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.
