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#### Hermitian and Non-Hermitian Topology from Photon-Mediated Interactions

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##### Citation

Roccati, F., Bello, M., Gong, Z., Ueda, M., Ciccarello, F., Chenu, A., et al. (2024).
Hermitian and Non-Hermitian Topology from Photon-Mediated Interactions.* Nature Communications,*
*15*: 2400. doi:10.1038/s41467-024-46471-w.

Cite as: https://hdl.handle.net/21.11116/0000-000C-E065-5

##### Abstract

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.

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.