Time reversal symmetry breaking of $p$-orbital bosons in a one-dimensional optical lattice

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.