  We study nonlinear optical responses in superconducting systems with
inversion ($\mathcal{I}$) symmetry-breaking order parameters. We first show
that any superconducting system with $\mathcal{I}$ and time-reversal
($\mathcal{T}$) symmetries requires an $\mathcal{I}$-breaking order parameter
to support optical transitions between particle-hole pair bands. We then use a
1D toy model of an $\mathcal{I}$-breaking superconductor to numerically
calculate linear and nonlinear conductivities, including shift current and
second harmonic generations (SHG) responses. We find that the magnitude of the
signal is significantly larger in shift current/SHG response compare to the
linear response due to the matrix element effect. We also present various
scaling behaviors of the SHG signal, which may be relevant to the recent
experimental observation of SHG in cuprates [1]. Finally, we confirm the
generality of our observations regarding nonlinear responses of
$\mathcal{I}$-breaking superconductors, by analyzing other models including a
1D three-band model and 2D square lattice model.
