such quantum spin fluctuations should be experimentally observable via either spin or single-component Bragg spectroscopy. The advantage of Bragg spectroscopy is that thermal contributions cancel, to first order [@Brunello2001; @Blakie2002], resulting in the observation of the quantum fluctuations. Moreover, we develop a variational model and use this to analytically predict the shift of the critical point, providing a simple means for finding suitable regimes of enhanced quantum spin fluctuations. We find good agreement between the predictions of the variational model and the numerical results.

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Formalism
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System and Parameters {#Sec:System}
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We investigate a binary condensate for which interspecies interconversion is prohibited and the populations are fixed. The trapping potential is harmonic $V({\mathbf{x}}) = m(\omega_x^2x^2 + \omega_y^2y^2+\omega_z^2z^2)/2$, where $\omega_j$ represents the trapping frequencies and $m$ is the mass, which we take to be the same for both components. We consider a cylindrically symmetric trap, $\omega_x=\omega_y\equiv\omega_\rho$, in the quasi-2D regime, $\omega_z \gg \omega_\rho$, where the $z$ direction is well described by the harmonic oscillator ground state. Scattering is still three-dimensional, i.e. $a_{\alpha\beta}<<a_z$, where $a_{\alpha\beta}$ is the s-wave scattering length between components $\alpha$ and $\beta$, and $a_z = \sqrt{\hbar/ m\omega_z}$. From here on we work with the planar coordinate, ${\boldsymbol\rho}= \{x,y\}$, 