pith. sign in

arxiv: 2605.06771 · v1 · submitted 2026-05-07 · 🪐 quant-ph · cond-mat.mes-hall

Giant-atom-enabled quantum optics with valley-polarized photons

Marcel A. Pinto , Giovanni Luca Sferrazza , Silvia Casulleras , Alejandro Gonzalez-Tudela , Daniele De Bernardis , Francesco Ciccarello This is my paper

Pith reviewed 2026-05-11 00:59 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords giant atomsvalleytronicsquantum opticshoneycomb latticeschiral emissionphotonic edge modesBerry curvaturedomain walls
0
0 comments X

The pith

A giant atom can emit photons selectively into one valley of a honeycomb lattice by tailoring its coupling geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work shows how a simple two-level qubit, when coupled nonlocally at several points to a honeycomb lattice of resonators with detuned sublattice frequencies, can emit light into just one of the lattice's two valleys. The choice of where the coupling points are placed determines which valley is selected, giving the photons a definite valley index along with the lattice's Berry curvature. Positioning the qubit near a domain wall that separates areas with opposite detuning turns the emission into chiral flow along the wall's edge modes. Such chiral single-photon channels arise without needing to break the time-reversal symmetry of the medium, opening possibilities in circuit quantum electrodynamics.

Core claim

Tailoring the geometry of the coupling points allows the giant atom to emit selectively into a single valley. The emitted photons thereby acquire a well-defined valley character and inherit the associated Berry curvature. By placing the qubit near a domain wall between regions of opposite sublattice detuning, whose interface supports valley-polarized edge modes, emission becomes chiral along the domain wall.

What carries the argument

The giant atom, realized as a qubit with multiple nonlocal coupling points to the resonator lattice, whose positions are chosen to break symmetry between the two valleys.

Load-bearing premise

The honeycomb lattice of resonators with controllable sublattice detuning can be fabricated and the giant atom's nonlocal couplings can be implemented with enough precision and low enough loss to achieve the valley selectivity.

What would settle it

If experiments with a fabricated circuit-QED device show that emitted photons do not preferentially follow one valley's propagation characteristics or fail to show chirality along the domain wall, the selective emission claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.06771 by Alejandro Gonzalez-Tudela, Daniele De Bernardis, Francesco Ciccarello, Giovanni Luca Sferrazza, Marcel A. Pinto, Silvia Casulleras.

Figure 1
Figure 1. Figure 1: Sketch of the setup: a two-level giant atom is cou [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bare lattice spectrum. (a) Density plot of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Homogeneous lattice: spontaneous emission of a normal and a giant atom for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Valley-polarized edge modes under a spatially vary [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Lattice with a non-uniform sublattice detuning [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

Valleytronics and valley photonics exploit the valley degree of freedom to encode and manipulate information. Here we show that photonic valleys can be selectively addressed in quantum optics using a simple two-level emitter, provided it is coupled nonlocally to the field, thereby realizing a so-called giant atom. Specifically, we consider a qubit coupled at multiple points to an engineered honeycomb lattice of resonators with detuned sublattice frequencies. By tailoring the geometry of the coupling points, the giant atom can be made to emit selectively into a single valley. The emitted photons thereby acquire a well-defined valley character and inherit the associated Berry curvature. By placing the qubit near a domain wall between regions of opposite sublattice detuning, whose interface supports valley-polarized edge modes, emission becomes chiral along the domain wall. This provides a promising route toward implementation of single-photon disorder-robust chiral emission without breaking time-reversal symmetry of the electromagnetic medium in platforms such as circuit QED.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using a giant atom (two-level qubit with nonlocal couplings) to a honeycomb lattice of resonators with sublattice detuning to achieve valley-selective photon emission. Tailoring the geometry of coupling points exploits interference to suppress emission into one valley (K or K') while allowing the other, so emitted photons acquire definite valley character and associated Berry curvature. Placing the qubit near a domain wall between regions of opposite detuning enables chiral propagation along the supported valley-polarized edge modes, all without breaking time-reversal symmetry of the medium. The approach is suggested for circuit-QED implementations of disorder-robust chiral single-photon sources.

Significance. If the interference-based valley selectivity and chiral edge-mode coupling hold, the result would be significant for quantum optics and valleytronics. It provides a route to chiral emission in time-reversal-symmetric photonic systems, leveraging giant-atom physics and valley degrees of freedom for potential applications in robust quantum information processing. The proposal avoids the usual requirement to break TRS, which is a practical advantage in resonator lattices.

major comments (2)
  1. [Abstract and §II] Abstract and §II (model): the central claim that geometric tailoring of coupling points produces perfect (or near-perfect) valley selectivity via phase cancellation at the K/K' points is load-bearing, yet no explicit derivation of the coupling amplitudes, decay rates into each valley, or the resulting master equation appears in the visible text. The interference argument is stated at a high level but must be shown quantitatively (e.g., via the Fourier transform of the coupling positions evaluated at the Dirac points) to confirm suppression of one valley.
  2. [§III] §III (domain-wall emission): the assertion that emission becomes chiral along the valley-polarized edge modes relies on the qubit coupling preferentially to those modes. The manuscript should provide the explicit overlap integrals or the projected master equation demonstrating unidirectional propagation, including any dependence on the distance to the domain wall and the sublattice detuning strength.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a schematic figure illustrating the giant-atom coupling points on the honeycomb lattice and the domain-wall geometry.
  2. [§II] Notation for the sublattice detuning and the valley index should be defined consistently when first introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the work and for the constructive comments. We have revised the manuscript to incorporate explicit derivations and quantitative details as requested, improving the clarity of the central claims.

read point-by-point responses

  1. Referee: [Abstract and §II] Abstract and §II (model): the central claim that geometric tailoring of coupling points produces perfect (or near-perfect) valley selectivity via phase cancellation at the K/K' points is load-bearing, yet no explicit derivation of the coupling amplitudes, decay rates into each valley, or the resulting master equation appears in the visible text. The interference argument is stated at a high level but must be shown quantitatively (e.g., via the Fourier transform of the coupling positions evaluated at the Dirac points) to confirm suppression of one valley.

    Authors: We agree that an explicit quantitative derivation is essential for rigor. In the revised manuscript we have expanded Section II with the full calculation of the coupling amplitudes obtained from the Fourier transform of the coupling positions evaluated at the K and K' points. This explicitly demonstrates the phase cancellation that suppresses emission into one valley while allowing the other. We also derive the valley-dependent decay rates and present the resulting master equation governing the giant-atom dynamics. revision: yes

  2. Referee: [§III] §III (domain-wall emission): the assertion that emission becomes chiral along the valley-polarized edge modes relies on the qubit coupling preferentially to those modes. The manuscript should provide the explicit overlap integrals or the projected master equation demonstrating unidirectional propagation, including any dependence on the distance to the domain wall and the sublattice detuning strength.

    Authors: We thank the referee for this suggestion. The revised Section III now includes the explicit overlap integrals between the giant atom's coupling points and the valley-polarized edge modes. We derive the projected master equation for the unidirectional propagation and analyze its dependence on the distance from the qubit to the domain wall as well as the sublattice detuning strength, confirming that chiral emission is robust within the relevant parameter regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives valley-selective emission and chiral domain-wall coupling directly from the tight-binding Hamiltonian of a detuned honeycomb lattice plus the standard nonlocal giant-atom master equation. Selection rules follow from explicit phase factors at the K/K' points due to coupling-point geometry; no parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled in. The logic is self-contained and externally falsifiable via the underlying lattice model and quantum-optics formalism.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proposal rests on standard assumptions of valley photonics and giant-atom physics; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption A honeycomb lattice with detuned sublattice frequencies supports valley modes carrying Berry curvature.
    Invoked to explain the valley character of emitted photons.
  • domain assumption Nonlocal coupling points of a giant atom can be chosen to address a single valley selectively.
    Core premise of the selective-emission claim.

pith-pipeline@v0.9.0 · 5482 in / 1336 out tokens · 41887 ms · 2026-05-11T00:59:01.893045+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 1 internal anchor

  1. [1]

    Giant-atom-enabled quantum optics with valley-polarized photons

    and the valley Hall effect [1, 5]. An analogous valley-Hall mechanism exists in pho- tonics: in engineered photonic crystals, inversion- symmetry breaking opens a topological band gap at the inequivalentKandK ′ valleys, which carry oppo- site valley character [6, 7]. This enables robust valley- selective transport through valley-Hall edge states, kink mod...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    and replacing∆→∆(x)in the effective massive 7 Dirac Hamiltonian (18) for each valleyτ, one can de- rive analytical expressions for the edge modes and their spectrum nearω=0, where the dispersion is approxi- mately linear [see Fig. 4(a)]. The resulting edge modes can be written in real space as single-photon Fock states Eτ,qy E = ∑ R h Ψτ,qy (R)a † R +Φ τ,...

    work page doi:10.13039/501100011033 2022

  3. [3]

    D. Xiao, W. Yao, and Q. Niu, Valley-contrasting physics in graphene: Magnetic moment and topological trans- port, Phys. Rev. Lett.99, 236809 (2007)

    work page 2007

  4. [4]

    J. R. Schaibley, H. Yu, G. Clark, P . Rivera, J. S. Ross, K. L. Seyler, W. Yao, and X. Xu, Valleytronics in 2d materials, Nature Reviews Materials1, 16055 (2016)

    work page 2016

  5. [5]

    X. Xu, W. Yao, D. Xiao, and T. F. Heinz, Spin and pseu- dospins in layered transition metal dichalcogenides, Na- ture Physics10, 343 (2014)

    work page 2014

  6. [6]

    K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS2 by optical helicity, Nature Nanotechnology7, 494 (2012)

    work page 2012

  7. [7]

    R. V . Gorbachev, J. C. W. Song, G. L. Yu, A. V . Kretinin, F. Withers, Y. Cao, A. Mishchenko, I. V . Grigorieva, K. S. Novoselov, L. S. Levitov, and A. K. Geim, Detecting topo- logical currents in graphene superlattices, Science346, 448 (2014)

    work page 2014

  8. [8]

    Ma and G

    T. Ma and G. Shvets, All-Si valley-Hall photonic topolog- ical insulator, New J. Phys.18, 025012 (2016)

    work page 2016

  9. [9]

    Dong, X.-D

    J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, Valley photonic crystals for control of spin and topology, Nat. Mater.16, 298 (2017)

    work page 2017

  10. [10]

    J. Noh, S. Huang, K. P . Chen, and M. C. Rechtsman, Ob- servation of photonic topological valley-Hall edge states, Phys. Rev. Lett.120, 063902 (2018)

    work page 2018

  11. [11]

    F. Gao, H. Xue, Z. Yang, K. Lai, Y. Yu, X. Lin, Y. Chong, G. Shvets, and B. Zhang, Topologically protected refrac- tion of robust kink states in valley photonic crystals, Nat. Phys.14, 140 (2018)

    work page 2018

  12. [12]

    Y. Kang, X. Ni, X. Cheng, A. B. Khanikaev, and A. Z. Genack, Pseudo-spin–valley coupled edge states in a photonic topological insulator, Nat. Commun.9, 3029 11 (2018)

    work page 2018

  13. [13]

    Barik, A

    S. Barik, A. Karasahin, C. Flower, T. Lu, H. Miyake, M. Hafezi, and E. Waks, Chiral quantum optics using a topological resonator, Phys. Rev. B101, 205303 (2020)

    work page 2020

  14. [14]

    M. J. Mehrabad, A. P . Foster, R. Dost, E. Clarke, P . K. Patil, A. M. Fox, M. S. Skolnick, and L. R. Wilson, Chiral topological photonics with an embedded quantum emit- ter, Optica7, 1690 (2020)

    work page 2020

  15. [15]

    M. J. Mehrabad, A. P . Foster, N. J. Martin, R. Dost, E. Clarke, P . K. Patil, M. S. Skolnick, and L. R. Wilson, Chiral topological add-drop filter for integrated quantum photonic circuits, Optica10, 415 (2023)

    work page 2023

  16. [16]

    S. Shi, X. Xie, S. Yan, J. Yang, J. Dang, S. Xiao, L. Yang, D. Dai, B. Fu, Y. Yuan, R. Zhu, X. Su, H. Liu, Z. Zuo, C. Wang, H. Ni, Z. Niu, Q. Gong, and X. Xu, Asymmetric chiral coupling in a topological resonator, Appl. Phys. Lett.122, 191104 (2023)

    work page 2023

  17. [17]

    Lodahl, S

    P . Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P . Schneeweiss, J. Volz, H. Pichler, and P . Zoller, Chiral quantum optics, Nature541, 473 (2017)

    work page 2017

  18. [18]

    D. G. Suárez-Forero, M. Jalali Mehrabad, C. Vega, A. González-Tudela, and M. Hafezi, Chiral quantum op- tics: Recent developments and future directions, PRX Quantum6, 020101 (2025)

    work page 2025

  19. [19]

    X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits, Physics Reports718, 1 (2017)

    work page 2017

  20. [20]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Reviews of Modern Physics93, 025005 (2021)

    work page 2021

  21. [21]

    A. F. Kockum, Quantum optics with giant atoms—the first five years, inInternational Symposium on Mathemat- ics, Quantum Theory, and Cryptography, edited by T. Tak- agi, M. Wakayama, K. Tanaka, N. Kunihiro, K. Kimoto, and Y. Ikematsu (Springer Singapore, Singapore, 2021) pp. 125–146

    work page 2021

  22. [22]

    A. F. Kockum, G. Johansson, and F. Nori, Decoherence- free interaction between giant atoms in waveguide quan- tum electrodynamics, Phys. Rev. Lett.120, 140404 (2018)

    work page 2018

  23. [23]

    Frisk Kockum, P

    A. Frisk Kockum, P . Delsing, and G. Johansson, Design- ing frequency-dependent relaxation rates and lamb shifts for a giant artificial atom, Phys. Rev. A90, 013837 (2014)

    work page 2014

  24. [24]

    L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Giant acoustic atom: A single quantum system with a deterministic time delay, Phys. Rev. A95, 053821 (2017)

    work page 2017

  25. [25]

    Y. Wang, J. J. García-Ripoll, and A. C. Santos, Decoherence-free qubit and chiral emission from a giant molecule in waveguide QED, arXiv preprint arXiv:2603.27443 10.48550/arXiv.2603.27443 (2026), arXiv:2603.27443 [quant-ph]

    work page doi:10.48550/arxiv.2603.27443 2026

  26. [26]

    Levy-Yeyati, T

    T. Levy-Yeyati, T. Ramos, and A. González-Tudela, Engi- neering giant transmon molecules as mediators of condi- tional two-photon gates, Phys. Rev. Appl. (2026)

    work page 2026

  27. [27]

    M. V . Gustafsson, T. Aref, A. F. Kockum, M. K. Ekström, G. Johansson, and P . Delsing, Propagating phonons cou- pled to an artificial atom, Science346, 207 (2014)

    work page 2014

  28. [28]

    Kannanet al., Waveguide quantum electrodynamics with superconducting artificial giant atoms, Nature583, 775 (2020)

    B. Kannanet al., Waveguide quantum electrodynamics with superconducting artificial giant atoms, Nature583, 775 (2020)

    work page 2020

  29. [29]

    A. M. Vadiraj, A. Ask, T. G. McConkey, I. Nsanzineza, C. W. S. Chang, A. F. Kockum, and C. M. Wilson, Engi- neering the level structure of a giant artificial atom in waveguide quantum electrodynamics, Physical Review A103, 023710 (2021)

    work page 2021

  30. [30]

    Wang, Y.-P

    Z.-Q. Wang, Y.-P . Wang, J. Yao, R.-C. Shen, W.-J. Wu, J. Qian, J. Li, S.-Y. Zhu, and J. Q. You, Single-magnon quantum acoustics, Nature Communications13, 7580 (2022)

    work page 2022

  31. [31]

    González-Tudela, C

    A. González-Tudela, C. Sánchez Muñoz, and J. I. Cirac, Engineering and harnessing giant atoms in high- dimensional baths: A proposal for implementation with cold atoms, Phys. Rev. Lett.122, 203603 (2019)

    work page 2019

  32. [32]

    Y.-T. Chen, L. Du, Y. Zhang, L. Guo, J.-H. Wu, M. Artoni, and G. C. La Rocca, Giant-atom effects on population and entanglement dynamics of rydberg atoms in the optical regime, Phys. Rev. Research5, 043135 (2023)

    work page 2023

  33. [33]

    Joshi, F

    C. Joshi, F. Yang, and M. Mirhosseini, Quantum control of giant atoms, Physical Review X13, 021039 (2023)

    work page 2023

  34. [34]

    W. P . Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Physical Review Letters42, 1698 (1979)

    work page 1979

  35. [35]

    Jouanny, L

    V . Jouanny, L. Peyruchat, M. Scigliuzzo, A. Mercurio, E. Di Benedetto, D. De Bernardis, D. Sbroggiò, S. Frasca, V . Savona, F. Ciccarello, and P . Scarlino, Superstrong dy- namics and chiral emission of a giant atom in a struc- tured bath, arXiv preprint arXiv:2509.01579 (2025)

    work page arXiv 2025

  36. [36]

    Ramos, B

    T. Ramos, B. Vermersch, P . Hauke, H. Pichler, and P . Zoller, Non-markovian dynamics in chiral quantum networks with spins and photons, Physical Review A93, 062104 (2016)

    work page 2016

  37. [37]

    Roccati and D

    F. Roccati and D. Cilluffo, Controlling markovianity with chiral giant atoms, Physical Review Letters133, 063603 (2024)

    work page 2024

  38. [38]

    González-Tudela and J

    A. González-Tudela and J. I. Cirac, Exotic quantum dy- namics and purely long-range coherent interactions in Dirac-cone-like baths, Phys. Rev. A97, 043831 (2018)

    work page 2018

  39. [39]

    Perczel and M

    J. Perczel and M. D. Lukin, Theory of dipole radiation near a dirac photonic crystal, Phys. Rev. A101, 033822 (2020)

    work page 2020

  40. [40]

    Redondo-Yuste, M

    J. Redondo-Yuste, M. B. de Paz, P . A. Huidobro, and A. González-Tudela, Quantum electrodynamics in anisotropic and tilted dirac photonic lattices, New J. Phys.23, 103018 (2021)

    work page 2021

  41. [41]

    Di Benedetto, X

    E. Di Benedetto, X. Sun, M. A. Pinto, L. Leonforte, C.-Y. Chang, V . Jouanny, L. Peyruchat, P . Scarlino, and F. Ci- ccarello, Emergent cavity-qed dynamics along the edge of a photonic lattice, arXiv 10.48550/arXiv.2507.13444 (2025), arXiv:2507.13444 [quant-ph]

    work page doi:10.48550/arxiv.2507.13444 2025

  42. [42]

    A. S. Sheremet, M. I. Petrov, I. V . Iorsh, A. V . Poshakin- skiy, and A. N. Poddubny, Waveguide quantum electro- dynamics: Collective radiance and photon-photon corre- lations, Rev. Mod. Phys.95, 015002 (2023)

    work page 2023

  43. [43]

    Ciccarello, P

    F. Ciccarello, P . Lodahl, and D. Schneble, Waveguide quantum electrodynamics, Opt. Photonics News35, 34 (2024)

    work page 2024

  44. [44]

    Liu and A

    Y. Liu and A. A. Houck, Quantum electrodynamics near a photonic bandgap, Nat. Phys.13, 48 (2017)

    work page 2017

  45. [45]

    Mirhosseini, E

    M. Mirhosseini, E. Kim, V . S. Ferreira, M. Kalaee, A. Sipahigil, A. J. Keller, and O. Painter, Superconducting metamaterials for waveguide quantum electrodynamics, Nat. Commun.9, 3706 (2018)

    work page 2018

  46. [46]

    N. M. Sundaresan, R. Lundgren, G. Zhu, A. V . Gorshkov, 12 and A. A. Houck, Interacting qubit-photon bound states with superconducting circuits, Phys. Rev. X9, 011021 (2019)

    work page 2019

  47. [47]

    E. Kim, X. Zhang, V . S. Ferreira, J. Banker, J. K. Iverson, A. Sipahigil, M. Bello, A. González-Tudela, M. Mirhos- seini, and O. Painter, Quantum electrodynamics in a topological waveguide, Phys. Rev. X11, 011015 (2021)

    work page 2021

  48. [48]

    Scigliuzzoet al., Controlling atom-photon bound states in an array of josephson-junction resonators, Phys

    M. Scigliuzzoet al., Controlling atom-photon bound states in an array of josephson-junction resonators, Phys. Rev. X12, 031036 (2022)

    work page 2022

  49. [49]

    J. C. Owenset al., Chiral cavity quantum electrodynam- ics, Nat. Phys.18, 1048 (2022)

    work page 2022

  50. [50]

    Zhang, E

    X. Zhang, E. Kim, D. K. Mark, S. Choi, and O. Painter, A superconducting quantum simulator based on a photonic-bandgap metamaterial, Science379, 278 (2023)

    work page 2023

  51. [51]

    J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P . Yu, H. J. Kimble, and D. E. Chang, Atom-atom interactions around the band edge of a photonic crystal waveguide, Proc. Natl. Acad. Sci. U.S.A.113, 10507 (2016)

    work page 2016

  52. [52]

    O’Brienet al., A circuit-qed lattice system with flexible connectivity and gapped flat bands for photon-mediated spin models, arXiv (2025), arXiv:2505.05559 [quant-ph]

    K. O’Brienet al., A circuit-qed lattice system with flexible connectivity and gapped flat bands for photon-mediated spin models, arXiv (2025), arXiv:2505.05559 [quant-ph]

    work page arXiv 2025

  53. [53]

    C. Vega, D. Porras, and A. González-Tudela, Topologi- cal multimode waveguide qed, Phys. Rev. Res.5, 023031 (2023)

    work page 2023

  54. [54]

    De Bernardis, F

    D. De Bernardis, F. S. Piccioli, P . Rabl, and I. Carusotto, Chiral quantum optics in the bulk of photonic quantum hall systems, PRX Quantum4, 030306 (2023)

    work page 2023

  55. [55]

    This choice is convenient becausee 2 is aligned with theyaxis

    The notation distinguishes betweenN 1, the number of unit cells along thee 1 direction, andN y, the number of unit cells along thee 2 direction. This choice is convenient becausee 2 is aligned with theyaxis

    work page

  56. [56]

    A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

    work page 2009

  57. [57]

    L. E. F. Foa Torres, S. Roche, and J.-C. Charlier,Intro- duction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport(Cambridge University Press, Cambridge, 2014)

    work page 2014

  58. [58]

    S. M. Girvin and K. Yang,Modern Condensed Matter Physics(Cambridge University Press, Cambridge, 2019)

    work page 2019

  59. [59]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    work page 1959

  60. [60]

    Setting a non-zero∆is not necessary for the essential qualitative results of the present section, which would hold even for∆=0 provided thatω 0 is sufficiently far from the Dirac points

    work page

  61. [61]

    J. Jung, F. Zhang, Z. Qiao, and A. H. MacDonald, Valley- hall kink and edge states in multilayer graphene, Physi- cal Review B84, 075418 (2011)

    work page 2011

  62. [62]

    J. Noh, S. Huang, K. P . Chen, and M. C. Rechtsman, Ob- servation of photonic topological valley hall edge states, Physical Review Letters120, 063902 (2018)

    work page 2018

  63. [63]

    J. K. Asbóth, L. Oroszlány, and A. Pályi,A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions(Springer International Publishing, Cham, 2016)

    work page 2016

  64. [64]

    The essential requirement is that its frequency lies within the gap and its coupling is small compared to the detuning from band edges

    We place the emitter at the center of the band gap for the sake of argument. The essential requirement is that its frequency lies within the gap and its coupling is small compared to the detuning from band edges

    work page

  65. [65]

    Leonforte, X

    L. Leonforte, X. Sun, D. Valenti, B. Spagnolo, F. Illumi- nati, A. Carollo, and F. Ciccarello, Quantum optics with giant atoms in a structured photonic bath, Quantum Sci. Technol.10, 015057 (2025)

    work page 2025