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We provide general theorems on the topology of photon-mediated interactions in terms of both Hermitian and non-Hermitian topological invariants, unveiling the phenomena of topological preservation and reversal, and revealing a system-bath topological correspondence. Depending on the Hermiticity of the environment and the parity of the spatial dimension, the atomic and photonic topological invariants turn out to be equal or opposite. Consequently, the emergence of atomic and photonic topological boundary modes with opposite group velocities in two-dimensional Hermitian topological systems is established. Owing to its general applicability, our results can guide the design of topological systems. Topological properties of a photonic environment are crucial to engineer robust photon-mediated interactions between quantum emitters. 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data-test="article-identifier"> <li class="c-article-identifiers__item" data-test="article-category">Article</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item">Published: <time datetime="2024-03-16">16 March 2024</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">Hermitian and non-Hermitian topology from photon-mediated interactions</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Federico-Roccati-Aff1" 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data-test="access-count"> <p class="c-article-metrics-bar__count">3486 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="citation-count"> <p class="c-article-metrics-bar__count">4 <span class="c-article-metrics-bar__label">Citations</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">5 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/s41467-024-46471-w/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/quantum-mechanics" data-track="click" data-track-action="view subject" data-track-label="link">Quantum mechanics</a></li><li class="c-article-subject-list__subject"><a href="/subjects/quantum-optics" data-track="click" data-track-action="view subject" data-track-label="link">Quantum optics</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>As light can mediate interactions between atoms in a photonic environment, engineering it for endowing the photon-mediated Hamiltonian with desired features, like robustness against disorder, is crucial in quantum research. We provide general theorems on the topology of photon-mediated interactions in terms of both Hermitian and non-Hermitian topological invariants, unveiling the phenomena of topological preservation and reversal, and revealing a system-bath topological correspondence. Depending on the Hermiticity of the environment and the parity of the spatial dimension, the atomic and photonic topological invariants turn out to be equal or opposite. Consequently, the emergence of atomic and photonic topological boundary modes with opposite group velocities in two-dimensional Hermitian topological systems is established. Owing to its general applicability, our results can guide the design of topological systems.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-021-22264-3/MediaObjects/41467_2021_22264_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-021-22264-3?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41467-021-22264-3">Topological protection versus degree of entanglement of two-photon light in photonic topological insulators </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">30 March 2021</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-020-14994-7/MediaObjects/41467_2020_14994_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-020-14994-7?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s41467-020-14994-7">Topological edge states of interacting photon pairs emulated in a topolectrical circuit </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">18 March 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41566-023-01360-4/MediaObjects/41566_2023_1360_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41566-023-01360-4?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41566-023-01360-4">Non-local skyrmions as topologically resilient quantum entangled states of light </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">08 January 2024</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732781756, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>The study of topological phases and related phenomena, including edge states protected against disorder, dates back to the 1980s when the quantum Hall effect was discovered<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="von Klitzing, K. et al. 40 years of the quantum Hall effect. Nat. Rev. Phys. 2, 397–401 (2020)." href="/articles/s41467-024-46471-w#ref-CR1" id="ref-link-section-d93114156e610">1</a></sup>. Since then, the field of topological phases of matter has expanded considerably and is now a prominent topic in contemporary condensed matter physics<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045 (2010)." href="/articles/s41467-024-46471-w#ref-CR2" id="ref-link-section-d93114156e614">2</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. 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Topologically protected quantum state transfer in a chiral spin liquid. Nat. Commun. 4, 1585 (2013)." href="#ref-CR12" id="ref-link-section-d93114156e649_1">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Almeida, G. M. A., Ciccarello, F., Apollaro, T. J. G. & Souza, A. M. C. Quantum-state transfer in staggered coupled-cavity arrays. Phys. Rev. A 93, 032310 (2016)." href="#ref-CR13" id="ref-link-section-d93114156e649_2">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Lemonde, M.-A., Peano, V., Rabl, P. & Angelakis, D. G. Quantum state transfer via acoustic edge states in a 2D optomechanical array. New J. Phys. 21, 113030 (2019)." href="/articles/s41467-024-46471-w#ref-CR14" id="ref-link-section-d93114156e652">14</a></sup> and multi-mode entanglement generation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Vega, C., Porras, D. & González-Tudela, A. Topological multimode waveguide QED. Phys. Rev. Research 5, 023031 (2023)." href="/articles/s41467-024-46471-w#ref-CR15" id="ref-link-section-d93114156e656">15</a></sup>. Notably, for a photonic Su–Schrieffer–Heeger (SSH) model, it was predicted that an atom, being a quantum zero-dimensional defect, can seed dressed bound states that are topologically protected<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Bello, M., Platero, G., Cirac, J. I. & González-Tudela, A. Unconventional quantum optics in topological waveguide QED. Sci. Adv. 5, eaaw0297 (2019)." href="/articles/s41467-024-46471-w#ref-CR16" id="ref-link-section-d93114156e660">16</a></sup>; the essential properties and occurrence criteria for such topological dressed states were then derived on a general basis<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Leonforte, L., Carollo, A. & Ciccarello, F. Vacancy-like dressed states in topological waveguide QED. Phys. Rev. Lett. 126, 063601 (2021)." href="/articles/s41467-024-46471-w#ref-CR17" id="ref-link-section-d93114156e664">17</a></sup>. Specific investigations on topological dressed states were performed in other photonic analogs of prototypical topological models, such as the Harper–Hofstadter<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="De Bernardis, D., Cian, Z. P., Carusotto, I., Hafezi, M. & Rabl, P. Light-matter interactions in synthetic magnetic fields: Landau-photon polaritons. Phys. Rev. Lett. 126, 103603 (2021)." href="/articles/s41467-024-46471-w#ref-CR18" id="ref-link-section-d93114156e669">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Bello, M. & Cirac, J. I. Topological effects in two-dimensional quantum emitter systems. Phys. Rev. B 107, 054301 (2023)." href="/articles/s41467-024-46471-w#ref-CR19" id="ref-link-section-d93114156e672">19</a></sup>, the Haldane model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Leonforte, L., Carollo, A. & Ciccarello, F. Vacancy-like dressed states in topological waveguide QED. Phys. Rev. Lett. 126, 063601 (2021)." href="/articles/s41467-024-46471-w#ref-CR17" id="ref-link-section-d93114156e676">17</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Bello, M. & Cirac, J. I. Topological effects in two-dimensional quantum emitter systems. Phys. Rev. B 107, 054301 (2023)." href="/articles/s41467-024-46471-w#ref-CR19" id="ref-link-section-d93114156e679">19</a></sup>, as well as lossy systems exhibiting non-Hermitian topology<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Anomalous behaviors of quantum emitters in non-hermitian baths. Phys. Rev. Lett. 129, 223601 (2022)." href="/articles/s41467-024-46471-w#ref-CR20" id="ref-link-section-d93114156e683">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Bound states and photon emission in non-hermitian nanophotonics. Phys. Rev. A 106, 053517 (2022)." href="/articles/s41467-024-46471-w#ref-CR21" id="ref-link-section-d93114156e686">21</a></sup>. Furthermore, the use of atomic emission properties was proposed to sense topological phases<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Vega, C., Bello, M., Porras, D. & González-Tudela, A. Qubit-photon bound states in topological waveguides with long-range hoppings. Phys. Rev. A 104, 053522 (2021)." href="/articles/s41467-024-46471-w#ref-CR22" id="ref-link-section-d93114156e690">22</a></sup>. Such investigations were motivated by recent technological advances to fabricate photonic lattices with engineered properties (large periodic one-dimensional (1D) or two-dimensional (2D) arrangements of coupled cavities/resonators) and to coherently couple them to a set of controllable atoms/quantum emitters. For example, there have been several demonstrations in various experimental platforms such as ultracold atoms<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Krinner, L., Stewart, M., Pazmiño, A., Kwon, J. & Schneble, D. Spontaneous emission of matter waves from a tunable open quantum system. Nature 559, 589–592 (2018)." href="/articles/s41467-024-46471-w#ref-CR23" id="ref-link-section-d93114156e694">23</a></sup>, circuit quantum electrodynamics<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Liu, Y. & Houck, A. A. Quantum electrodynamics near a photonic bandgap. Nat. Phys. 13, 48–52 (2017)." href="#ref-CR24" id="ref-link-section-d93114156e698">24</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Sundaresan, N. M., Lundgren, R., Zhu, G., Gorshkov, A. V. & Houck, A. A. Interacting qubit-photon bound states with superconducting circuits. Phys. Rev. X 9, 011021 (2019)." href="#ref-CR25" id="ref-link-section-d93114156e698_1">25</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kim, E. et al. Quantum electrodynamics in a topological waveguide. Phys. Rev. X 11, 011015 (2021)." href="#ref-CR26" id="ref-link-section-d93114156e698_2">26</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Scigliuzzo, M. et al. Controlling atom-photon bound states in an array of Josephson-junction resonators. Phys. Rev. X 12, 031036 (2022)." href="#ref-CR27" id="ref-link-section-d93114156e698_3">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Owens, J. C. et al. Chiral cavity quantum electrodynamics. Nat. Phys. 18, 1048–1052 (2022)." href="/articles/s41467-024-46471-w#ref-CR28" id="ref-link-section-d93114156e701">28</a></sup>, and coupled-resonator optical waveguides<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Tang, J. S. et al. Nonreciprocal single-photon band structure. Phys. Rev. Lett. 128, 203602 (2022)." href="/articles/s41467-024-46471-w#ref-CR29" id="ref-link-section-d93114156e706">29</a></sup>. In these setups, the photonic lattice acts as an artificially <i>engineered bath</i>, or <i>environment</i>, for the quantum emitters as their decay rate into the lattice guided modes is larger than their decay rate in free space<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Chang, D. E., Douglas, J. S., González-Tudela, A., Hung, C.-L. & Kimble, H. J. Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons. Rev. Mod. Phys. 90, 031002 (2018)." href="/articles/s41467-024-46471-w#ref-CR30" id="ref-link-section-d93114156e716">30</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Sheremet, A. S., Petrov, M. I., Iorsh, I. V., Poshakinskiy, A. V. & Poddubny, A. N. Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations. Rev. Mod. Phys. 95, 015002 (2023)." href="/articles/s41467-024-46471-w#ref-CR31" id="ref-link-section-d93114156e719">31</a></sup>.</p><p>Here, a fundamental question remains unanswered: do atoms coupled to a photonic bath with known topological properties inherit any of those topological properties? If so, how are the symmetry class and topological phase of the atoms related to those of the field?</p><p>To address this issue, we adopt the standard Altland–Zirnbauer classification of topological insulators<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997)." href="/articles/s41467-024-46471-w#ref-CR32" id="ref-link-section-d93114156e729">32</a></sup> and consider a general model consisting of a photonic lattice weakly coupled to a periodic arrangement of two-level emitters, such that the total system is translationally invariant (with a unit cell potentially larger than that of the bare engineered bath), see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig1">1A</a>. We obtain general results linking the photonic and atomic topological invariants. On the basis of the bulk–edge correspondence in the Hermitian case<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)." href="/articles/s41467-024-46471-w#ref-CR33" id="ref-link-section-d93114156e736">33</a></sup>, we reveal the relationship between photonic and atomic boundary modes under open boundary conditions, and the relationship between skin modes or more general bulk anomalous dynamics in the non-Hermitian case<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020)." href="/articles/s41467-024-46471-w#ref-CR34" id="ref-link-section-d93114156e740">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between hermitian and non-hermitian systems: Anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019)." href="/articles/s41467-024-46471-w#ref-CR35" id="ref-link-section-d93114156e743">35</a></sup>. The key results of this system–bath topological correspondence are summarized in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41467-024-46471-w#Tab1">1</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Setup."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1: Setup.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="320"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p><b>A</b> Two-level quantum emitters (blue spheres) coupled to an engineered photonic bath (lattice with red spheres). The full light-atom system is translationally invariant. For concreteness, the figure shows a specific 2D example with two emitters per unit cell (light shaded area). Note how, in this case, the bare photonic lattice has a smaller unit cell (dark shaded area). Important parameters of the model are the emitter transition frequency <span class="mathjax-tex">${\omega }_{e}$</span> and the light-matter coupling strength <span class="mathjax-tex">$g$</span>. In the single-excitation sector, dissipation and decoherence due to coupling to uncontrolled environmental modes (wiggly arrows) can be modeled through the use of non-Hermitian Hamiltonians. <b>B</b> Engineered bath spectrum. Depending on the geometry and parameters of the system, one or more (Hermitian or non-Hermitian) photonic bandgaps can emerge, i.e., photons of frequency within certain ranges cannot propagate through the engineered bath. <b>C</b> Emitter-emitter, bath-mediated interactions. If the emitters are spectrally tuned to one of the bandgaps (as shown in <b>B</b>), and weakly coupled to the environment, such that the spectral distance <span class="mathjax-tex">$\Delta$</span> to the nearest photonic bands is larger than the light-matter coupling, the bath degrees of freedom can be traced out, leading to effective exchange interactions among the emitters.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1 System-bath topological correspondence</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/articles/s41467-024-46471-w/tables/1" aria-label="Full size table 1"><span>Full size table</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Results"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Results</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3">System–bath topological correspondence</h3><p>Photons in these engineered photonic baths can mediate second-order interactions between atoms; this interaction can be described by an effective many-body atomic Hamiltonian<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Bello, M., Platero, G., Cirac, J. I. & González-Tudela, A. Unconventional quantum optics in topological waveguide QED. Sci. Adv. 5, eaaw0297 (2019)." href="/articles/s41467-024-46471-w#ref-CR16" id="ref-link-section-d93114156e1385">16</a></sup> in the commonly studied regime of weak-coupling and Markovian dynamics. We investigated the topological properties of the latter Hamiltonian and demonstrated that they depend on the detuning between the atomic frequency and mean photonic frequency (typically located in the middle of the central photonic bandgap).</p><p>In the following we detail the type of systems to which our theory is applicable. To be precise, we distinguish between photonic modes of the engineered bath and those of the surrounding space. Coupling to the latter can be modeled through non-Hermitian Hamiltonians, both for the engineered bath and the quantum emitters. We may also neglect the surrounding environmental modes and use Hermitian Hamiltonians instead, assuming that the emitters couple more strongly to the engineered photonic bath modes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Douglas J. S. et al. Quantum many-body models with cold atoms coupled to photonic crystals. Nat. Photonics 9, 326–331 (2015)." href="/articles/s41467-024-46471-w#ref-CR36" id="ref-link-section-d93114156e1392">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="González-Tudela, A. & Cirac, J. I. Markovian and non-Markovian dynamics of quantum emitters coupled to two-dimensional structured reservoirs. Phys. Rev. A 96, 043811 (2017)." href="/articles/s41467-024-46471-w#ref-CR37" id="ref-link-section-d93114156e1395">37</a></sup>. Thus, the whole system is modeled by the Hamiltonian <span class="mathjax-tex">$\hat{H}=$</span> <span class="mathjax-tex">${\hat{H}}_{e}+{\hat{H}}_{p}+{\hat{H}}_{{{{{\mathrm{int}}}}}}$</span>. The free atomic Hamiltonian can be written as <span class="mathjax-tex">${\hat{H}}_{e}={\omega }_{e}{\sum }_{n}{\hat{\sigma }}_{n}^{{{\dagger}} }{\hat{\sigma }}_{n}$</span>, where <span class="mathjax-tex">${\hat{\sigma }}_{n}={{{{{\rm{|}}}}}}g{{{\rangle }}}_{n}{{\langle }}e{{{{{\rm{|}}}}}}$</span>. Under periodic boundary conditions, the bare photonic Hamiltonian can be expressed in terms of bath modes with a definite quasimomentum <span class="mathjax-tex">${{{{{\bf{k}}}}}}$</span> as <span class="mathjax-tex">${\hat{H}}_{p}={\sum }_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}{\hat{A}}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }{H}_{p}({{{{{\bf{k}}}}}}){\hat{A}}_{{{{{{\bf{k}}}}}}}$</span>, where BZ represents the first Brillouin zone, <span class="mathjax-tex">${\hat{A}}_{{{{{{\rm{k}}}}}}}^{{{\dagger}} }=$</span> <span class="mathjax-tex">$({\hat{a}}_{{{{{{\bf{k}}}}}},1}^{{{\dagger}} },\ldots,\, {\hat{a}}_{{{{{{\bf{k}}}}}},{N}_{b}}^{{{\dagger}} }),\, {\hat{a}}_{{{{{{\bf{k}}}}}},s}$</span> ‘s are the bosonic annihilation operators of the field’s normal modes, and <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> is the <span class="mathjax-tex">${N}_{b}\times {N}_{b}$</span> Bloch Hamiltonian matrix (see Methods). We denote the bare resonator frequency as <span class="mathjax-tex">${\omega }_{0}$</span> which we choose to be the reference energy, i.e., we set <span class="mathjax-tex">${\omega }_{0}=0$</span>. According to the standard rotating-wave approximation, the interaction between quantum emitters (QEs) and the field is described by the last term of the total Hamiltonian <span class="mathjax-tex">${\hat{H}}_{{{{{\mathrm{int}}}}}}=\mathop{\sum }\nolimits_{n=1}^{{N}_{c}}\mathop{\sum }\nolimits_{s=1}^{{N}_{b}}{g}_{s}({\hat{\sigma }}_{{ns}}^{{{\dagger}} }{\hat{a}}_{{ns}}+{{{{{\rm{H.c.}}}}}})$</span>. Here, <span class="mathjax-tex">${\hat{a}}_{{ns}}$</span> is the real-space annihilation operator of the resonator located in the <i>n</i>th unit cell, belonging to the <span class="mathjax-tex">$s$</span>-sublattice <span class="mathjax-tex">$\left(s=1,\ldots,\, {N}_{b}\right)$</span>, and <span class="mathjax-tex">${N}_{c}$</span> is the total number of unit cells. The atomic operator, <span class="mathjax-tex">${\hat{\sigma }}_{{ns}}$</span>, in <span class="mathjax-tex">${\hat{H}}_{{{{{\mathrm{int}}}}}}$</span> has two indices to specify the resonator to which it is coupled. The coupling strength <span class="mathjax-tex">${g}_{s}$</span> satisfies <span class="mathjax-tex">${g}_{s}=g$</span> if a QE is coupled to the resonator <span class="mathjax-tex">${\hat{a}}_{{ns}}$</span>, and is set to zero otherwise. Note that <span class="mathjax-tex">${g}_{s}$</span> is independent on the cell index <span class="mathjax-tex">$n$</span>, ensuring translational invariance.</p><p>Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig1">1A</a> presents a specific instance of the studied setup. It consists of two-level quantum emitters, <span class="mathjax-tex">${N}_{e}$</span> in total, with a ground state <span class="mathjax-tex">${{{{{\rm{|}}}}}}g{{\rangle }}$</span> and an excited state <span class="mathjax-tex">${{{{{\rm{|}}}}}}e{{\rangle }}$</span>, which are separated by the Bohr frequency <span class="mathjax-tex">${\omega }_{e}$</span>. The QEs are locally coupled to a translationally invariant photonic lattice implemented by coupled single-mode resonators. The lattice unit cell contains <span class="mathjax-tex">${N}_{b}$</span> resonators. Hence, there are (generally) as many sublattices as photonic bands.</p><p>We consider an emitter arrangement with a spatial period equal to or greater than that of the photonic lattice that is translationally invariant. This is a translationally-invariant setup, featuring no randomness (in the position of the emitters) whatsoever.</p><p>In the Markovian regime, the degrees of freedom of the bath can be traced out. When the emitters’ frequency <span class="mathjax-tex">${\omega }_{e}$</span> lies within a photonic bandgap and the coupling constant <span class="mathjax-tex">$g$</span> is small (smaller than the spectral distance between <span class="mathjax-tex">${\omega }_{e}$</span> and the photonic bands), the photonic lattice then induces effective coherent interactions between the emitters, described by an effective atomic Hamiltonian <span class="mathjax-tex">${\hat{H}}_{a}\sim {\hat{H}}_{e}+{g}^{2}{\hat{G}}_{p}\left({\omega }_{e}\right)$</span> (see Methods), where the latter is the resolvent operator of the bath<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Economou, E. N. Green’s functions in quantum physics. Springer S. Solid State Sci. Berlin Heidelberg, Berlin, Heidelberg: Springer 7 (2006)." href="/articles/s41467-024-46471-w#ref-CR38" id="ref-link-section-d93114156e3126">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="González-Tudela, A., Hung, C.-L., Chang, D. E., Cirac, J. I. & Kimble, H. J. Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals. Nat. Photonics 9, 320–325 (2015)." href="/articles/s41467-024-46471-w#ref-CR39" id="ref-link-section-d93114156e3129">39</a></sup>, see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig1">1</a>.</p><p>We established a <i>system–bath topological correspondence</i> by relating the topological properties of the bare bath and those of the system dressed by the bath. At resonance, <span class="mathjax-tex">${\omega }_{e}={\omega }_{0}$</span>, both the photonic and the effective emitters’ Hamiltonians possess identical symmetries (see below). In addition, when the system and bath have the same number of degrees of freedom (i.e., <span class="mathjax-tex">${g}_{s}=g$</span> for all <span class="mathjax-tex">$s$</span>), we reveal <i>the topological preservation</i> and <i>reversal</i> based on the Altland–Zirnbauer (AZ) classification<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997)." href="/articles/s41467-024-46471-w#ref-CR32" id="ref-link-section-d93114156e3244">32</a></sup>. Specifically, the topological invariants <span class="mathjax-tex">${\nu }_{a(p)}$</span> of <span class="mathjax-tex">${\hat{H}}_{a(p)}$</span> are related to each other as follows:</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\nu }_{a}=\left\{\begin{array}{cc}{\nu }_{p} \hfill& {{\mbox{for}}}\,{{\mathbb{Z}}}_{2}\, {{\mbox{phases}}}{{{{{\rm{;}}}}}}\\ {\nu }_{p}{(-1)}^{D+{\mathfrak{h}}} & {{\mbox{for}}}\,{\mathbb{Z}}\, {{\mbox{phases}}}.\end{array}\right.$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>Here, <span class="mathjax-tex">$D$</span> is the spatial dimension, and <span class="mathjax-tex">${\mathfrak{h}}$</span> is <span class="mathjax-tex">$1(0)$</span> if <span class="mathjax-tex">${\hat{H}}_{p}$</span> is (non-)Hermitian and belongs to the (non-)Hermitian <span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> (<span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> and <span class="mathjax-tex">${{{{{{\rm{AZ}}}}}}}^{{{\dagger}} }$</span>) classes. In the <span class="mathjax-tex">${\mathbb{Z}}$</span> phases, the system preserves or reverses the integer topological invariant of the bath depending on the dimension and Hermiticity. By contrast, <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span>-protected bath phases are always inherited by the system. The proof of Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ1">1</a> is provided in Methods.</p><p>In addition, this result conforms to the known Hermitian–non-Hermitian correspondence<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between hermitian and non-hermitian systems: Anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019)." href="/articles/s41467-024-46471-w#ref-CR35" id="ref-link-section-d93114156e3732">35</a></sup>. The boundary mode of a <span class="mathjax-tex">$D+1$</span> dimensional Hermitian topological system characterized by the topological invariant <span class="mathjax-tex">$\nu$</span> can be mapped into a <span class="mathjax-tex">$D$</span>-dimensional non-Hermitian system on a closed manifold, with an identical topological invariant. In fact, we demonstrate that topological preservation (reversal), i.e., <span class="mathjax-tex">${\nu }_{a}={\nu }_{p}$</span> <span class="mathjax-tex">$({\nu }_{a}=-{\nu }_{p})$</span>, occurs in both <span class="mathjax-tex">$D+1$</span> dimensional Hermitian systems as well as in <span class="mathjax-tex">$D$</span>-dimensional non-Hermitian ones (Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41467-024-46471-w#Tab1">1</a>).</p><p>The topological correspondence we found holds in the case of one emitter per resonator, namely <span class="mathjax-tex">${g}_{s}=g$</span> for all <span class="mathjax-tex">$s$</span>. In the case where there are fewer emitters than resonators (<span class="mathjax-tex">${g}_{s}=0$</span> for some <span class="mathjax-tex">$s$</span>), this general correspondence does not hold anymore, cf. Supplementary Information.</p><p>We note that (i) our result is not limited to quantum optical systems, but holds for any open quantum system satisfying the same conditions, (ii) the photonic bath Hamiltonian can be fully general, as long as it can be written in Bloch form (see above). Also, it is important to note that the topological invariants we discuss correspond to non-interacting models, and therefore, for the atomic subsystem they only classify the single-particle/linear regime. In this sense, our results are also applicable or could be extended to classical systems (<i>e.g</i>., topological mechanical/acoustic systems). We now proceed illustrating one example of topological preservation and reversal in each of the four possible cases (Hermitian/non-Hermitian and odd/even spatial dimension). The proofs are detailed in Methods.</p><h3 class="c-article__sub-heading" id="Sec4">Topological preservation and reversal</h3><p>The Bloch Hamiltonian of the entire system is</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H({{{{{\bf{k}}}}}})=\left[\begin{array}{cc}{\omega }_{e}I & {gI}\\ {gI} & {H}_{p}({{{{{\bf{k}}}}}})\end{array}\right],$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>where <span class="mathjax-tex">$I$</span> is the <span class="mathjax-tex">${N}_{b}$</span>-dimensional identity matrix. Remarkably, the entire atom–light Hamiltonian is topologically trivial (see Methods). The effective atomic Hamiltonian is obtained using the standard perturbation theory up to the second order (see Methods) as</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${H}_{a}({{{{{\bf{k}}}}}})={\omega }_{e}+\frac{{g}^{2}}{{\omega }_{e}-{H}_{p}({{{{{\bf{k}}}}}})}.$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>The real space form, <span class="mathjax-tex">${\hat{H}}_{a}$</span>, is recovered using the inverse Fourier transform.</p><p>Depending on the spatial dimension, the symmetries, and whether the photonic Hamiltonian is Hermitian or not, we discovered that the topology of this effective system can either be preserved or reversed at least for the fundamental symmetry classes. Here, we present the general results for the Hermitian and non-Hermitian cases as summarized in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41467-024-46471-w#Tab1">1</a>, together with representative examples.</p><p>First we consider the Hermitian case. Altland and Zirnbauer<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997)." href="/articles/s41467-024-46471-w#ref-CR32" id="ref-link-section-d93114156e4392">32</a></sup> were the first to identify the 10 fundamental symmetry classes (AZ classes). The topological classification of Hermitian noninteracting systems (insulators and superconductors) was subsequently developed on the basis of their classification<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008)." href="/articles/s41467-024-46471-w#ref-CR40" id="ref-link-section-d93114156e4396">40</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Kitaev, A. Periodic table for topological insulators and superconductors. A.I.P. Conf. Proc. 1134, 22 (2009)." href="/articles/s41467-024-46471-w#ref-CR41" id="ref-link-section-d93114156e4399">41</a></sup>. Only five of these symmetry classes are relevant for the number-conserving Hamiltonians<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019)." href="/articles/s41467-024-46471-w#ref-CR5" id="ref-link-section-d93114156e4403">5</a></sup>, (see Supplementary Table <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-024-46471-w#MOESM1">S1</a>). The classification is based on time–reversal symmetry (TRS or equivalently <span class="mathjax-tex">$T$</span>), particle–hole symmetry (PHS or equivalently <span class="mathjax-tex">$C$</span>), and chiral symmetry (<span class="mathjax-tex">$S$</span>)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019)." href="/articles/s41467-024-46471-w#ref-CR5" id="ref-link-section-d93114156e4462">5</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997)." href="/articles/s41467-024-46471-w#ref-CR32" id="ref-link-section-d93114156e4465">32</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)." href="/articles/s41467-024-46471-w#ref-CR33" id="ref-link-section-d93114156e4468">33</a></sup>. Their explicit effect on the Bloch Hamiltonian <span class="mathjax-tex">$H({{{{{\bf{k}}}}}})$</span> is given by <span class="mathjax-tex">${TH}({{{{{\bf{k}}}}}}){T}^{-1}=H(-{{{{{\bf{k}}}}}})$</span>, <span class="mathjax-tex">${CH}({{{{{\bf{k}}}}}}){C}^{-1}=-H(-{{{{{\bf{k}}}}}})$</span> and <span class="mathjax-tex">${SH}({{{{{\bf{k}}}}}}){S}^{-1}=-H({{{{{\bf{k}}}}}})$</span>, respectively.</p><p>Both <span class="mathjax-tex">$T$</span> and <span class="mathjax-tex">$C$</span> are antiunitary operators, <i>i.e.</i>, <span class="mathjax-tex">$T={U}_{{{{{{\rm{TRS}}}}}}}K$</span> and <span class="mathjax-tex">$C={U}_{{{{{{\rm{PHS}}}}}}}K$</span>, where <span class="mathjax-tex">${U}_{{{{{{\rm{TRS}}}}}}}$</span> and <span class="mathjax-tex">${U}_{{{{{{\rm{PHS}}}}}}}$</span> are unitaries and <span class="mathjax-tex">$K$</span> denotes complex conjugation. By applying these to <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span>, one can show that TRS is never broken in the following sense: <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> has TRS if and only if <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> has TRS for any atomic frequency <span class="mathjax-tex">${\omega }_{e}$</span> (provided that it lies within a bandgap of <span class="mathjax-tex">${\hat{H}}_{p}$</span>). In turn, PHS, and therefore chiral symmetry<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)." href="/articles/s41467-024-46471-w#ref-CR33" id="ref-link-section-d93114156e5091">33</a></sup>, can be broken. Indeed, for <span class="mathjax-tex">${\omega }_{e}={\omega }_{0}=0$</span>, <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> has PHS if and only if <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> has PHS, the same holding for chiral symmetry.</p><p>Conversely, in the absence of photonic symmetries, no new symmetry can be generated at the atomic level whatever the value of <span class="mathjax-tex">${\omega }_{e}$</span>.</p><p>This demonstrates that, on resonance <span class="mathjax-tex">$\left({\omega }_{e}={\omega }_{0}=\right.$</span> 0), <span class="mathjax-tex">${\hat{H}}_{a}$</span> and <span class="mathjax-tex">${\hat{H}}_{p}$</span> belong to the same symmetry class; off resonance, the following transitions of symmetry classes occur when going from <span class="mathjax-tex">${\hat{H}}_{p}$</span> to <span class="mathjax-tex">${\hat{H}}_{a}:{{{{{\rm{AIII}}}}}}\to {{{{{\rm{A}}}}}}$</span>, <span class="mathjax-tex">${{{{{\rm{BDI}}}}}} \to {{{{{\rm{AI}}}}}}$</span> and <span class="mathjax-tex">${{{{{\rm{D}}}}}}\to {{{{{\rm{A}}}}}}$</span> (we refer to the standard terminology of AZ classes, see Supplementary Table <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-024-46471-w#MOESM1">S1</a>). Topologically distinct phases within the same symmetry class are characterized by different values of the topological invariant (<i>e.g</i>., Zak phase, Chern number, and Chern–Simons invariants), which we generally denote as <span class="mathjax-tex">${\nu }_{l}$</span>, with <span class="mathjax-tex">$l=p,a$</span> that refer to the photonic and atomic Hamiltonians, respectively. According to the bulk–edge correspondence, <span class="mathjax-tex">${\nu }_{l}$</span> represents the number of edge modes in the system under open boundary conditions, where the trivial phase has a topological invariant equal to zero<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)." href="/articles/s41467-024-46471-w#ref-CR33" id="ref-link-section-d93114156e5619">33</a></sup>.</p><p>To ensure that the PHS and chiral symmetry are inherited, we focus on the case <span class="mathjax-tex">${\omega }_{e}={\omega }_{0}=0$</span>, such that <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})=-{g}^{2}{H}_{p}{({{{{{\bf{k}}}}}})}^{-1}$</span>.</p><p>Our main observation is that <i>Hermitian topology is preserved</i> <span class="mathjax-tex">$({\nu }_{a}={\nu }_{p})$</span> <i>for</i> <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> <i>phases in all spatial dimensions and for</i> <span class="mathjax-tex">${\mathbb{Z}}$</span> <i>phases in odd dimensions</i> (in particular 1D and 3D). Instead, <span class="mathjax-tex">${\mathbb{Z}}$</span> <i>phases in even dimensions</i> undergo <i>a topological reversal</i> <span class="mathjax-tex">$({\nu }_{a}=-{\nu }_{p})$</span> (proof given in Methods). Remarkably, the topological reversal has direct observable consequences on the basis of the bulk–boundary correspondence, as we discuss later on.</p><p>As a minimal example of 1D Hermitian topological preservation, we consider the case of QEs coupled to a Su–Schrieffer–Heeger (SSH) lattice (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig2">2</a>), whose Hamiltonian is</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\hat{H}}_{p}=\mathop{\sum}\limits_{n}\, (v{\hat{a}}_{n,1}^{{{\dagger}} }{\hat{a}}_{n,2}+w{\hat{a}}_{n,2}^{{{\dagger}} }{\hat{a}}_{n+1,1})+{{{{{\rm{H.c.}}}}}}\:.$$</span></div><div class="c-article-equation__number"> (4) </div></div><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Hermitian topological preservation."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2: Hermitian topological preservation.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig2_HTML.png?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="240"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p><b>A</b> Scheme of the photonic Su–Schrieffer–Heeger (SSH) lattice with staggered <span class="mathjax-tex">$v$</span> and <span class="mathjax-tex">$w$</span> <span class="mathjax-tex">${{{{{\rm{couplings}}}}}}$</span>. The coupling strength between each quantum emitter and each resonator is <span class="mathjax-tex">$g$</span>. <b>B</b> A sketch of the mediated emitters’ Hamiltonian <span class="mathjax-tex">${\hat{H}}_{a}$</span> is shown in purple (where the multiple links highlight its high connectivity). Open boundary conditions for the atomic system are obtained by removing quantum emitters, but leaving the photonic lattice unaffected (hence it remains translationally invariant). <b>C</b> Modulus of the wave function of the photonic edge states with <span class="mathjax-tex">$N=60$</span> resonators (top), and atomic edge states with <span class="mathjax-tex">${N}_{e}=44$</span> emitters (bottom) coupled to a periodic SSH lattice with <span class="mathjax-tex">$N$</span> resonators (top) and <span class="mathjax-tex">$d=8$</span> sites. The resonators are numbered in increasing order including both types (1 and 2, cf. Equation <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ4">4</a>) of resonators. Atomic open boundary conditions are obtained by removing 2<span class="mathjax-tex">$d$</span> quantum emitters (outer violet stripes) while maintaining the periodic photonic structure. Atomic edge states are mostly localized on the first and last sites (notice the logarithmic scale). The nonzero amplitude on the remaining sites is a finite-size effect. The insets show the photonic and atomic energy spectra under open boundary conditions in units of <span class="mathjax-tex">$v$</span>. Parameters: <span class="mathjax-tex">$w=1.5v,g=0.1v$</span>, and <span class="mathjax-tex">${\omega }_{e}=0$</span>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>This model belongs to the BDI class, admitting <span class="mathjax-tex">${\mathbb{Z}}$</span> phases<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Bello, M., Platero, G., Cirac, J. I. & González-Tudela, A. Unconventional quantum optics in topological waveguide QED. Sci. Adv. 5, eaaw0297 (2019)." href="/articles/s41467-024-46471-w#ref-CR16" id="ref-link-section-d93114156e6557">16</a></sup>. Our hypothesis <span class="mathjax-tex">${\omega }_{e}={\omega }_{0}=0$</span> entails that the effective Hamiltonian between QEs preserves the chiral symmetry, and therefore, the same topology as the underlying photonic lattice based on our results, see Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41467-024-46471-w#Tab1">1</a>. This is confirmed by the bulk–edge correspondence. In fact, when <span class="mathjax-tex">${\hat{H}}_{a}$</span> is subject to open boundary conditions, which is indeed the case for a finite array of QEs in a larger periodic photonic SSH lattice, the effective atomic Hamiltonian supports topological edge states in the nontrivial phase despite the high connectivity of the mediated interactions.</p><p>The Hermitian reversal of topology occurs in two dimensions. We consider QEs coupled to a photonic 2D Chern insulator described by the Qi–Wu–Zhang (QWZ) model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Qi, X.-L., Wu, Y.-S. & Zhang, S.-C. Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors. Phys. Rev. B 74, 085308 (2006)." href="/articles/s41467-024-46471-w#ref-CR42" id="ref-link-section-d93114156e6656">42</a></sup> (class A) to illustrate its implications. For this bipartite lattice (featuring two sublattices), the Bloch Hamiltonian <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> is</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${H}_{{{{{{\rm{QWZ}}}}}}}\left({{{{{\bf{k}}}}}}\right)=J\sin \left({k}_{x}\right){\tau }_{x}+J\sin ({k}_{y}){\tau }_{y}+J\left[u+\cos \left({k}_{x}\right)+\cos ({k}_{y})\right]{\tau }_{z},$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>where <span class="mathjax-tex">${\tau }_{\alpha }$</span> <span class="mathjax-tex">$(\alpha \in \{x,y,z\})$</span> are the Pauli matrices and the two fictitious spin states correspond to the two sublattices. The system is in the nontrivial phase whenever <span class="mathjax-tex">$0 < {{{{{\rm{|}}}}}}u{{{{{\rm{|}}}}}} < 2$</span>. Assuming open boundary conditions along the <span class="mathjax-tex">$x$</span>-direction only, the system then supports chiral photonic boundary modes propagating along <span class="mathjax-tex">$y$</span>, in one direction in one boundary and in the opposite direction in the other boundary. In this case, based on Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="/articles/s41467-024-46471-w#Tab1">1</a>, topological reversal occurs so that the photonic and atomic Chern numbers have opposite sign, resulting in atomic boundary modes featuring atomic excitations that travel with opposite group velocity compared to their photonic counterparts. This is indeed the case when considering a finite array of QEs in the <span class="mathjax-tex">$x$</span> direction, see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig3">3</a>. When the set of QEs is deeply embedded in the photonic bulk, each photonic boundary mode has a corresponding atomic mode with opposite chirality on the same boundary.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Hermitian topological reversal."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3: Hermitian topological reversal.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig3_HTML.png?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="203"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p><b>A</b> Quantum emitters (green spheres) coupled to a QWZ 2D photonic lattice (yellow plane); see Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ5">5</a>). Photonic open boundary conditions are imposed only in the <span class="mathjax-tex">$x$</span> direction. Here, <span class="mathjax-tex">$d$</span> is the thickness (in photonic unit cells) of the outer stripes separating the photonic and atomic boundaries; it interpolates between the case of a system full of emitters <span class="mathjax-tex">$(d=0)$</span>, and that of a finite array of quantum emitters in a finite but larger translationally invariant array of resonators along the <span class="mathjax-tex">$x$</span> direction <span class="mathjax-tex">$(d\gg 0)$</span>, at fixed <span class="mathjax-tex">$y$</span>. We emphasize that when <span class="mathjax-tex">$d$</span> is sufficiently large the topological properties do not depend on it anymore. <b>B</b> Energy spectrum of the full system. Emitters are coupled to the photonic bath, except for two stripes of size <span class="mathjax-tex">$d$</span> along its edges. Top row: photonic spectrum. Bottom row: zoom in of the top one, displaying the atomic spectrum. The colors denote the degree of localization in the <span class="mathjax-tex">$x$</span> direction according to the legend (bottom left). The triviality of the full light–matter system is evident from the absence of in-gap edge states (leftmost plot, <span class="mathjax-tex">$d=0$</span>). The effects of the topological reversal are apparent for large <span class="mathjax-tex">$d$</span> (rightmost plot): at each boundary the photonic and atomic boundary modes have opposite group velocities. Parameters: <span class="mathjax-tex">$L=50$</span> unit cells in the <span class="mathjax-tex">$x$</span> direction, <span class="mathjax-tex">$u=1.2$</span>, <span class="mathjax-tex">${\omega }_{e}=0$</span> and <span class="mathjax-tex">$g=0.1J$</span>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>We consider now the non-Hermitian case. The number of fundamental symmetry classes increases from 10 (AZ classes) to 38 (Bernard–LeClair classes) when the Hamiltonians are allowed to be non-Hermitian<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="D. Bernard, A. LeClair. in Statistical Field Theories, A. Cappelli, G. Mussardo, Eds. (Berlin: Springer, 2002), pp. 207–214." href="/articles/s41467-024-46471-w#ref-CR43" id="ref-link-section-d93114156e7454">43</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and Topology in Non-Hermitian Physics. Phys. Rev. X 9, 041015 (2019)." href="/articles/s41467-024-46471-w#ref-CR44" id="ref-link-section-d93114156e7457">44</a></sup>. Here, we focus only on a subclass of the latter that by definition generalizes the former. In particular, complex conjugation is no longer equivalent to transposition, which is a significant difference. The non-Hermitian equivalent of the Hermitian symmetries yield the following constraints for the non-Hermitian AZ classes <span class="mathjax-tex">${U}_{{{{{{\rm{TRS}}}}}}}{H}^{*}({{{{{\bf{k}}}}}}){U}_{{{{{{\rm{TRS}}}}}}}^{-1}=H(-{{{{{\bf{k}}}}}})$</span>, <span class="mathjax-tex">${U}_{{{{{{\rm{PHS}}}}}}}{H}^{{{{{{\rm{T}}}}}}}({{{{{\bf{k}}}}}}){U}_{{{{{{\rm{PHS}}}}}}}^{-1}=-H(-{{{{{\bf{k}}}}}})$</span>, and <span class="mathjax-tex">$S{H}^{{{\dagger}} }({{{{{\bf{k}}}}}}){S}^{-1}=-H({{{{{\bf{k}}}}}})$</span> <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and Topology in Non-Hermitian Physics. Phys. Rev. X 9, 041015 (2019)." href="/articles/s41467-024-46471-w#ref-CR44" id="ref-link-section-d93114156e7739">44</a></sup>, where *, T, and † represent complex conjugate, transpose, and Hermitian conjugate, respectively.</p><p>Furthermore, there are the non-Hermitian <span class="mathjax-tex">${{{{{{\rm{AZ}}}}}}}^{{{\dagger}} }$</span> classes<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and Topology in Non-Hermitian Physics. Phys. Rev. X 9, 041015 (2019)." href="/articles/s41467-024-46471-w#ref-CR44" id="ref-link-section-d93114156e7775">44</a></sup>, for which the symmetry constraints are <span class="mathjax-tex">${U}_{{{{{{\rm{TRS}}}}}}}{H}^{{{{{{\rm{T}}}}}}}({{{{{\bf{k}}}}}}){U}_{{{{{{\rm{TRS}}}}}}}^{-1}=H(-{{{{{\bf{k}}}}}})$</span>, <span class="mathjax-tex">${U}_{{{{{{\rm{PHS}}}}}}}{H}^{*}({{{{{\bf{k}}}}}}){U}_{{{{{{\rm{PHS}}}}}}}^{-1}=-H(-{{{{{\bf{k}}}}}})$</span> and <span class="mathjax-tex">$S{H}^{{{\dagger}} }({{{{{\bf{k}}}}}}){S}^{-1}=-H({{{{{\bf{k}}}}}})$</span>.</p><p>According to<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between hermitian and non-hermitian systems: Anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019)." href="/articles/s41467-024-46471-w#ref-CR35" id="ref-link-section-d93114156e8060">35</a></sup>, the topological classification of a Hermitian AZ class in <span class="mathjax-tex">$D$</span> dimensions coincides with that of a non-Hermitian <span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> <span class="mathjax-tex">$({{{{{{\rm{AZ}}}}}}}^{{{\dagger}}})$</span> class in <span class="mathjax-tex">$D+1\ (D-1)$</span> dimensions. This Hermitian–non-Hermitian correspondence is entirely consistent with the topological preservation and reversal found in this work.</p><p>For the non-Hermitian <span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> and <span class="mathjax-tex">${{{{{{\rm{AZ}}}}}}}^{{{\dagger}} }$</span> classes, we find that the non-Hermitian topology is always maintained with the exception of <span class="mathjax-tex">${\mathbb{Z}}$</span> phases in odd dimensions, where topological reversal occurs (see Methods). Here, we discuss two case studies illustrating the non-Hermitian topological reversal and preservation. We recall that the only requirement on <span class="mathjax-tex">${{{{{{\rm{\omega }}}}}}}_{{{{{{\rm{e}}}}}}}$</span> is that it does not belong to the photonic spectrum.</p><p>Topological reversal has non-trivial consequences for 1D systems such as those considered in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Anomalous behaviors of quantum emitters in non-hermitian baths. Phys. Rev. Lett. 129, 223601 (2022)." href="/articles/s41467-024-46471-w#ref-CR20" id="ref-link-section-d93114156e8281">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Roccati, F. et al. Exotic interactions mediated by a non-Hermitian photonic bath. Optica 9, 565 (2022)." href="/articles/s41467-024-46471-w#ref-CR45" id="ref-link-section-d93114156e8284">45</a></sup>. The topological origin of the largely investigated 1D non-Hermitian skin effect<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020)." href="/articles/s41467-024-46471-w#ref-CR34" id="ref-link-section-d93114156e8288">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Borgnia, D. S., Kruchkov, A. J. & Slager, R.-J. Non-Hermitian boundary modes and topology. Phys. Rev. Lett. 124, 056802 (2020)." href="/articles/s41467-024-46471-w#ref-CR46" id="ref-link-section-d93114156e8291">46</a></sup> is a point gap spectrum characterized by a nontrivial winding number <span class="mathjax-tex">$\nu$</span>. The sign of <span class="mathjax-tex">$\nu$</span> can in general be related to the boundary on which the skin states accumulate.</p><p>In addition, there exist more complex symmetry-protected variants, such as the <span class="mathjax-tex">${{{{{{\rm{TRS}}}}}}}^{{{\dagger}} }$</span>-protected <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> skin effect in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020)." href="/articles/s41467-024-46471-w#ref-CR34" id="ref-link-section-d93114156e8390">34</a></sup>. This provides an example in which the spectral winding always disappears, but a nontrivial <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> number indicates the presence of two skin modes localized at both boundaries. Notwithstanding, a nonzero winding number is characteristic of nonreciprocal models and typically results from the combination of a broken TRS and dissipation<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Clerk, A. A. Introduction to quantum non-reciprocal interactions: from non-Hermitian Hamiltonians to quantum master equations and quantum feedforward schemes. SciPost Phys. Lect. 44 
 https://doi.org/10.21468/SciPostPhysLectNotes.44
 
 (2022)." href="/articles/s41467-024-46471-w#ref-CR47" id="ref-link-section-d93114156e8423">47</a></sup>.</p><p>When occurring, the topological reversal ensures that photonic skin states on one edge correspond to atomic skin states on the opposite edge.</p><p>Similar to<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Bound states and photon emission in non-hermitian nanophotonics. Phys. Rev. A 106, 053517 (2022)." href="/articles/s41467-024-46471-w#ref-CR21" id="ref-link-section-d93114156e8433">21</a></sup>, we consider a photonic Hatano–Nelson 1D array with nonreciprocal right <span class="mathjax-tex">${J}_{R}=J(1+\delta )$</span> and left <span class="mathjax-tex">${J}_{L}=J(1-\delta )$</span> couplings, and uniform local dissipation <span class="mathjax-tex">$\gamma=2\delta J$</span></p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\hat{H}}_{p}=\mathop{\sum}\limits_{n}{J}_{R}{\hat{a}}_{n+1}^{{{\dagger}} }{\hat{a}}_{n}+{J}_{L}{\hat{a}}_{n}^{{{\dagger}} }{\hat{a}}_{n+1}-i\gamma {a}_{n}^{{{\dagger}} }{\hat{a}}_{n}$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>with QEs coupled to all resonators. Since only one resonator is present per unit cell, we drop the sublattice index in this discussion. Under open boundary conditions, the photonic skin modes accumulate on the right for <span class="mathjax-tex">$\delta > 0$</span>. The atomic periodic system possesses reversed topology (opposite windings), and therefore, its skin modes accumulate to the left (Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig4">4</a>).</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Non-Hermitian topological reversal."><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4: Non-Hermitian topological reversal.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig4_HTML.png?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="591"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p><b>A</b> A bare photonic 1D Hatano–Nelson model, Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ6">6</a> with <span class="mathjax-tex">$N=20$</span> resonators, where photonic skin states accumulate on the right edge (<span class="mathjax-tex">$\,\, {J}_{R} > {J}_{L}$</span>). When emitters are coupled to the same photonic lattice, under periodic boundary conditions, the atomic skin states accumulate on the left edge (<b>B</b>). Both figures show the normalized average of all skin modes <span class="mathjax-tex">$\left|{\psi }_{i}\right\rangle$</span>, <i>i.e.</i>, <span class="mathjax-tex">${\sum }_{i}{|\langle n\, {{{{{\rm{| }}}}}}\, {\psi }_{i}\rangle |}^{2}$</span>, with <span class="mathjax-tex">${{|}}n{{\rangle }}$</span> being the state where the photon (excitation) is located at the <i>n</i>th resonator (atom). Atomic open boundary conditions, inducing the skin effect, are obtained by removing <span class="mathjax-tex">$2d=10$</span> quantum emitters (outed violet stripes) while maintaining the periodic photonic structure. The inset in A shows the photonic (blue) and atomic (red) complex spectrum under periodic boundary conditions in units of <span class="mathjax-tex">$J$</span>. The inset in B shows a magnified view of the inset in A. We find opposite windings, witnessing the topological reversal. The vertical dashed orange axis is <span class="mathjax-tex">${{{{\mathrm{Re}}}}}E={{{{\mathrm{Re}}}}}\,{\omega }_{e}$</span>. The dashed black circle centered at <span class="mathjax-tex">${\omega }_{e}$</span> indicates the strength of the atom–photon interaction <span class="mathjax-tex">$g$</span> (the radius). This is highlighted because the reversal in <span class="mathjax-tex">$1{{{{{\rm{D}}}}}}$</span> can be described by a circular inversion with respect to this circle<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="H. Schwerdtfeger. Geometry of complex numbers (University of Toronto press, 2020)." href="/articles/s41467-024-46471-w#ref-CR57" id="ref-link-section-d93114156e9216">57</a></sup>. Parameters: <span class="mathjax-tex">$\delta=0.5,\, g=0.5J,\, {{{{{\rm{and}}}}}}$</span> <span class="mathjax-tex">${\omega }_{e}=-{iJ}$</span>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>The topology may be preserved in non-Hermitian cases as well. Consider a 2D chiral symmetric non-Hermitian photonic bath as an example. This system resembles the gapless surface states of three-dimensional chiral topological insulators<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Hosur, P., Ryu, S. & Vishwanath, A. Chiral topological insulators, superconductors, and other competing orders in three dimensions. Phys. Rev. B 81, 045120 (2010)." href="/articles/s41467-024-46471-w#ref-CR48" id="ref-link-section-d93114156e9318">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Asbóth, J. K., Oroszlány, L. & Pályi, A. A short course on topological insulators. Lect. Notes Phys. 919, 166 (2016)." href="/articles/s41467-024-46471-w#ref-CR49" id="ref-link-section-d93114156e9321">49</a></sup> from the perspective of the Hermitian–non-Hermitian correspondence<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between hermitian and non-hermitian systems: Anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019)." href="/articles/s41467-024-46471-w#ref-CR35" id="ref-link-section-d93114156e9325">35</a></sup>. Accordingly, it can be characterized by the net chiral charge of Dirac cones above the base energy. In our configuration, this is equal to <span class="mathjax-tex">${\omega }_{e}$</span> and is constrained by chiral symmetry to be entirely imaginary. We considered the model examined in<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Lee, J. Y., Ahn, J., Zhou, H. & Vishwanath, A. Topological correspondence between hermitian and non-hermitian systems: Anomalous dynamics. Phys. Rev. Lett. 123, 206404 (2019)." href="/articles/s41467-024-46471-w#ref-CR35" id="ref-link-section-d93114156e9357">35</a></sup>:</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${H}_{p}\left({{{{{\bf{k}}}}}}\right)=J\sin \left({k}_{x}\right){\tau }_{x}+J\sin \left({k}_{y}\right){\tau }_{y}+{iJ}\left(2\cos \left({k}_{x}\right)+\cos \left({k}_{y}\right)-3\right){I}_{2},$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>where <span class="mathjax-tex">${I}_{2}$</span> is the two-dimensional identity matrix. Its complex spectrum is shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig5">5A</a>. With <span class="mathjax-tex">${\omega }_{e}$</span> chosen as <span class="mathjax-tex">$-{iJ}$</span>, the topological invariant is 1 because there exists a single Dirac cone with a positive chiral charge located above <span class="mathjax-tex">${\omega }_{e}$</span>. The corresponding effective Hamiltonian for the emitters can then be derived from Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ3">3</a>. Its spectrum is shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-024-46471-w#Fig5">5B</a>. The same reasoning yields a topological invariant of 1 for the emitters’ Hamiltonian, indicating that the topology is preserved.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="Non-Hermitian topological preservation."><figure><figcaption><b id="Fig5" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 5: Non-Hermitian topological preservation.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/5" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig5_HTML.png?as=webp"><img aria-describedby="Fig5" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-024-46471-w/MediaObjects/41467_2024_46471_Fig5_HTML.png" alt="figure 5" loading="lazy" width="685" height="480"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-5-desc"><p><b>A</b> Spectrum of the <span class="mathjax-tex">$2{{{{{\rm{D}}}}}}$</span> chiral symmetric non-Hermitian photonic lattice<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Anomalous behaviors of quantum emitters in non-hermitian baths. Phys. Rev. Lett. 129, 223601 (2022)." href="/articles/s41467-024-46471-w#ref-CR20" id="ref-link-section-d93114156e9713">20</a></sup> (blue) and spectrum of the coupled emitters (red) obtained from Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ3">3</a>) (<b>B</b>) is a magnified image of (<b>A</b>). The dots show the Dirac cones with either <span class="mathjax-tex">$(+)$</span> or <span class="mathjax-tex">$(-)$</span> chiral charge. Each photonic Dirac cone above (below) <span class="mathjax-tex">${\omega }_{e}$</span> is mapped to an atomic Dirac cone above (below) <span class="mathjax-tex">${\omega }_{e}$</span> with the same chiral charge according to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ3">3</a>). The topological preservation follows from the fact that the topological invariant is equal to the total chiral charge above <span class="mathjax-tex">${\omega }_{e}$</span>. Dark-shaded areas are swiped twice as <span class="mathjax-tex">${{{{{\bf{k}}}}}}$</span> varies in the BZ. Parameters: <span class="mathjax-tex">${\omega }_{e}=-{iJ}$</span> and <span class="mathjax-tex">$g=0.5J$</span>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-024-46471-w/figures/5" data-track-dest="link:Figure5 Full size image" aria-label="Full size image figure 5" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div></div></div></section><section data-title="Discussion"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5">Discussion</h2><div class="c-article-section__content" id="Sec5-content"><p>The above results introduce a fundamental topological reversal/preservation principle that predicts, in a universal way, the occurrence of both Hermitian and non-Hermitian topological phases of emitters effectively interacting via photon exchange. Besides its conceptual importance, our theory provides a general model-independent recipe to engineer topological phases, which can be carried out in a variety of experimental scenarios ranging from superconducting circuits and quantum optical platforms to classical oscillators and nanophotonics. Indeed, arrays of coupled superconducting resonators represent a tunable and flexible platform to implement topological photonic lattices, as it was recently demonstrated<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Bello, M., Platero, G., Cirac, J. I. & González-Tudela, A. Unconventional quantum optics in topological waveguide QED. Sci. Adv. 5, eaaw0297 (2019)." href="/articles/s41467-024-46471-w#ref-CR16" id="ref-link-section-d93114156e9967">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Kim, E. et al. Quantum electrodynamics in a topological waveguide. Phys. Rev. X 11, 011015 (2021)." href="/articles/s41467-024-46471-w#ref-CR26" id="ref-link-section-d93114156e9970">26</a></sup>, in which case the emitters can well be implemented by superconducting resonators themselves or superconducting qubits. In addition, several recent works have demonstrated the non-Hermitian winding topology and the associated skin effect in setups such as robotic metamaterials<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat Commun 10, 4608 (2019)." href="/articles/s41467-024-46471-w#ref-CR50" id="ref-link-section-d93114156e9974">50</a></sup>, acoustic platforms<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Zhang, L. et al. Acoustic non-Hermitian skin effect from twisted winding topology. Nat Commun 12, 6297 (2021)." href="/articles/s41467-024-46471-w#ref-CR51" id="ref-link-section-d93114156e9978">51</a></sup>, and topolelectrical circuits<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Zou, D. et al. Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat Commun 12, 7201 (2021)." href="/articles/s41467-024-46471-w#ref-CR52" id="ref-link-section-d93114156e9982">52</a></sup> (the latter in higher dimensions as well). All these setups are perfectly suited to probe the topological correspondence that we unveil, and the reversed non-Hermitian skin effect in 1D in particular.</p><p>Of course, in any physical implementation, coupling to unwanted environmental modes is inevitable, resulting in some amount of photon loss in both the emitters and the engineered bath. As a consequence, the single-particle eigenstates that we have discussed will acquire a lifetime. To observe some nontrivial dynamics this lifetime should be larger than the characteristic timescale of the bath-mediated interactions <span class="mathjax-tex">$( \sim J/{g}^{2})$</span>. This requirement could be relaxed in the Hermitian case by post-selecting the measurements where the number of excitations is conserved. We note that the dynamics produced by the kind of bath-mediated interactions we consider have already been observed in experiments<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title="Zhang, X., Kim, E., Mark, D. K., Choi, S. & Painter, O. A superconducting quantum simulator based on a photonic-bandgap metamaterial. Science 379, 278–283 (2023)." href="/articles/s41467-024-46471-w#ref-CR53" id="ref-link-section-d93114156e10033">53</a></sup>.</p><p>The general principle we provide predicts the topological properties of the emitters’ Hamiltonian based solely on the topological properties of the photonic bath, its Hermitian/non-Hermitian nature, and its dimensionality. This system-bath topological correspondence sheds a new light on the emergence of exotic nonreciprocal interactions mediated by nonreciprocal 1D photonic baths observed in recent studies<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Anomalous behaviors of quantum emitters in non-hermitian baths. Phys. Rev. Lett. 129, 223601 (2022)." href="/articles/s41467-024-46471-w#ref-CR20" id="ref-link-section-d93114156e10040">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Bound states and photon emission in non-hermitian nanophotonics. Phys. Rev. A 106, 053517 (2022)." href="/articles/s41467-024-46471-w#ref-CR21" id="ref-link-section-d93114156e10043">21</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Roccati, F. et al. Exotic interactions mediated by a non-Hermitian photonic bath. Optica 9, 565 (2022)." href="/articles/s41467-024-46471-w#ref-CR45" id="ref-link-section-d93114156e10046">45</a></sup>. To showcase the effectiveness of our theory, we have considered 1D and 2D Hermitian and non-Hermitian models, thereby unveiling the occurrence of remarkable effects on the basis of the bulk-edge correspondence. For example, in a 2D Hermitian system, topological reversal enforces atomic edge modes featuring opposite group velocity compared to the photonic edge modes.</p><p>Our general classification requires that the system and the bath have the same number of degrees of freedom. By breaking this condition, we were able to show a rich variety of interesting cases (see Supplementary Information) which go beyond our topological correspondence principle. We thus provide a general criterion and fundamental hindsight that set down a cornerstone for the design of topological systems immersed in topological/non-topological Hermitian/non-Hermitian environments.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6">Methods</h2><div class="c-article-section__content" id="Sec6-content"><h3 class="c-article__sub-heading" id="Sec7">Photonic Hamiltonian</h3><p>The photonic Hamiltonian in real space is</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\hat{H}}_{p}=\mathop{\sum }\limits_{n,m=1}^{{N}_{c}}\mathop{\sum }\limits_{s,{s}^{{\prime} }=1}^{{N}_{b}}\langle {{{{{{\bf{r}}}}}}}_{n},s|{\hat{H}}_{p}|{{{{{{\bf{r}}}}}}}_{n+m},{s}^{{\prime} }\rangle {\hat{a}}_{{ns}}^{{{\dagger}} }{\hat{a}}_{n+m,{s}^{{\prime} }},$$</span></div><div class="c-article-equation__number"> (8) </div></div><p>where <span class="mathjax-tex">${N}_{c}$</span> is the number of unit cells, <span class="mathjax-tex">$|{{{{{{\bf{r}}}}}}}_{n},s\rangle={\hat{a}}_{{ns}}^{{{\dagger}} }{{{{{\rm{|vac}}}}}}{{\rangle }}$</span>, and the closure relation is <span class="mathjax-tex">${{\mathbb{1}}}_{p}={\sum }_{n,s}|{{{{{{\bf{r}}}}}}}_{n},s\rangle \langle {{{{{{\bf{r}}}}}}}_{n},s|$</span>. Because of translational invariance, the couplings are independent of the cell position, i.e., <span class="mathjax-tex">$\langle {{{{{{\bf{r}}}}}}}_{n},s|{\hat{H}}_{p}|{{{{{{\bf{r}}}}}}}_{n+m},{s}^{{\prime} }\rangle=$</span> <span class="mathjax-tex">$\langle {{{{{{\bf{r}}}}}}}_{0},s|{\hat{H}}_{p}|{{{{{{\bf{r}}}}}}}_{m},{s}^{{\prime} }\rangle$</span>.</p><p>Assuming periodic boundary conditions and using the closure relation <span class="mathjax-tex">${{\mathbb{1}}}_{p}={\sum }_{{{{{{\bf{k}}}}}},s}{{{{{\rm{|}}}}}}{{{{{\bf{k}}}}}},s{{\rangle }}{{\langle }}{{{{{\bf{k}}}}}},s{{{{{\rm{|}}}}}}$</span>, where <span class="mathjax-tex">${{{{{\rm{|}}}}}}{{{{{\bf{k}}}}}},s{{\rangle }}={\hat{a}}_{{{{{{\bf{k}}}}}}s}^{{{\dagger}} }{{{{{\rm{|vac}}}}}}{{\rangle }}$</span>, and <span class="mathjax-tex">$\langle {{{{{{\bf{r}}}}}}}_{n},\, s\, {{{{{\rm{| }}}}}} \, {{{{{\bf{k}}}}}},\, {s}^{{\prime} }\rangle={\delta }_{s{s}^{{\prime} }}{e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{r}}}}}}}_{n}}/\sqrt{{N}_{c}}$</span>, we have <span class="mathjax-tex">${\hat{a}}_{{ns}}={\sum }_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}{e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{r}}}}}}}_{n}}{\hat{a}}_{{{{{{\bf{k}}}}}}s}/\sqrt{{N}_{c}}$</span>. Thus, the photonic Hamiltonian of the periodic lattice is obtained as</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\hat{H}}_{p}=\mathop{\sum}\limits_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}\mathop{\sum }\limits_{s,{s}^{{\prime} }=1}^{{N}_{b}}\langle {{{{{{\bf{r}}}}}}}_{0},\, s|{\hat{H}}_{p}|{{{{{{\bf{r}}}}}}}_{n},\, {s}^{{\prime} }\rangle \, {e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{r}}}}}}}_{n}}{\hat{a}}_{{{{{{\bf{k}}}}}}s}^{{{\dagger}} }{\hat{a}}_{{{{{{\bf{k}}}}}}{s}^{{\prime} }}.$$</span></div><div class="c-article-equation__number"> (9) </div></div><p>By introducing the vector operator <span class="mathjax-tex">${\hat{A}}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }=({\hat{a}}_{{{{{{\bf{k}}}}}},1}^{{{\dagger}} },\ldots,\, {\hat{a}}_{{{{{{\bf{k}}}}}},{N}_{b}}^{{{\dagger}} })$</span> and denoting the matrix elements <span class="mathjax-tex">${[{H}_{p}({{{{{\bf{k}}}}}})]}_{s{s}^{{\prime} }}=\mathop{\sum}\nolimits_{n}\langle {{{{{{\bf{r}}}}}}}_{0},s|{\hat{H}}_{p}|{{{{{{\bf{r}}}}}}}_{n},{s}^{{\prime} }\rangle {e}^{-i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{r}}}}}}}_{n}}$</span> Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ9">9</a>) can be rewritten as <span class="mathjax-tex">${\hat{H}}_{p}={\sum }_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}{\hat{A}}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }{H}_{p}({{{{{\bf{k}}}}}}){\hat{A}}_{{{{{{\bf{k}}}}}}}$</span>, cf. Results section.</p><h3 class="c-article__sub-heading" id="Sec8">Effective Mediated Hamiltonian</h3><p>Consider the atomic frequency <span class="mathjax-tex">${\omega }_{e}$</span> to be shifted by <span class="mathjax-tex">$\Delta$</span> from the photonic continuum. If the atom–photon coupling <span class="mathjax-tex">$g$</span> is weak so that <span class="mathjax-tex">$g/\Delta \ll 1$</span>, it is possible to adiabatically eliminate the photonic bath and derive an effective photon-mediated atomic Hamiltonian <span class="mathjax-tex">${\hat{H}}_{a}$</span><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, P. Thickstun, Atom-photon interactions: basic processes and applications. (Wiley Online Library, 1992)." href="/articles/s41467-024-46471-w#ref-CR54" id="ref-link-section-d93114156e12199">54</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Sánchez-Burillo, E., Porras, D. & González-Tudela, A. Limits of photon-mediated interactions in one-dimensional photonic baths. Phys. Rev. A 102, 013709 (2020)." href="/articles/s41467-024-46471-w#ref-CR55" id="ref-link-section-d93114156e12202">55</a></sup><span class="mathjax-tex">$.$</span> The explicit expression for the case of one emitter per resonator is</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\hat{H}}_{a}={\hat{H}}_{e}+\mathop{\sum}\limits_{{nm}}\mathop{\sum}\limits_{s{s}^{{\prime} }}{h}_{{ns},m{s}^{{\prime} }}{\hat{\sigma }}_{{ns}}^{{{\dagger}} }{\hat{\sigma }}_{m{s}^{{\prime} }},$$</span></div><div class="c-article-equation__number"> (10) </div></div><p>where</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${h}_{{ns},m{s}^{{\prime} }}={g}^{2}\langle {{{{{{\bf{r}}}}}}}_{m},{s}^{{\prime} }|{\hat{G}}_{p}({\omega }_{e})|{{{{{{\bf{r}}}}}}}_{n},s\rangle,$$</span></div><div class="c-article-equation__number"> (11) </div></div><p><span class="mathjax-tex">${\hat{G}}_{p}\left(z\right)={\left(z-{\hat{H}}_{p}\right)}^{-1}$</span> is the Green’s function of the bare photonic Hamiltonian, and <span class="mathjax-tex">$|{{{{{{\bf{r}}}}}}}_{m},s\rangle$</span> is the state with one excitation in the <i>s</i>th resonator of the <i>m</i>th unit cell of the photonic lattice. The double index in the atomic operators specifies both the cell <span class="mathjax-tex">$(n,m)$</span> and sublattice <span class="mathjax-tex">$\left(s,{s}^{{\prime} }\right)$</span> the emitter is coupled to.</p><p>As <span class="mathjax-tex">${\hat{H}}_{p}$</span> is translationally invariant, so is its resolvent operator<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Economou, E. N. Green’s functions in quantum physics. Springer S. Solid State Sci. Berlin Heidelberg, Berlin, Heidelberg: Springer 7 (2006)." href="/articles/s41467-024-46471-w#ref-CR38" id="ref-link-section-d93114156e12873">38</a></sup> and</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\left\langle {{{{{{\bf{r}}}}}}}_{m}\left|{\hat{G}}_{p}\left({\omega }_{e}\right)\right|{{{{{{\bf{r}}}}}}}_{n}\right\rangle=\frac{1}{N}\mathop{\sum}\limits_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}\frac{{e}^{i{{{{{\bf{k}}}}}}\cdot \left({{{{{{\bf{r}}}}}}}_{m}-{{{{{{\bf{r}}}}}}}_{n}\right)}}{{\omega }_{e}-{H}_{p}({{{{{\bf{k}}}}}})},$$</span></div><div class="c-article-equation__number"> (12) </div></div><p>where <span class="mathjax-tex">$\langle {{{{{{\bf{r}}}}}}}_{m}|{\hat{G}}_{p}({\omega }_{e})|{{{{{{\bf{r}}}}}}}_{n}\rangle$</span> is the <span class="mathjax-tex">${N}_{b}\times {N}_{b}$</span> matrix in the sublattice space. Therefore, the atomic Hamiltonian can be written in Bloch form as <span class="mathjax-tex">${\hat{H}}_{a}=\mathop{\sum}\nolimits_{{{{{{\bf{k}}}}}}\in {{{{{\rm{BZ}}}}}}}{\hat{S}}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }{H}_{a}({{{{{\bf{k}}}}}}){\hat{S}}_{{{{{{\bf{k}}}}}}}$</span>, where <span class="mathjax-tex">${\hat{S}}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }=({\hat{\sigma }}_{{{{{{\bf{k}}}}}}1}^{{{\dagger}} },\ldots,{\hat{\sigma }}_{{{{{{\bf{k}}}}}}{N}_{b}}^{{{\dagger}} }),\, {H}_{a}({{{{{\bf{k}}}}}})$</span> is the Bloch Hamiltonian as in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ3">3</a> and <span class="mathjax-tex">${\hat{\sigma }}_{{{{{{\bf{k}}}}}}s}=\mathop{\sum}\nolimits_{n}{e}^{i{{{{{\bf{k}}}}}}\cdot {{{{{{\bf{r}}}}}}}_{n}}{\hat{\sigma }}_{{ns}}/\sqrt{N}$</span>, with <span class="mathjax-tex">${{{{{{\bf{r}}}}}}}_{n}$</span> being the atomic operator position in real space<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Gong, Z., Bello, M., Malz, D. & Kunst, F. K. Bound states and photon emission in non-hermitian nanophotonics. Phys. Rev. A 106, 053517 (2022)." href="/articles/s41467-024-46471-w#ref-CR21" id="ref-link-section-d93114156e13742">21</a></sup>. When there are fewer emitters than resonators that are still arranged so as to preserve translational invariance, one can repeat the above arguments with a small modification: the indices <span class="mathjax-tex">$s$</span> and <span class="mathjax-tex">${s}^{{\prime} }$</span> in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ11">11</a> belong only to the sublattices coupled to quantum emitters. This directly leads to the insertion of a projection operator in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ12">12</a> cf. Supplementary Information.</p><h3 class="c-article__sub-heading" id="Sec9">Triviality of the full atom-light Hamiltonian</h3><p>Here, we prove that the entire atom–light Hamiltonian in Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ2">2</a>) is topologically trivial. Its spectrum and eigenstates can be computed analytically as follows. Suppose <span class="mathjax-tex">${U}_{{{{{{\bf{k}}}}}}}$</span> is the unitary that diagonalizes <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}}),\, {U}_{{{{{{\bf{k}}}}}}}^{{{\dagger}} }{H}_{p}({{{{{\bf{k}}}}}}){U}_{{{{{{\bf{k}}}}}}}={{{{{\rm{diag}}}}}}\left({\omega }_{1}({{{{{\bf{k}}}}}}),\, {\omega }_{2}({{{{{\bf{k}}}}}}),\ldots,\, {\omega }_{N}({{{{{\bf{k}}}}}})\right)\equiv \Lambda ({{{{{\bf{k}}}}}})$</span>. Then,</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${\left({I}_{2}\otimes {U}_{{{{{{\bf{k}}}}}}}\right)}^{{{\dagger}} }H({{{{{\bf{k}}}}}})\left({I}_{2}\otimes {U}_{{{{{{\bf{k}}}}}}}\right)=\left[\begin{array}{cc}{\omega }_{e}I & {gI}\\ {gI} & \Lambda ({{{{{\bf{k}}}}}})\end{array}\right].$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>Thus, for each band of the bare bath, <span class="mathjax-tex">${\omega }_{j}({{{{{\bf{k}}}}}})$</span>, there are two bands <span class="mathjax-tex">${\omega }_{\pm , j}\left({{{{{\bf{k}}}}}}\right)=[{\omega }_{e}+{\omega }_{j}\left({{{{{\bf{k}}}}}}\right)]/2$</span> <span class="mathjax-tex">$\sqrt{{[{\omega }_{e}-{\omega }_{j}({{{{{\bf{k}}}}}})]}^{2}/4+{g}^{2}}$</span>, which are the eigenvalues of <span class="mathjax-tex">${H}_{j}({{{{{\bf{k}}}}}})=[{\omega }_{e}+{\omega }_{j}\left({{{{{\bf{k}}}}}}\right)]I/2+[{\omega }_{e}-{\omega }_{j}\left({{{{{\bf{k}}}}}}\right)]{\tau }_{z}/2+g{\tau }_{x}$</span>.</p><p>The corresponding eigenvectors are <span class="mathjax-tex">$|{v}_{\pm , j}({{{{{\bf{k}}}}}})\rangle \otimes$</span> <span class="mathjax-tex">$|{u}_{j}({{{{{\bf{k}}}}}})\rangle$</span>, where <span class="mathjax-tex">$|{u}_{j}({{{{{\bf{k}}}}}})\rangle$</span> is the eigenstate of <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> with eigenvalue <span class="mathjax-tex">${\omega }_{j}({{{{{\bf{k}}}}}})$</span>, while <span class="mathjax-tex">$|{v}_{\pm , j}({{{{{\bf{k}}}}}})\rangle$</span> is the eigenstate of <span class="mathjax-tex">${H}_{j}({{{{{\bf{k}}}}}})$</span> with eigenvalue <span class="mathjax-tex">${\omega }_{\pm , j}({{{{{\bf{k}}}}}})$</span>. Note that regardless the value of <span class="mathjax-tex">${\omega }_{e}$</span>, as long as it lies in the gap of the spectrum of the bare bath, half of the spectrum is above it and half below it, i.e., <span class="mathjax-tex">${\omega }_{- ,j}({{{{{\bf{k}}}}}}) < {\omega }_{e} < {\omega }_{+,j}({{{{{\bf{k}}}}}})$</span> for all <span class="mathjax-tex">$j$</span> and <span class="mathjax-tex">${{{{{\bf{k}}}}}}$</span>. If we now consider the bands below <span class="mathjax-tex">${\omega }_{e}$</span> and compute the topological invariant, we can consider instead the topologically equivalent Hamiltonian <span class="mathjax-tex">$H=I-2P({{{{{\bf{k}}}}}})$</span>, where <span class="mathjax-tex">$P({{{{{\bf{k}}}}}})={\sum }_{j}|{u}_{j}({{{{{\bf{k}}}}}})\rangle \langle {u}_{j}({{{{{\bf{k}}}}}}){{{{{\rm{|}}}}}}\otimes {{{{{\rm{|}}}}}}-\rangle {{\langle }}-{{{{{\rm{|}}}}}}=I\otimes {{{{{\rm{|}}}}}}-{{\rangle }}{{\langle }}-{{{{{\rm{|}}}}}}$</span>, with a constant <span class="mathjax-tex">${{{{{\rm{|}}}}}}-{{\rangle }}$</span>; therefore <span class="mathjax-tex">${dH}=0$</span>, so <span class="mathjax-tex">${{{{{{\rm{Ch}}}}}}}_{n}=0$</span>, cf. Equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ20">20</a>) below.</p><p>For chiral systems in odd dimensions, the Bloch Hamiltonian of the bath can be written as<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016)." href="/articles/s41467-024-46471-w#ref-CR33" id="ref-link-section-d93114156e15539">33</a></sup></p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${H}_{p}({{{{{\bf{k}}}}}})=\left[\begin{array}{cc}0 & {Q}_{p}({{{{{\bf{k}}}}}})\\ {Q}_{p}^{{{\dagger}} }({{{{{\bf{k}}}}}}) & 0\end{array}\right],$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>with <span class="mathjax-tex">${Q}_{p}({{{{{\bf{k}}}}}})$</span> being a suitable matrix. Then, the Bloch Hamiltonian of the bath with emitters <span class="mathjax-tex">$\left({\omega }_{e}=0\right)$</span> can also be written in the same block-off-diagonal form with</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$Q({{{{{\bf{k}}}}}})=\left[\begin{array}{cc}{Q}_{p}({{{{{\bf{k}}}}}}) & {gI}\\ {gI} & 0\end{array}\right].$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>Note that the inverse is given by</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$Q{({{{{{\bf{k}}}}}})}^{-1}=\left[\begin{array}{cc}0 & {g}^{-1}I\\ {g}^{-1}I & -{g}^{-2}{Q}_{p}({{{{{\bf{k}}}}}})\end{array}\right].$$</span></div><div class="c-article-equation__number"> (16) </div></div><p>Thus,</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${Q}^{-1}{dQ}=\mathop{\sum}\limits_{j}\left[\begin{array}{cc}0 & 0\\ {g}^{-1}{\partial }_{j}Q & 0\end{array}\right]d{k}_{j}$$</span></div><div class="c-article-equation__number"> (17) </div></div><p>As a consequence, <span class="mathjax-tex">${{{{{\rm{Tr}}}}}}[{({Q}^{-1}{dQ})}^{2n+1}]=0$</span>, so <span class="mathjax-tex">${\nu }_{2n+1}=0$</span>, cf. Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ19">19</a>).</p><p>Finally, we provide an alternative proof showing that the entire system is trivial without referring to any formulas of topological invariants. Equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ2">2</a>) can be continuously deformed into <span class="mathjax-tex">${H}_{1}=\left({\omega }_{e}{I}_{2}+g{\tau }_{x}\right)\otimes I$</span>, with <span class="mathjax-tex">${I}_{2}$</span> being the 2 × 2 identity matrix, via a linear interpolation <span class="mathjax-tex">${H}_{\lambda }({{{{{\bf{k}}}}}})=(1-\lambda )H({{{{{\bf{k}}}}}})+\lambda {H}_{1},\, \lambda \in [0,1]$</span>.</p><p>Further, <span class="mathjax-tex">$\det \left({H}_{\lambda }({{{{{\bf{k}}}}}})-{\omega }_{e}{I}_{2}\otimes I\right)=$</span> <span class="mathjax-tex">$\det \left(-{g}^{2}I\right)\ne 0$</span>, so the Hamiltonian remains gapped near <span class="mathjax-tex">${\omega }_{e}$</span> during the deformation. Note that any time–reversal symmetry is preserved, and so is the particle–hole (chiral) symmetry if it is extended as <span class="mathjax-tex">$(-C)\oplus C((-S)\oplus S)$</span>. Since <span class="mathjax-tex">${H}_{1}$</span> does not depend on <span class="mathjax-tex">${{{{{\bf{k}}}}}}$</span> and is thus trivial, we conclude that <span class="mathjax-tex">$H({{{{{\bf{k}}}}}})$</span>, which is continuously connected to <span class="mathjax-tex">${H}_{1}$</span>, is also trivial. Note that the above proof applies equally to Hermitian and non-Hermitian systems. Moreover, the fact that an appropriately extended chiral symmetry requires a minus sign on the emitter side explains why the triviality of the entire system does not contradict the topological preservation in chiral symmetric systems.</p><h3 class="c-article__sub-heading" id="Sec10">Proof of topological preservation and reversal</h3><p>Here, we provide a general analysis of the fundamental symmetry classes (in Hermitian AZ, non-Hermitian AZ, or non-Hermitian <span class="mathjax-tex">${{{{{\rm{A}}}}}}{{{{{{\rm{Z}}}}}}}^{{{\dagger}} }$</span>) that exhibit topological reversal or otherwise topological preservation for the one-emitter-per-resonator setup. To ensure that the PHS and chiral symmetry are inherited, we focused on the case of <span class="mathjax-tex">${\omega }_{e}={\omega }_{0}=0$</span> (<span class="mathjax-tex">${\omega }_{0}$</span> is the bare resonator frequency), so that <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})=-{g}^{2}{H}_{p}{({{{{{\bf{k}}}}}})}^{-1}$</span> with both <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> and <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> gapped near 0. We observed that for the non-Hermitian case, both Bloch Hamiltonians must be point-gapped around <span class="mathjax-tex">${\omega }_{e}$</span> with a negative imaginary part so that their spectra lie below the real axis in the complex energy plane. This is a rigid shift along the imaginary axis that does not affect the following discussion.</p><p>One obvious observation is that the mapping from <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> to <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> is invertible. This immediately implies that after obtaining the topological equivalence classes of <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> and <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span>, we obtain an automorphism on the classification group. Recalling that nontrivial Hermitian <span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> classes are classified by <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> or <span class="mathjax-tex">${\mathbb{Z}},$</span> and so are the non-Hermitian AZ <span class="mathjax-tex">$({{{{{{\rm{AZ}}}}}}}^{{{\dagger}} })$</span> classes, it suffices to consider the automorphisms on <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> or <span class="mathjax-tex">${\mathbb{Z}}$</span> (with respect to addition). We note here that in the literature, the topological classifications of some classes are usually denoted as <span class="mathjax-tex">$2{\mathbb{Z}}$</span>, meaning that the winding number or Chern number can only be an even integer. Nevertheless, since <span class="mathjax-tex">$2{\mathbb{Z}}$</span> is isomorphic to <span class="mathjax-tex">${\mathbb{Z}}$</span>, the convention <span class="mathjax-tex">${\mathbb{Z}}$</span> is also used. In the former case <span class="mathjax-tex">$\left({{\mathbb{Z}}}_{2}\right)$</span>, the only automorphism is the identity map, implying that all the <span class="mathjax-tex">${{\mathbb{Z}}}_{2}$</span> phases exhibit topological preservation. In the latter case <span class="mathjax-tex">$({\mathbb{Z}})$</span>, the only two possibilities of an automorphism are the identity map and inversion <span class="mathjax-tex">$(n\mapsto -n)$</span>, corresponding to topological preservation and reversal, respectively. We emphasize that the above results apply to both Hermitian and non-Hermitian systems. The problem is thus reduced to distinguishing the <span class="mathjax-tex">${\mathbb{Z}}$</span> phases exhibiting topological reversal from those exhibiting topological preservation.</p><p>We first consider the Hermitian case. Using the band flattening <span class="mathjax-tex">$(H\to {{{{\mathrm{sgn}}}}}H)$</span> technique<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Kitaev, A. Periodic table for topological insulators and superconductors. A.I.P. Conf. Proc. 1134, 22 (2009)." href="/articles/s41467-024-46471-w#ref-CR41" id="ref-link-section-d93114156e17755">41</a></sup>, the map from <span class="mathjax-tex">${H}_{p}({{{{{\bf{k}}}}}})$</span> to <span class="mathjax-tex">${H}_{a}({{{{{\bf{k}}}}}})$</span> can be simplified into a simple inversion <span class="mathjax-tex">$(H\to -H)$</span> at the level of topological equivalence classes. If the spatial dimension is odd, all the <span class="mathjax-tex">${\mathbb{Z}}$</span> phases are chiral symmetric, and thus, the Bloch Hamiltonian takes the following form:</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H({{{{{\bf{k}}}}}})=\left[\begin{array}{cc}0 & Q({{{{{\bf{k}}}}}})\\ Q{({{{{{\bf{k}}}}}})}^{{{\dagger}} } & 0\end{array}\right].$$</span></div><div class="c-article-equation__number"> (18) </div></div><p>The integer topological invariant is the winding number given by</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\nu \propto {\int }_{{\!\!\!\!{{{{\rm{BZ}}}}}}}{{{{{\rm{Tr}}}}}}{\left({Q}^{-1}{dQ}\right)}^{D}.$$</span></div><div class="c-article-equation__number"> (19) </div></div><p>Obviously, this topological invariant does not change under inversion <span class="mathjax-tex">$H\to -H$</span> (leading to <span class="mathjax-tex">$\to -Q$</span>). Otherwise, in even spatial dimensions, the topological invariant is the Chern number, which is determined by the flattened Bloch Hamiltonian <span class="mathjax-tex">$H({{{{{\bf{k}}}}}})$</span> via</p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$${{{{{\rm{Ch}}}}}}\propto {\int }_{{\!\!\!\!{{{{\rm{BZ}}}}}}}{{{{{\rm{Tr}}}}}}\left(H{({dH})}^{D}\right).$$</span></div><div class="c-article-equation__number"> (20) </div></div><p>However, unlike the winding number, the Chern number is inversed upon the inversion of the Hamiltonian.</p><p>We now move to the non-Hermitian case. Here, the counterpart of band flattening is unitarization<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="Gong, Z. et al. Topological phases of Non-Hermitian systems. Phys. Rev. X 8, 031079 (2018)." href="/articles/s41467-024-46471-w#ref-CR56" id="ref-link-section-d93114156e18263">56</a></sup>, <span class="mathjax-tex">$H\to V=H{\left(\sqrt{{H}^{{{\dagger}} }H}\right)}^{-1}$</span>, upon which the photon–atom map is simplified to <span class="mathjax-tex">$V\to -{V}^{{{\dagger}} }$</span>. The topological invariants in odd dimensions are the winding numbers given in Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ19">19</a> and we always have a topological reversal. In even dimensions, any <span class="mathjax-tex">${\mathbb{Z}}$</span> topological phase exhibits a chiral symmetry <span class="mathjax-tex">$S$</span>, <i>i.e.</i>, <span class="mathjax-tex">$S{H}^{{{\dagger}} }({{{{{\bf{k}}}}}}){S}^{-1}=-H({{{{{\bf{k}}}}}})$</span>. The integer topological invariant is then given by the Chern number, Eq. <a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-024-46471-w#Equ20">20</a> for <i>H = iSV</i> <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and Topology in Non-Hermitian Physics. Phys. Rev. X 9, 041015 (2019)." href="/articles/s41467-024-46471-w#ref-CR44" id="ref-link-section-d93114156e18499">44</a></sup>, which is confirmed to be Hermitian and flattened (i.e., square to identity). After the operation <span class="mathjax-tex">$V\to -{V}^{{{\dagger}} }$</span>, this quantity turns out to undergo a unitary conjugation, <span class="mathjax-tex">${iSV}\to -{iS}{V}^{{{\dagger}} }={iVS}={S}^{-1}({iSV})S$</span>, leaving the Chern number unchanged.</p><p>In summary, for the Hermitian AZ classes, a topological reversal occurs only for <span class="mathjax-tex">${\mathbb{Z}}$</span> phases in even dimensions. For the non-Hermitian <span class="mathjax-tex">${{{{{\rm{AZ}}}}}}$</span> and <span class="mathjax-tex">${{{{{{\rm{AZ}}}}}}}^{{{\dagger}} }$</span> classes, a topological reversal occurs only for <span class="mathjax-tex">${\mathbb{Z}}$</span> phases in odd dimensions.</p></div></div></section> </div> <div> <section data-title="Data availability"><div class="c-article-section" id="data-availability-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="data-availability">Data availability</h2><div class="c-article-section__content" id="data-availability-content"> <p>All data needed to evaluate the conclusions in this study are present in the paper and in the Supplementary Information.</p> </div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">von Klitzing, K. et al. 40 years of the quantum Hall effect. <i>Nat. 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Schwerdtfeger. <i>Geometry of complex numbers</i> <i>(</i>University of Toronto press, <i>2020</i><i>)</i>.</p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/s41467-024-46471-w?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>We acknowledge Diego Porras and Pablo Martínez-Azcona for useful discussions. F.R. and A. Ch. acknowledge support from the Luxembourg National Research Fund (FNR, Attract grant 15382998). M.B. was supported by the projects FermiQP and EQUAHUMO of the Bildungsministerium für Bildung und Forschung (BMBF). Z.G. was supported by the Max-Planck-Harvard Research Center for Quantum Optics (MPHQ). M.B. and Z.G. acknowledge financial support from the Munich Center for Quantum Science and Technology (MCQST), funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy (EXC2111-390814868). M.U. acknowledges the support by KAKENHI Grant No. JP22H01152 from the Japan Society for the Promotion of Science. A.Ca. and F.C. acknowledge support from European Union – Next Generation EU through Project Eurostart 2022 Topological atom-photon interactions for quantum technologies (MUR D.M. 737/2021) and through Project PRIN 2022-PNRR no. P202253RLY “Harnessing topological phases for quantum technologies”.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Department of Physics and Materials Science, University of Luxembourg, L-1511, Luxembourg, Luxembourg</p><p class="c-article-author-affiliation__authors-list">Federico Roccati & Aurélia Chenu</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, Garching, 85748, Germany</p><p class="c-article-author-affiliation__authors-list">Miguel Bello & Zongping Gong</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">Munich Center for Quantum Science and Technology, Schellingstraße 4, 80799, München, Germany</p><p class="c-article-author-affiliation__authors-list">Miguel Bello & Zongping Gong</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama, 351-0198, Japan</p><p class="c-article-author-affiliation__authors-list">Zongping Gong</p></li><li id="Aff5"><p class="c-article-author-affiliation__address">Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan</p><p class="c-article-author-affiliation__authors-list">Zongping Gong</p></li><li id="Aff6"><p class="c-article-author-affiliation__address">Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan</p><p class="c-article-author-affiliation__authors-list">Masahito Ueda</p></li><li id="Aff7"><p class="c-article-author-affiliation__address">RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan</p><p class="c-article-author-affiliation__authors-list">Masahito Ueda</p></li><li id="Aff8"><p class="c-article-author-affiliation__address">Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan</p><p class="c-article-author-affiliation__authors-list">Masahito Ueda</p></li><li id="Aff9"><p class="c-article-author-affiliation__address">Università degli Studi di Palermo, Dipartimento di Fisica e Chimica-Emilio Segrè, via Archirafi 36, I-90123, Palermo, Italy</p><p class="c-article-author-affiliation__authors-list">Francesco Ciccarello & Angelo Carollo</p></li><li id="Aff10"><p class="c-article-author-affiliation__address">NEST, Istituto Nanoscienze-CNR, Piazza S. 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F.R., M.B. and Z.G. developed the theory. All authors discussed the results and contributed to writing the manuscript. M.U, F.C. and A.Ch supervised the work.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:federico.roccati@uni.lu">Federico Roccati</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar2">Competing interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Peer review"><div class="c-article-section" id="peer-review-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="peer-review">Peer review</h2><div class="c-article-section__content" id="peer-review-content"> <h3 class="c-article__sub-heading" id="FPar1">Peer review information</h3> <p><i>Nature Communications</i> thanks the anonymous reviewers for their contribution to the peer review of this work. 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