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We theoretically investigate an ultracold spin-polarized atomic Fermi gaswith resonant odd-channel ($p$-wave) interactions trapped in one-dimensionalharmonic traps. We solve the Yang-Yang thermodynamic equations based on theexact Bethe ansatz solution, and predict the finite-temperature density profileand breathing mode frequency, by using a local density approximation to takeinto account the harmonic trapping potential. The system features an exoticsuper fermionic Tonks-Girardeau (super-fTG) phase, due to the large effectiverange of the interatomic interactions. We explore the parameter space for sucha fascinating super-fTG phase at finite temperature and provide smoking-gunsignatures of its existence in both breathing mode frequencies and densityprofiles. Our results suggest that the super-fTG phase can be readily probed attemperature at about $0.1T_{F}$, where $T_{F}$ is the Fermi temperature. Theseresults are to be confronted with future cold-atom experiments with $^{6}$Liand $^{40}$K atoms.

Probing an effective-range-induced super fermionic Tonks-Girardeau gas with ultracold atoms in one-dimensional harmonic traps