and time, i.e. $C=-\mu_\alpha^\mathcal{V} t/\hbar$.

As detailed in Appendix \[Sec:AppendixAna\], we use the Euler-Lagrange equations to derive equations of motion for the ansatz parameters. We then go on to derive analytical stationary-state predictions for the density, $n_\alpha^\mathcal{V}=|\psi_\alpha^\mathcal{V}|^2$, in the miscible regime \[Eqs. (\[Eq:Amisc\])-(\[Eq:muMisc\])\] and a set of transcendental equations for the immiscible regime \[Eqs. (\[Eq:cp0Imisc\])-(\[Eq:muImisc\])\], both of which we will compare with our numerical results. Finally, the variational prediction for the critical value of the intercomponent interaction strength, above which the system becomes immiscible, takes the form $$g_{12}^{\rm c,\mathcal{V}} = g + 2\pi . \label{Eq:gc}$$ The thermodynamic-limit result $ g_{12}^{\rm c,TD}=\sqrt{g_{11}g_{22}}=g$ is recovered for large particle number or intracomponent scattering length (recall that $g_{\alpha\beta}=N_\beta\sqrt{8\pi}a_{\alpha\beta}/a_z$).

Two-Component Structure Factor {#Sec:SFformalism}
------------------------------

Bragg spectroscopy furnishes a means by which to selectively probe the $T=0$ component of the structure factor. Naturally, this provides a useful avenue for the study of quantum fluctuations by *seeing through* the thermal fluctuations. More specifically, Bragg spectroscopy measures the imaginary part of the response function and this relates to the dynamic structure factor (which will be defined shortly) as $$\mathrm{Im}[\chi({\mathbf{k}_\rho},\omega)] = -\frac{\pi}{\hbar}[S({\mathbf{k}_\rho},\omega)-S(-{\mathbf{k}_\rho},-\omega)] ,$$ where the finite-temperature contributions cancel to leading order [@Brunello2001; @Blakie2002]. Several groups have proposed schemes for 