  Bloch electrons in multiorbital systems carry nontrivial quantum geometric
information characteristic of their orbital composition as a function of their
wave vector. When such electrons form Cooper pairs, the resultant
superconducting state naturally inherits aspects of the quantum geometry. In
this paper, we study how this geometric character is revealed in the intrinsic
optical response of the superconducting state. In particular, due to the
superconducting gap opening, interband optical transitions involving states
around the Fermi level are forbidden. This generally leads to an anomalous
suppression of the optical conductivity at frequencies matching the band
separation -- which could be significantly higher than the superconducting gap
energy. We discuss how the predicted anomaly may have already emerged in two
earlier measurements on an iron-based superconductor. When interband Cooper
pairing is present, intraband optical transitions may be allowed and finite
conductivity emerges at low frequencies right above the gap edge. These
conductivities depend crucially on the off-diagonal elements of the velocity
matrix in the band representation, i.e. the interband velocity -- which is
related to the non-Abelian Berry connection of the Bloch states.
