We investigate the prevalence of galactic-scale outflows in post-starburst (PSB) galaxies at high redshift ($1 < z < 1.4$), using the deep optical spectra available in the UKIDSS Ultra Deep Survey (UDS). We use a sample of $sim40$ spectroscopically confirmed PSBs, recently identified in the UDS field, and perform a stacking analysis in order to analyse the structure of strong interstellar absorption features such as Mg ii ($lambda2800$ Ang.). We find that for massive ($M_* > 10^{10}rm,M_{odot}$) PSBs at $z > 1$, there is clear evidence for a strong blue-shifted component to the Mg ii absorption feature, indicative of high-velocity outflows ($v_{rm out}sim1150pm160rm,km,s^{-1}$) in the interstellar medium. We conclude that such outflows are typical in massive PSBs at this epoch, and potentially represent the residual signature of a feedback process that quenched these galaxies. Using full spectral fitting, we also obtain a typical stellar velocity dispersion $sigma_*$ for these PSBs of $sim200rm,km,s^{-1}$, which confirms they are intrinsically massive in nature (dynamical mass $M_{rm d}sim10^{11}rm,M_{odot}$). Given that these high-$z$ PSBs are also exceptionally compact ($r_{rm e}sim1$--$2rm,kpc$) and spheroidal (Sersic index $nsim3$), we propose that the outflowing winds may have been launched during a recent compaction event (e.g. major merger or disc collapse) that triggered either a centralised starburst or active galactic nuclei (AGN) activity. Finally, we find no evidence for AGN signatures in the optical spectra of these PSBs, suggesting they were either quenched by stellar feedback from the starburst itself, or that if AGN feedback is responsible, the AGN episode that triggered quenching does not linger into the post-starburst phase.
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