High resolution spectral imaging of CO(7-6), [CI](2-1) and continuum of three high-z lensed dusty star-forming galaxies using ALMA

High-redshift dusty star-forming galaxies with very high star formation rates (500 -- 3000 M$_{\odot}$ yr$^{-1}$) are key to understanding the formation of the most extreme galaxies in the early Universe. Characterising the gas reservoir of these systems can reveal the driving factor behind the high star formation. Using molecular gas tracers like high-J CO lines, neutral carbon lines and the dust continuum, we can estimate the gas density and radiation field intensity in their interstellar media. In this paper, we present high resolution ($\sim$0.4$^{\prime\prime}$) observations of CO(7-6), [CI](2-1) and dust continuum of 3 lensed galaxies from the SPT-SMG sample at $z\sim$ 3 with the Atacama Large Millimeter/submillimeter Array. Our sources have high intrinsic star-formation rates ($>$850 M$_{\odot}$yr$^{-1}$) and rather short depletion timescales ($<$100 Myr). Based on the L$_{[\rm CI](2-1)}$/L$_{\rm CO(7-6)}$ and L$_{[\rm CI](2-1)}$/L$_{\rm IR}$ ratios, our galaxy sample has similar radiation field intensities and gas densities compared to other submillimetre galaxies. We perform visibility-based lens modelling on these objects to reconstruct the kinematics in the source plane. We find that the cold gas masses of the sources are compatible with simple dynamical mass estimates using ULIRG-like values of the CO-H$_2$ conversion factor $\alpha_{\rm CO}$ but not Milky Way-like values. We find diverse source kinematics in our sample: SPT0103-45 and SPT2147-50 are likely rotating disks while SPT2357-51 is possibly a major merger. The analysis presented in the paper could be extended to a larger sample to determine better statistics of morphologies and interstellar medium properties of high-$z$ dusty star-forming galaxies.