  We theoretically study the nonlinear optical response of ABC trilayer
graphene with inversion symmetry broken by the application of a displacement
field perpendicular to the trilayer. We show that rhombohedral trilayer
graphene exhibits a large bulk photovoltaic effect arising from a DC shift
current response. The conductivity of the trilayer contains features similar to
AB bilayer graphene as well as features distinct from AB bilayer graphene. The
new features of ABC trilayer graphene relative to AB bilayer graphene arise
from the quantum geometric features of the electronic band structure and can be
tuned by varying the displacement field. We focus on a regime of displacement
field where certain band gaps close and reopen away from the charge neutrality
point leading to drastic changes in the quantum geometric structure in momentum
space, a feature characteristic of the trilayer graphene band structure. These
features manifest as a sign change in shift current conductivity in a certain
frequency window and can thus serve as a probe of quantum geometry.
