Certify vacuum origin and entanglement of analog cosmological pair creation

In a 2D BEC or a quantum fluid of light with an effective scale factor $a(t)$ set by expansion or an interaction ramp, phonons at momenta $k$ and $-k$ acquire occupation $n_k$ and anomalous correlation $c_k = \langle b_k b_{-k}\rangle$. Determine whether $\mid c_k\mid > n_k$ (non-separability for the symmetric two-mode state) holds after accounting for the initial thermal occupation, and whether $n_k$ matches the vacuum Bogoliubov coefficient $\mid\beta_k\mid^2$ for the imposed $a(t)$. An answer is a measurement establishing or excluding the inequality with stated confidence and a measured initial temperature.

Quadrature-resolved or momentum-correlation measurements of phonon pairs violating the separability bound, with the initial thermal population measured independently.