Post-starburst (PSB), or E+A, galaxies represent a rapid transitional phase between major, gas-rich mergers and gas-poor, quiescent early-type galaxies. Surprisingly, many PSBs have been shown to host a significant interstellar medium (ISM), despite theoretical predictions that the majority of star-forming gas should be expelled in AGN- or starburst-driven outflows. To-date, the resolved properties of this surviving ISM have remained unknown. We present high resolution ALMA continuum and CO(2$-$1) observations in six gas- and dust-rich PSBs, revealing for the first time the spatial and kinematic structure of their ISM on sub-kpc scales. We find extremely compact molecular reservoirs, with dust and gas surface densities rivaling those found in (ultra-)luminous infrared galaxies. We observe spatial and kinematic disturbances in all sources, with some also displaying disk-like kinematics. Estimates of the internal turbulent pressure in the gas exceed those of normal star-forming disks by 2$-$4 orders of magnitude, and rival the turbulent gas found in local interacting galaxies, such as the Antennae. Though the source of this high turbulent pressure remains uncertain, we suggest that the high incidence of tidal disruption events (TDEs) in PSBs could play a role. The star formation in these PSBs turbulent central molecular reservoirs is suppressed, forming stars $<$10% as efficiently as galaxies with similar gas surface densities. The fall of star formation in these galaxies was not precipitated by complete gas expulsion or redistribution. Rather, this high-resolution view of PSBs ISM indicates that star formation in their remaining compact gas reservoirs is suppressed by significant turbulent heating.
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