[Paper Review] Hermitian and Non-Hermitian Topology from Photon-Mediated Interactions

Federico Roccati, Miguel Bello+5 more|arXiv (Cornell University)|Mar 1, 2023

Advanced Fiber Laser Technologies48 references4 citations

TL;DR

This paper establishes a general system-bath topological correspondence for photon-mediated interactions in quantum emitters coupled to photonic lattices, showing that atomic topological invariants are linked to photonic ones via invertible mappings. Key results reveal topological preservation or reversal depending on Hermiticity and spatial dimension, with non-Hermitian systems exhibiting reversal in odd dimensions and Hermitian systems in even dimensions.

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.

Motivation & Objective

• To establish general criteria for engineering desired effective Hamiltonians in atomic systems via photon-mediated interactions.
• To determine whether atomic systems inherit topological properties from a photonic bath, especially in the presence of disorder.
• To derive a system-bath topological correspondence linking atomic and photonic topological invariants using the Altland-Zirnbauer classification.
• To investigate how Hermiticity and spatial dimension affect topological inheritance, including edge modes and skin effects.
• To extend the framework to non-Hermitian systems and identify conditions for topological reversal or preservation.

Proposed method

• Adopts the Altland-Zirnbauer (AZ) classification to analyze topological invariants in both Hermitian and non-Hermitian systems.
• Models a photonic lattice with weakly coupled two-level emitters, preserving translational invariance with a larger unit cell.
• Derives the atomic Hamiltonian as $ H_a( extbf{k}) = -g^2 H_p( extbf{k})^{-1} $, establishing an invertible map between photonic and atomic Bloch Hamiltonians.
• Applies band-flattening (Hermitian) and unitarization (non-Hermitian) techniques to simplify topological classification.
• Uses winding numbers and Chern numbers as topological invariants in odd and even dimensions, respectively.
• Analyzes symmetry inheritance (chiral and particle-hole symmetry) under the condition $ u_e = 0 $, ensuring gapped spectra.

Experimental results

Research questions

• RQ1 Can the topological properties of a photonic bath be transferred to an array of quantum emitters via photon-mediated interactions?
• RQ2 Under what conditions does the atomic system inherit the same topology as the photonic bath (topological preservation) or the opposite (topological reversal)?
• RQ3 How do Hermiticity and spatial dimension affect the topological correspondence between the photonic bath and atomic system?
• RQ4 What is the role of non-Hermitian physics in determining the topological invariants of the atomic system?
• RQ5 Can bulk-edge correspondence and skin effects be predicted from the photonic bath's topology?

Key findings

• For one emitter per resonator, a system-bath topological correspondence exists, with topological invariants of the atomic system linked to those of the photonic bath via an invertible map.
• In Hermitian systems, topological reversal occurs only for $ bZ $-classified phases in even spatial dimensions, while $ bZ_2 $ phases always exhibit topological preservation.
• In non-Hermitian systems, topological reversal occurs only for $ bZ $-classified phases in odd spatial dimensions, due to the behavior of winding numbers under $ V o -V^ op $.
• The Chern number in even-dimensional non-Hermitian systems remains invariant under the photon-to-atom map due to unitary conjugation, preserving the topological invariant.
• In 2D Hermitian topological systems, bulk-edge correspondence implies the existence of atomic edge modes with group velocity opposite to photonic edge modes.
• When there are fewer emitters than photonic modes, no general topological correspondence exists, as shown by two counterexamples in the paper.