Collective Oscillations of Ultracold Fermi Gases Across a Feshbach Resonance

Ultracold atomic gases, enabled by advanced laser-cooling techniques, provide an exceptional platform for studying strongly interacting quantum matter with tunable interactions. Motivated by recent experimental advances, we investigate the collective oscillation frequencies of harmonically trapped Fermi gases across a Feshbach resonance. Using hydrodynamic equations with a density-dependent polytropic index, we first obtain analytical results for collective mode frequencies. We then employ a sum-rule approach combined with mean-field theory to calculate the monopole and quadrupole modes of single-band Fermi systems. Extending this framework, we generalize the sum-rule method to two-band Fermi systems, relevant for alkaline-earth-like atoms. Our results reveal the dependence of collective oscillations on interaction strength and provide theoretical predictions directly applicable to ongoing and future experiments probing the dynamics of strongly correlated Fermi gases.