Pi-Josephson Junction and Spontaneous Superflow in Rings from Ultracold Fermionic Atomic Gases

he BCS-like pairing in ultracold fermionic atomic ($UCFAG$) gases is studied in the model of “isotopic-spin” pairing proposed in 1991 \cite% {Ku-Hof-SSC}. This model assumes a mismatch ($\delta$) in chemical potentials of pairing fermionic atoms. It is shown that a $\pi$-Josephson junction can be realized in $UCFAG$ systems, where the left and right banks $$ are the $UCFAG$ superfluids. The weak link $M$ consists from the normal $$ with the finite mismatch $\delta$. If the $\pi$-junction is a part of a closed ring the superfluid mass-current flows spontaneously in the ring, i.e., the time-reversal symmetry is broken spontaneously. This is realized if the radius of the ring $R$ is larger than the critical one $$. All these effects exist also in the case when $\delta \gg \mathrm{\Delta }$, where $\mathrm{\Delta }$ is the superfluid gap, but with the reduced thickness of the weak link. It is also discussed, that if junctions $S{M}_1{M}_{2}S$ and trilayers $$ from $UCFAG$ are realizable this renders a possibility for a novel electronics - $\mathit{\text{hypertronics}}$.