Bogoliubov analysis of Higgs mode in trapped Fermi superfluids with spatial inhomogeneity

The Higgs mode is a key component in the spontaneous breaking of a continuous symmetry along with the Nambu-Goldstone mode, and has been studied extensively for homogeneous systems. We consider it for inhomogeneous systems, using the superfluid of harmonically trapped ultracold Fermi atomic gas. The Fermionic field operators are expanded in a complete set of wave functions corresponding to inhomogeneous situation. Within the Hatree-Fock approximation, we derive integral equations from the Bogoliubov-de Gennes equations, which lead to the frequencies of the collective modes, including the Higgs and Nambu-Goldstone modes. The results show that the frequency of the Higgs mode equals twice the absolute value of the order parameter at the center of trap. This feature is robust against variations in the interaction strength, trap potential, and temperature. These results are consistent with previous theoretical and experimental studies.