  We present a numerical study of a confined ultracold Fermi gas showing that
the Goldstone mode of the BCS gap is directly visible in the dynamics of the
single-particle excitations. To this end, we investigate the low-energy dynamic
response of a confined Fermi gas to a rapid change of the scattering length
(i.e., an interaction quench). Based on a fully microscopic time-dependent
density-matrix approach within the Bogoliubov-de Gennes formalism that includes
a 3D harmonic confinement we simulate and identify the emergence of the
Goldstone mode in a cigar-shaped $^6$Li gas. We show that the quench leads to a
low-frequency in-phase oscillation of the single-particle occupations. Complete
inversion is achieved for occupations corresponding to the lowest-lying
single-particle states.
