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We study the dynamical, momentum dependent two- and four-spin responsefunctions in doped and undoped 1D cuprates, as probed by resonant inelasticx-ray scattering, using an exact numerical diagonalization procedure. In theundoped $t-J$ system the four-spin response vanishes at $\pi$, whereas thetwo-spin correlator is peaked around $\pi/2$, with generally larger spectralweight. Upon doping spectra tend to soften and broaden, with a transfer ofspectral weight towards higher energy. However, the total spectral weight andaverage peak position of either response are only weakly affected by doping upto a concentration of 1/8. Only the two-spin response at $\pi$ changesstrongly, with a large reduction of spectral weight and enhancement ofexcitation energy. At other momenta the higher-energy, generic features of themagnetic response are robust against doping. It signals the presence of strongshort-range antiferromagnetic correlations, even after doping mobile holes intothe system. We expect this to hold also in higher dimensions.

Doping dependence of magnetic excitations of 1D cuprates as probed by Resonant Inelastic x-ray Scattering