  We study ultracold superfluid Bose-Fermi mixtures in three dimensions, with
stronger confinement along one or two directions, using a non-perturbative
beyond-mean-field model for bulk chemical potential valid along the
weak-coupling to unitarity crossover. Although bosons are considered to be in a
superfluid state, we consider two possibilities for the fermions --
spin-polarized degenerate state and superfluid state. Simplified reduced
analytic lower-dimensional models are derived along the weak-coupling to
unitarity crossover in quasi-one-dimensional (quasi-1D) and
quasi-two-dimensional (quasi-2D) settings. The only parameters in these models
are the constants of the beyond-mean-field Bose-Bose and Fermi-Fermi
Lee-Huang-Yang interactions and the respective universal Bertsch parameter at
unitarity. In addition to the numerical results for a fully-trapped system, we
also present results for quasi-2D Bose-Fermi mixtures where one of the
components is untrapped but localized due to the interaction mediated by the
other component. We demonstrate the validity of the reduced quasi-1D and
quasi-2D models via a comparison of the numerical solutions for the ground
states obtained from the reduced models and the full three-dimensional (3D)
model.
