  In this letter we present a coherent picture for the evolution of Higgs mode
in both neutral and charged $s$-wave fermion superfluids, as the strength of
attractive interaction between fermions increases from the BCS to the BEC
regime. In the case of neutral fermionic superfluid, such as ultracold
fermions, the Higgs mode is pushed to higher energy while at the same time,
gradually loses its spectral weight as interaction strength increases toward
the BEC regime, because the system is further tuned away from Lorentz
invariance. On the other hand, when damping is taken into account, Higgs mode
is significantly broadened due to coupling to phase mode in the whole BEC-BCS
crossover. In the charged case of electron superconductor, the Anderson-Higgs
mechanism gaps out the phase mode and suppresses the coupling between the Higgs
and the phase modes, and consequently, stabilizes the Higgs mode.
