First- and second-sound-like modes at finite temperature in trapped Fermi gases from BCS to BEC

We determine the temperature ($T$) dependence of first- and second-sound-like mode frequencies for trapped Fermi gases undergoing the BCS to Bose-Einstein condensation (BEC) crossover. Our results are based on numerical solution of the two-fluid equations in conjunction with a microscopic calculation of thermodynamical variables. As in experiment and at unitarity, we show that the lowest radial breathing mode is $T$ independent. At finite $T$, higher-order breathing modes strongly mix with second sound. Their complex $T$ dependence should provide an alternative way of measuring the transition temperature $T_c$. We will also discuss collective mode frequancy for polarized Fermi gas.