the implementation of spin (and density) Bragg spectroscopy in binary condensates [@Rodriguez2002; @Bruun2006; @Carusotto2006; @Baillie2016]. The general idea is to engineer a different coupling of the Bragg lasers to each component. This has been experimentally realized recently by the Vale group [@Hoinka2012].

We find the excitations of the BEC by linearizing about the ground state using a small parameter $\eta$. This amounts to inserting the time-dependent ansatz $$\Psi_\alpha({\boldsymbol\rho}) = \left\{\psi_\alpha({\boldsymbol\rho}) + \eta \left[ u_\alpha({\boldsymbol\rho})e^{-i\omega t} + v_\alpha^*({\boldsymbol\rho})e^{i\omega^*t} \right] \right\} e^{-i\mu_\alpha t/\hbar}$$ into the GPE (Eq. \[Eq:GPE\]) and solving the resulting BdG equations. Further details are provided in Appendix \[Sec:AppendixBog\] (also see Refs. [@Pu1998A; @Timmermans1998; @Ticknor2013; @Ticknor2014]).

The $T=0$ component of the density $S_\mathcal{D}$ [@Zambelli2000] and spin $S_\mathcal{S}$ [@Abad2013; @Symes2014A] structure factors are calculated from the BdG excitations, with each excitation labeled with superscript $\kappa$. Explicitly, we evaluate $$\begin{aligned}
S_{\{\mathcal{D,S}\}} &({\mathbf{k}_\rho},\omega) = \sum_\kappa \Big| \int d^2{\boldsymbol\rho}e^{i{\mathbf{k}_\rho}\cdot{\boldsymbol\rho}} \Big([u^\kappa_1({\boldsymbol\rho})+v^\kappa_1({\boldsymbol\rho})]\psi_1({\boldsymbol\rho}) \notag \\
& \pm [u^\kappa_2({\boldsymbol\rho})+v^\kappa_2({\boldsymbol\rho})]\psi_2({\boldsymbol\rho})\Big) \Big|^2 \delta(\omega-\omega^\kappa) , \label{Eq:SF}\end{aligned}$$ where the $+~(-)$ indicates $S_\mathcal{D}$ ($S_\mathcal{S}$), $\omega^\kappa = \epsilon^\kappa/\hbar$ (for BdG energy $\epsilon^\kappa$) and we have used the planar momentum coordinate ${\mathbf{k}_\rho}= \{k_x,k_y\}$. The corresponding static structure factors relate as $S_{\{\mathcal{D,S}\}}({\mathbf{k}_\rho}) = \int d\omega S_{\{\mathcal{D,S}\}}({\mathbf{k}_\rho},\omega)$.

![Density profiles for (a) the miscible phase for $g_{12}/g_{12}^{\rm c}=0.9$, 