We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies ${\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the
z direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.
We study fermionic superfluidity in a boson-single-species-fermion cold-atom mixture. We argue that apart from the standard
p-wave fermion pairing mediated by the phonon field of the boson gas, the system also exhibits
s-wave pairing with the anomalous correlator being an odd function of time or frequency. We show that such a superfluid phase can have a much higher transition temperature than the
p-wave and may exist for sufficiently strong couplings between fermions and bosons. These conditions for odd-frequency pairing are favorable close to the value of the coupling at which the mixture phase separates. We evaluate the critical temperatures for this system and discuss the experimental realization of this superfluid in ultracold atomic gases.