Possibility of $\pi$-Josephson junction and spontaneous current in a spin-polarized Fermi gas

We theoretically propose an idea to realize a $\pi$-phase in a superfluid Fermi gas, where the phase of the superfluid order parameter differs by $\pi$ across a Josephson junction. When a weak nonmagnetic potential barrier is embedded in a superfluid Fermi gas with population imbalance ($N_\uparrow>N_\downarrow$, where $N_\sigma$ is the number of atoms with pseudospin $\sigma=\uparrow, \downarrow$), this barrier may be {\it magnetized} in the sense that some of excess atoms $N_\uparrow-N_\downarrow>0$ are localized around it. This magnetic barrier behaves like a {\it ferromagnetic junction} discussed in superconductivity literature, which twists the phase of superfluid order parameter by $\pi$. We confirm this idea by solving an attractive Hubbard model within the mean-field theory at $T=0$. We also show that, when this ferromagnetic barrier is realized in a ring-shaped (or torus) trap, the system becomes the so-called $\pi$-Josephson junction, where spontaneous circulating current flows due to the phase twist at the juntion.