  We investigate the optical response to the THz pulses in the s-wave and
($s$+$p$)-wave superconducting semiconductor quantum wells by using the
gauge-invariant optical Bloch equations, in which the gauge structure in the
superconductivity is explicitly retained. By using the gauge transformation,
not only can the microscopic description for the quasiparticle dynamics be
realized, but also the dynamics of the condensate is included, with the
superfluid velocity and the effective chemical potential naturally
incorporated. We reveal that the superfluid velocity itself can contribute to
the pump of quasiparticles (pump effect), with its rate of change acting as the
drive field to drive the quasiparticles (drive effect). Specifically, the drive
effect can contribute to the formation of the blocking region for the
quasiparticle, which directly suppresses the anomalous correlation of the
Cooper pairs. We find that both the pump and drive effects contribute to the
oscillations of the Higgs mode with twice the frequency of the optical field.
However, it is shown that the contribution from the drive effect to the
excitation of Higgs mode is dominant as long as the driven superconducting
momentum is less than the Fermi momentum. This is in contrast to the conclusion
from the Liouville or Bloch equations in the literature, in which the drive
effect on the anomalous correlation is overlooked with only the pump effect
considered.Furthermore, in the gauge-invariant optical Bloch equations, the
charge neutrality condition is {\em consistently} considered based on the
two-component model for the charge, in which the charge imbalance of
quasiparticles can cause the fluctuation of the effective chemical potential.
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