Emergence of $eta$-pairing ground-state in population-imbalanced attractive Fermi-gases filling $p$ orbitals on 1-D optical lattice

We explore the ground states in population-imbalanced attractive 1-D fermionic optical lattice filling $p$ orbitals over the lowest $s$ one by using the density-matrix-renormalization-group (DMRG) method. The DMRG calculations find the occurrence of spatially non-uniform off-diagonal long-range order. In contrast to Fulde-Ferrel Larkin-Ovchinikov pair as observed in the single-band Hubbard model. The spatial oscillation period of the pair correlation function is widely fixed to be $pi$ irrespective of the mismatch between spin-split Fermi surfaces. The ground-state $pi$ order corresponds to $eta$-pair condensate predicted by Yang [Phys. Rev. Lett. textbf{63}, 2144 (1989)]. Taking account of the effects of harmonic traps, we confirm that the $eta$-pair state distinctly emerges at the center of the trap potential surrounded by perfectly-polarized states even in the trapped cases.