  We study bosons loaded in a one-dimensional optical lattice of two-fold
$p$-orbital degeneracy at each site. Our numerical simulations find an
anti-ferro-orbital p$_x$+ip$_y$, a homogeneous p$_x$ Mott insulator phase and
two kinds of superfluid phases distinguished by the orbital order
(anti-ferro-orbital and para-orbital). The anti-ferro-orbital order breaks time
reversal symmetry. Experimentally observable evidence is predicted for the
phase transition between the two different superfluid phases. We also discover
that the quantum noise measurement is able to provide a concrete evidence of
time reversal symmetry breaking in the first Mott phase.
