  Light can mediate effective dipole-dipole interactions between atoms or
quantum emitters coupled to a common environment. Exploiting them to tailor a
desired effective Hamiltonian can have major applications and advance the
search for many-body phases. Quantum technologies are mature enough to engineer
large photonic lattices with sophisticated structures coupled to quantum
emitters. In this context, a fundamental problem is to find general criteria to
tailor a photonic environment that mediates a desired effective Hamiltonian of
the atoms. Among these criteria, topological properties are of utmost
importance since an effective atomic Hamiltonian endowed with a non-trivial
topology can be protected against disorder and imperfections. Here, we find
general theorems that govern the topological properties (if any) of
photon-mediated Hamiltonians in terms of both Hermitian and non-Hermitian
topological invariants, thus unveiling a system-bath topological
correspondence. The results depend on the number of emitters relative to the
number of resonators. For a photonic lattice where each mode is coupled to a
single quantum emitter, the Altland-Zirnbauer classification of topological
insulators allows us to link the topology of the atoms to that of the photonic
bath: we unveil the phenomena of topological preservation and reversal to the
effect that the atomic topology can be the same or opposite to the photonic
one, depending on Hermiticity of the photonic system and on the parity of the
spatial dimension. As a consequence, the bulk-edge correspondence implies the
existence of atomic boundary modes with the group velocity opposite to the
photonic ones in a 2D Hermitian topological system. If there are fewer emitters
than photonic modes, the atomic system is less constrained and no general
photon-atom topological correspondence can be found. We show this with two
counterexamples.
