  We study the quantum phases of mixtures of ultra-cold bosonic atoms held in
an optical lattice that confines motion or hopping to one spatial dimension.
The phases are found by using Tomonaga-Luttinger liquid theory as well as the
numerical method of time evolving block decimation (TEBD). We consider a binary
mixture with repulsive intra-species interactions, and either repulsive or
attractive inter-species interaction. For a homogeneous system, we find paired-
and counterflow-superfluid phases at different filling and hopping energies. We
also predict parameter regions in which these types of superfluid order coexist
with charge density wave order. We show that the Tomonaga-Luttinger liquid
theory and TEBD qualitatively agree on the location of the phase boundary to
superfluidity. We then describe how these phases are modified and can be
detected when an additional harmonic trap is present. In particular, we show
how experimentally measurable quantities, such as time-of-flight images and the
structure factor, can be used to distinguish the quantum phases. Finally, we
suggest applying a Feshbach ramp to detect the paired superfluid state, and a
$\pi/2$ pulse followed by Bragg spectroscopy to detect the counterflow
superfluid phase.
