  Cold gas experiments can be tuned to achieve strongly-interacting regimes
such as that of low-density neutron matter found in neutron-stars' crusts. We
report $T$=0 diffusion Monte Carlo results (i) for the ground state of both
spin-1/2 fermions with short-range interactions and low-density neutron matter
in a cylindrical container, and (ii) properties of these systems with a vortex
line excitation. We calculate the equation of state for cold atoms and
low-density neutron matter in the bulk systems, and we contrast it to our
results in the cylindrical container. We compute the vortex line excitation
energy for different interaction strengths, and we find agreement between cold
gases and neutron matter for very low densities. We also calculate density
profiles, which allow us to determine the density depletion at the vortex core,
which depends strongly on the short-ranged interaction in cold atomic gases,
but it is of $\approx$ 25% for neutron matter in the density regimes studied in
this work. Our results can be used to constrain neutron matter properties by
using measurements from cold Fermi gases experiments.
