  We study the Higgs mode of superfluid Bose gases in a three dimensional
optical lattice, which emerges near the quantum phase transition to the Mott
insulator at commensurate fillings. Specifically, we consider responses of the
Higgs mode to temporal modulations of the onsite interaction and the hopping
energy. In order to calculate the response functions including the effects of
quantum and thermal fluctuations, we map the Bose-Hubbard model onto an
effective pseudospin-one model and use a perturbative expansion based on the
imaginary-time Green's function theory. We also include the effects of an
inhomogeneous trapping potential by means of a local density approximation. We
find that the response function for the hopping modulation is equal to that for
the interaction modulation within our approximation. At the unit filling rate
and in the absence of a trapping potential, we show that the Higgs mode can
exist as a sharp resonance peak in the dynamical susceptibilities at typical
temperatures. However, the resonance peak is significantly broadened due to the
trapping potential when the modulations are applied globally to the entire
system. We suggest that the Higgs mode can be detected as a sharp resonance
peak by partial modulations around the trap center.
