  We investigate the effects of quantum fluctuations on the low-energy
collective modes of two-dimensional (2D) $s$-wave Fermi superfluids from the
BCS to the Bose limit. We compare our results to recent Bragg scattering
experiments in 2D box potentials, with very good agreement. We show that
quantum fluctuations in the phase and modulus of the pairing order parameter
are absolutely necessary to give physically acceptable chemical potential and
dispersion relation of the low-energy collective mode throughout the BCS to
Bose evolution. Furthermore, we demonstrate that the dispersion of the
collective modes change from concave to convex as interactions are tuned from
the BCS to the Bose regime, and never crosses the two-particle continuum,
because arbitrarily small attractive interactions produce bound states in 2D.
