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High-redshift dusty star-forming galaxies with very high star formation rates (500−3000 M ⊙ 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 such as, 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 (∼0.4″) observations of CO(7−6), [CI](2−1), and dust continuum of three lensed galaxies from the South pole telescope – sub-millimetre galaxies (SPT-SMG) sample at z  ∼ 3 with the Atacama Large Millimetre/submillimetre Array. Our sources have high intrinsic star formation rates (> 850 M ⊙ yr −1 ) and rather short depletion timescales (< 100 Myr). Based on the L [CI](2−1) / L CO(7 − 6) and L [CI](2−1) / L IR ratios, our galaxy sample has similar radiation field intensities and gas densities compared to other submillimetre galaxies. We performed 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 α 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. 

ABSTRACT We present Atacama Compact Array and Atacama Pathfinder Experiment observations of the [N ii] 205 μm fine-structure line in 40 sub-millimetre galaxies lying at redshifts z = 3–6, drawn from the 2500 deg2 South Pole Telescope survey. This represents the largest uniformly selected sample of high-redshift [N ii] 205 μm measurements to date. 29 sources also have [C ii] 158 μm line observations allowing a characterization of the distribution of the [C ii] to [N ii] luminosity ratio for the first time at high redshift. The sample exhibits a median L$_{{\rm{[C\,{\small II}]}}}$/L$_{{\rm{[N\,{\small II}]}}}$ ≈ 11.0 and interquartile range of 5.0 –24.7. These ratios are similar to those observed in local (Ultra)luminous infrared galaxies (LIRGs), possibly indicating similarities in their interstellar medium. At the extremes, we find individual sub-millimetre galaxies with L$_{{\rm{[C\,{\small II}]}}}$/L$_{{\rm{[N\,{\small II}]}}}$ low enough to suggest a smaller contribution from neutral gas than ionized gas to the [C ii] flux and high enough to suggest strongly photon or X-ray region dominated flux. These results highlight a large range in this line luminosity ratio for sub-millimetre galaxies, which may be caused by variations in gas density, the relative abundances of carbon and nitrogen, ionization parameter, metallicity, and a variation in the fractional abundance of ionized and neutral interstellar medium.