, not the actual interacting-cloud edges; holding particle numbers fixed and removing cross-coupling generally changes those chemical potentials. The approximation also fails in the healing regions at the edges.