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Motivated by a recent experiment [Revelle et al. Phys. Rev. Lett. 117, 235301(2016)] that characterized the one- to three-dimensional crossover in aspin-imbalanced ultracold gas of $^6$Li atoms trapped in a two-dimensionalarray of tunnel-coupled tubes, we calculate the phase diagram for this systemusing Hartree-Fock Bogoliubov-de Gennes mean-field theory, and compare theresults with experimental data. Mean-field theory predicts fully spin-polarizednormal, partially spin-polarized normal, spin-polarized superfluid, andspin-balanced superfluid phases in a homogeneous system. We use the localdensity approximation to obtain density profiles of the gas in a harmonic trap.We compare these calculations with experimental measurements in Revelle {\em etal.} as well as previously unpublished data. Our calculations qualitativelyagree with experimentally-measured densities and coordinates of the phaseboundaries in the trap, and quantitatively agree with experimental measurementsat moderate-to-large polarizations. Our calculations also reproduce theexperimentally-observed universal scaling of the phase boundaries for differentscattering lengths at a fixed value of scaled inter-tube tunneling. However,our calculations have quantitative differences with experimental measurementsat low polarization, and fail to capture important features of the one- tothree-dimensional crossover observed in experiments. These suggest theimportant role of physics beyond-mean-field theory in the experiments. Weexpect that our numerical results will aid future experiments in narrowing thesearch for the FFLO phase.

Spin-imbalanced ultracold Fermi gases in a two-dimensional array of tubes