  Quantum nonlinear optics is a quickly growing field with large technological
promise, at the same time involving complex and novel many-body phenomena. In
the usual scenario, optical nonlinearities originate from the interactions
between polaritons, which are hybrid quasi-particles mixing matter and light
degrees of freedom. Here we introduce a type of polariton which is
intrinsically nonlinear and emerges as the natural quasi-particle in presence
quantum degenerate fermionic matter. It is a composite object made of a fermion
trapped inside an optical soliton forming a topological defect in a
spontaneously formed crystalline structure. Each of these soliton-polaritons
carries a $\textbf{Z}_2$ topological quantum number, as they create a domain
wall between two crystalline regions with opposite dimerization so that the
fermion is trapped in an interphase state. These composite objects are formally
equivalent to those appearing in the Su-Schrieffer-Heeger (SSH) model for
electrons coupled to lattice phonons.
