\int_0^{cT} \hat s_i (ct-\xi,t) \,d\xi$. For a strong classical amplitude populating the $x$-polarized light component, the meter system is the $y$-polarized quantum field and its quadrature operators can be defined as $\hat X_L \equiv \hat S_y / (c\sqrt{N_p/2})$ and $\hat P_L \equiv \hat S_z / (c\sqrt{N_p/2})$, where $N_p$ is the total photon number in the probe pulse. Initially, the meter is in the vacuum state and $(\Delta X_L^{\text{in}})^2 = (\Delta P_L^{\text{in}})^2 = \tfrac12$.

Regarding the atomic ensemble, we start from the generic reference product state $\bigotimes^N {|{\phi_0}\rangle}$ and assume that the system stays in the vicinity of this state. We further assume that the polarization in this reference state points in the $x$ direction ($\langle J_y \rangle_0 = \langle J_z \rangle_0 = 0$) and that ${|{\phi_0}\rangle}$ is not an eigenstate of $\hat J_z$. According to our results in Sec. \[sec:linarization\], we assign to $\hat J_z$ a collective atomic oscillator with quadrature variable $\hat P_1 = \hat J_z / {\sqrt{2(\Delta J_z)_0^2}}$. This is the atomic oscillator the light is coupled to and, for the moment, it is enough to consider only this mode. It is initially in the vacuum state with $(\Delta X_1^{\text{in}})^2 = (\Delta P_1^{\text{in}})^2 = \tfrac12$. In this generalized 