We propose a pump-probe set-up to analyse the properties of the collective excitation spectrum of a spinor polariton fluid. By using a linear response approximation scheme, we carry on a complete classification of all excitation spectra, as well as their intrinsic degree of polarisation, in terms of two experimentally tunable parameters only, the mean-field polarisation angle and a rescaled pump detuning. We evaluate the system response to the external probe, and show that the transmitted light can undergo a spin rotation along the dispersion for spectra that we classify as diffusive-like. We show that in this case, the spin flip predicted along the dispersion is enhanced when the system is close to a parametrically amplified instability.
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