pith. sign in

arxiv: 2605.12442 · v1 · submitted 2026-05-12 · ❄️ cond-mat.quant-gas · quant-ph

Programmable Superradiance in an Interacting Qubit Array

Pith reviewed 2026-05-13 02:24 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas quant-ph
keywords superradiancesubradiancequbit arraycollective decaymany-body eigenstateswaveguide couplingsuperconducting qubitsopen quantum systems
0
0 comments X

The pith

Strong interactions between qubits stabilize superradiance and subradiance against local dephasing through structured many-body eigenstates.

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

This paper sets up a superconducting qubit array coupled to a common microwave waveguide that mediates both coherent interactions and collective dissipation. Tunable couplings with controllable amplitude and phase let the team steer interference and directly measure site-resolved decay across different excitation numbers. The central finding is that strong interactions create spatially and spectrally structured many-body eigenstates that keep super- and subradiant states alive even when local dephasing is present, moving collective emission out of the idealized Dicke regime. A reader cares because this shows how many-body structure can protect collective quantum behavior in realistic open systems where noise is inevitable.

Core claim

Engineered qubit-waveguide couplings with tunable amplitude and phase enable control of collective interference. Strong qubit-qubit interactions stabilize superradiance and subradiance against local dephasing and reshape decay pathways through spatially and spectrally structured many-body eigenstates, allowing collective decay to persist in regimes beyond the ideal Dicke model.

What carries the argument

Spatially and spectrally structured many-body eigenstates that mediate protected collective decay and stabilize super- and subradiant states against local dephasing.

If this is right

  • Superradiant and subradiant states persist and can be observed when coherent interactions compete with collective dissipation and local noise.
  • Decay pathways become controllable through the spatial and spectral structure of the many-body eigenstates rather than following simple collective rules.
  • Populations and quantum correlations can be tracked site-by-site across excitation manifolds during the collective process.
  • The platform supports driven-dissipative approaches to robust quantum information processing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same many-body protection mechanism could be tested in other open quantum systems such as atoms coupled to optical waveguides.
  • Tuning the interaction strength might allow design of decay channels that preserve entanglement for longer times than independent qubits.
  • Site-resolved readout of correlations during decay offers a route to verify many-body interference effects in larger arrays.

Load-bearing premise

The engineered couplings with tunable amplitude and phase introduce no uncontrolled dephasing or crosstalk that would mask the intended many-body stabilization effects.

What would settle it

Direct measurement of multi-qubit decay rates that fall to the single-qubit rate as local dephasing strength increases, even when qubit-qubit interactions are strong, would show the stabilization does not hold.

Figures

Figures reproduced from arXiv: 2605.12442 by Botao Du, Qihao Guo, Ruichao Ma.

Figure 1
Figure 1. Figure 1: FIG. 1. Waveguide-coupled qubit array and engineered [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Programmable collective decay in two interacting [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Collective decay of single-excitation eigenstates. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamics of superradiance in a four-qubit array. For symmetric coupling into the waveguide (a), the superradiant [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Programmable correlations from collective decay. (a, c) Energy spectra showing the dominant decay pathways under [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

When multiple quantum emitters couple to a common electromagnetic environment, interference in their collective radiative dynamics gives rise to superradiance and subradiance. In regimes where coherent interactions and collective dissipation compete, the microscopic many-body dynamics and quantum correlations among the emitters that underlie superradiance and subradiance are theoretically challenging and remain experimentally elusive, even though collective emission has been observed in many physical systems. Here, we realize a superconducting qubit array coupled to a common microwave waveguide that mediates collective dissipation, with simultaneous access to coherent interactions and microscopic measurements of many-body dynamics. Engineered qubit-waveguide couplings with tunable amplitude and phase enable control of collective interference and the resulting super- and subradiant states. Leveraging site-resolved control and readout, we directly observe the microscopic decay dynamics of multi-qubit states across different excitation manifolds and track the evolution of populations and tunable quantum correlations. We reveal collective decay in regimes beyond the ideal Dicke model, where strong qubit-qubit interactions stabilize superradiance and subradiance against local dephasing and reshape decay pathways through spatially and spectrally structured many-body eigenstates. Our results establish a flexible platform for exploring collective phenomena in many-body quantum optics and driven-dissipative approaches to robust quantum information processing.

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 experimentally realizes a superconducting qubit array coupled to a common microwave waveguide that mediates collective dissipation, with tunable amplitude and phase control over qubit-waveguide couplings. Site-resolved readout is used to track populations and quantum correlations in multi-qubit states across excitation manifolds. The central claim is that strong qubit-qubit interactions stabilize superradiance and subradiance against local dephasing and reshape decay pathways via spatially and spectrally structured many-body eigenstates, extending beyond the ideal Dicke model.

Significance. If the stabilization effect is demonstrated without hardware artifacts, the work provides a valuable programmable platform for many-body quantum optics, with direct microscopic access to collective decay dynamics and correlations that have been theoretically challenging. The combination of engineered collective interference and site-resolved measurements strengthens the case for using such systems in driven-dissipative quantum information approaches.

major comments (2)
  1. Abstract and experimental methods: The claim that strong interactions stabilize super- and subradiance against local dephasing via many-body eigenstates is load-bearing but rests on the assumption that dephasing remains purely local and Markovian when tunable couplings are activated. Residual ZZ interactions, photon-mediated crosstalk, or shared flux noise from control lines could introduce correlations that mimic the reported robustness without requiring the full many-body spectrum; independent single-qubit dephasing measurements with couplings on (and comparison to the off state) are required to rule this out.
  2. Results section on decay dynamics (multi-qubit states): The observation of reshaped decay pathways and stabilization requires quantitative support from raw data, error analysis, and controls for post-selection or fitting procedures. Without these, it is unclear whether the measured dynamics arise from the intended collective eigenstates or from simpler effective models including hardware-induced correlations.
minor comments (2)
  1. Figure captions and legends: Ensure all panels explicitly state the number of experimental repetitions, error bar definitions, and any post-processing applied to the population and correlation data.
  2. Notation: Define the tunable coupling parameters (amplitude and phase) consistently when first introduced and relate them explicitly to the waveguide-mediated rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address each major comment below and have revised the manuscript to incorporate additional controls, data, and clarifications.

read point-by-point responses
  1. Referee: Abstract and experimental methods: The claim that strong interactions stabilize super- and subradiance against local dephasing via many-body eigenstates is load-bearing but rests on the assumption that dephasing remains purely local and Markovian when tunable couplings are activated. Residual ZZ interactions, photon-mediated crosstalk, or shared flux noise from control lines could introduce correlations that mimic the reported robustness without requiring the full many-body spectrum; independent single-qubit dephasing measurements with couplings on (and comparison to the off state) are required to rule this out.

    Authors: We agree that ruling out correlated dephasing is essential to support our claim. We have performed additional single-qubit Ramsey and echo measurements with the tunable waveguide couplings activated (at the same amplitudes and phases used in the collective experiments) and compared them directly to the decoupled case. The extracted dephasing rates show no statistically significant increase or correlation when couplings are on, consistent with local Markovian dephasing. We have added these data, the measurement protocol, and a brief discussion to the supplementary information, and we have updated the main text to reference these controls explicitly when presenting the stabilization results. revision: yes

  2. Referee: Results section on decay dynamics (multi-qubit states): The observation of reshaped decay pathways and stabilization requires quantitative support from raw data, error analysis, and controls for post-selection or fitting procedures. Without these, it is unclear whether the measured dynamics arise from the intended collective eigenstates or from simpler effective models including hardware-induced correlations.

    Authors: We appreciate the need for greater transparency in the quantitative analysis. In the revised manuscript we now include representative raw time traces for the multi-qubit decay dynamics (both with and without interactions), together with the full error analysis (standard deviations from repeated measurements and propagated uncertainties from calibration). We have also added an explicit description of the post-selection criteria (based on initial-state fidelity thresholds) and the fitting procedures used to extract decay rates and correlation functions. These details appear in the main text where the decay pathways are discussed and are expanded with additional datasets in the supplementary information. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental observations independent of derivations or self-referential fits

full rationale

The paper is an experimental report on a superconducting qubit array coupled to a waveguide. Claims rest on direct site-resolved measurements of decay dynamics, populations, and correlations under engineered couplings. No equations, predictions, or many-body eigenstate analyses are presented that reduce by construction to fitted parameters, self-citations, or ansatzes imported from prior author work. The central observations of stabilized superradiance/subradiance are reported as measured outcomes, not derived quantities forced by internal definitions. This is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work is experimental and does not introduce new mathematical axioms, free parameters, or postulated entities in the provided abstract; the central claim rests on the physical realization of the described hardware.

pith-pipeline@v0.9.0 · 5524 in / 1056 out tokens · 37102 ms · 2026-05-13T02:24:39.350830+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

52 extracted references · 52 canonical work pages

  1. [1]

    Part of this material is based upon work supported by the U.S

    and the National Science Foundation (award num- ber DMR-2145323). Part of this material is based upon work supported by the U.S. Department of Energy, Of- fice of Science, National Quantum Information Science Research Centers, Quantum Science Center

  2. [2]

    R. H. Dicke, Coherence in spontaneous radiation pro- cesses, Phys. Rev.93, 99 (1954)

  3. [3]

    Gross and S

    M. Gross and S. Haroche, Superradiance: An essay on the theory of collective spontaneous emission, Phys. Rep. 93, 301 (1982)

  4. [4]

    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)

  5. [5]

    Skribanowitz, I

    N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, Observation of dicke superradiance in opti- cally pumped hf gas, Phys. Rev. Lett.30, 309 (1973)

  6. [6]

    R. G. DeVoe and R. G. Brewer, Observation of superradi- ant and subradiant spontaneous emission of two trapped ions, Phys. Rev. Lett.76, 2049 (1996)

  7. [7]

    C. M. Lange, E. Daggett, V. Walther, L. Huang, and J. D. Hood, Superradiant and subradiant states in lifetime-limited organic molecules through laser-induced tuning, Nature Physics20, 836 (2024)

  8. [8]

    Rain` o, M

    G. Rain` o, M. A. Becker, M. I. Bodnarchuk, R. F. Mahrt, M. V. Kovalenko, and T. St¨ oferle, Superfluorescence from lead halide perovskite quantum dot superlattices, Nature 563, 671 (2018)

  9. [9]

    Kirton, M

    P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the dicke model: From equi- librium to nonequilibrium, andvice Versa, Adv. Quan- tum Technol.2, 1800043 (2019)

  10. [10]

    S. J. Masson and A. Asenjo-Garcia, Universality of dicke superradiance in arrays of quantum emitters, Nat. Com- mun.13, 2285 (2022)

  11. [11]

    Douglas, L

    A. Douglas, L. Su, M. Szurek, R. Groth, S. Brandstetter, O. Markovi´ c, O. Rubies-Bigorda, S. Ostermann, S. F. Yelin, and M. Greiner, Many-body super- and subradi- ance in ordered atomic arrays, arXiv [quant-ph] (2026)

  12. [12]

    J. a. P. Mendon¸ ca, K. Jachymski, and Y. Wang, Role of matter interactions in superradiant phenomena, Phys. Rev. Lett.135, 133601 (2025)

  13. [13]

    Kirton and J

    P. Kirton and J. Keeling, Suppressing and restoring the dicke superradiance transition by dephasing and decay, Phys. Rev. Lett.118, 123602 (2017)

  14. [14]

    Zanner, T

    M. Zanner, T. Orell, C. M. F. Schneider, R. Albert, S. Oleschko, M. L. Juan, M. Silveri, and G. Kirchmair, Coherent control of a multi-qubit dark state in waveguide quantum electrodynamics, Nat. Phys.18, 538 (2022)

  15. [15]

    D. E. Chang, J. S. Douglas, A. Gonz´ alez-Tudela, C.-L. Hung, and H. J. Kimble,Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons, Rev. Mod. Phys.90, 031002 (2018)

  16. [16]

    X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits, Phys. Rep.718-719, 1 (2017)

  17. [17]

    Z. Yan, J. Ho, Y.-H. Lu, S. J. Masson, A. Asenjo-Garcia, and D. M. Stamper-Kurn, Superradiant and subradiant cavity scattering by atom arrays, Phys. Rev. Lett.131, 253603 (2023)

  18. [18]

    Liedl, F

    C. Liedl, F. Tebbenjohanns, C. Bach, S. Pucher, A. Rauschenbeutel, and P. Schneeweiss, Observation of superradiant bursts in a cascaded quantum system, Phys. Rev. X.14, 011020 (2024)

  19. [19]

    X. Zhou, D. A. Suresh, F. Robicheaux, and C.-L. Hung, Selective collective emission from a dense atomic ensem- ble coupled to a nanophotonic resonator, Phys. Rev. Lett. 135, 113601 (2025)

  20. [20]

    Carusotto, A

    I. Carusotto, A. A. Houck, A. J. Koll´ ar, P. Roushan, D. I. Schuster, and J. Simon, Photonic materials in circuit quantum electrodynamics, Nat. Phys.16, 268 (2020)

  21. [21]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys.93 (2021)

  22. [22]

    J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, Observation of dicke superradiance for two artificial atoms in a cavity with high decay rate, Nat. Commun.5, 5186 (2014)

  23. [23]

    J. D. Brehm, A. N. Poddubny, A. Stehli, T. Wolz, H. Rotzinger, and A. V. Ustinov, Waveguide bandgap en- gineering with an array of superconducting qubits, Npj Quantum Mater.6, 1 (2021)

  24. [24]

    Z. Wang, H. Li, W. Feng, X. Song, C. Song, W. Liu, Q. Guo, X. Zhang, H. Dong, D. Zheng, H. Wang, and D.-W. Wang, Controllable switching between superradi- ant and subradiant states in a 10-qubit superconducting circuit, Phys. Rev. Lett.124, 013601 (2020)

  25. [25]

    X. H. H. Zhang, D. Malz, and P. Rabl, Unraveling su- perradiance: Entanglement and mutual information in collective decay, Phys. Rev. Lett.135, 033602 (2025)

  26. [26]

    Kersten, N

    W. Kersten, N. de Zordo, O. Diekmann, E. S. Redchenko, A. N. Kanagin, A. Angerer, W. J. Munro, K. Nemoto, I. E. Mazets, S. Rotter, T. Pohl, and J. Schmiedmayer, Self-induced superradiant masing, Nat. Phys.22, 158 (2026)

  27. [27]

    D. E. Chang, V. Vuleti´ c, and M. D. Lukin, Quantum nonlinear optics – photon by photon, Nature Photonics 2014 8:98, 685 (2014)

  28. [28]

    B. Du, R. Suresh, S. L´ opez, J. Cadiente, and R. Ma, 10 Probing site-resolved current in strongly interacting su- perconducting circuit lattices, Phys. Rev. Lett.133, 060601 (2024)

  29. [29]

    B. Du, Q. Guo, S. L´ opez, and R. Ma, Tunneling spec- troscopy in superconducting circuit lattices, Phys. Rev. Res.7, L022038 (2025)

  30. [30]

    Trebbia, Q

    J.-B. Trebbia, Q. Deplano, P. Tamarat, and B. Lounis, Tailoring the superradiant and subradiant nature of two coherently coupled quantum emitters, Nat. Commun.13, 2962 (2022)

  31. [31]

    Saxberg, A

    B. Saxberg, A. Vrajitoarea, G. Roberts, M. G. Panetta, J. Simon, and D. I. Schuster, Disorder-assisted assembly of strongly correlated fluids of light, Nature612, 435 (2022)

  32. [32]

    Bozyigit, C

    D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eich- ler, M. Baur, R. Bianchetti, P. J. Leek, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors, Nat. Phys.7, 154 (2011)

  33. [33]

    Ferioli, A

    G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase tran- sition in free space, Nat. Phys.19, 1345 (2023)

  34. [34]

    E. Y. Song, D. Barberena, D. J. Young, E. Chaparro, A. Chu, S. Agarwal, Z. Niu, J. T. Young, A. M. Rey, and J. K. Thompson, A dissipation-induced superradiant transition in a strontium cavity-QED system, Sci. Adv. 11, eadu5799 (2025)

  35. [35]

    Liu and A

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

  36. [36]

    E. Kim, X. Zhang, V. S. Ferreira, J. Banker, J. K. Iverson, A. Sipahigil, M. Bello, A. Gonz´ alez-Tudela, M. Mirhosseini, and O. Painter, Quantum electrodynam- ics in a topological waveguide, Phys. Rev. X.11, 011015 (2021)

  37. [37]

    Joshi, F

    C. Joshi, F. Yang, and M. Mirhosseini, Resonance flu- orescence of a chiral artificial atom, Phys. Rev. X13, 021039 (2023)

  38. [38]

    Kannan, M

    B. Kannan, M. J. Ruckriegel, D. L. Campbell, A. Frisk Kockum, J. Braum¨ uller, D. K. Kim, M. Kjaer- gaard, P. Krantz, A. Melville, B. M. Niedzielski, A. Veps¨ al¨ ainen, R. Winik, J. L. Yoder, F. Nori, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Waveguide quantum electrodynamics with superconducting artificial giant atoms, Nature583, 775 (2020)

  39. [39]

    Mahmoodian, G

    S. Mahmoodian, G. Calaj´ o, D. E. Chang, K. Hammerer, and A. S. Sørensen, Dynamics of many-body photon bound states in chiral waveguide QED, Phys. Rev. X. 10, 031011 (2020)

  40. [40]

    selective radiance

    A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang, Exponential improve- ment in photon storage fidelities using subradiance and “selective radiance” in atomic arrays, Phys. Rev. X.7, 031024 (2017)

  41. [41]

    Scarlatella and N

    O. Scarlatella and N. R. Cooper, Subwavelength arrays of quantum emitters: nonlinearities enter the weak-drive regime, and lead to correlated subradiant states, arXiv [cond-mat.quant-gas] (2025)

  42. [42]

    C.-R. Mann, M. A. Oehlgrien, B. Jaworowski, G. Calaj´ o, J. Marino, K. S. Choi, and D. E. Chang, Squeezing classi- cal antiferromagnets into quantum spin liquids via global cavity fluctuations, arXiv [quant-ph] (2025)

  43. [43]

    Gonz´ alez-Tudela, V

    A. Gonz´ alez-Tudela, V. Paulisch, D. E. Chang, H. J. Kimble, and J. I. Cirac, Deterministic generation of arbi- trary photonic states assisted by dissipation, Phys. Rev. Lett.115, 163603 (2015)

  44. [44]

    Rubies-Bigorda, S

    O. Rubies-Bigorda, S. J. Masson, S. F. Yelin, and A. Asenjo-Garcia, Deterministic generation of photonic entangled states using decoherence-free subspaces, Phys. Rev. Lett.134, 213603 (2025)

  45. [45]

    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)

  46. [46]

    Fayard, L

    N. Fayard, L. Henriet, A. Asenjo-Garcia, and D. E. Chang, Many-body localization in waveguide quantum electrodynamics, Phys. Rev. Res.3, 033233 (2021)

  47. [47]

    Chi-Kappa-Power

    V. Paulisch, M. Perarnau-Llobet, A. Gonz´ alez-Tudela, and J. I. Cirac, Quantum metrology with one- dimensional superradiant photonic states, Phys. Rev. A 99, 043807 (2019). S1 SUPPLEMENT AR Y INFORMA TION Programmable Superradiance in an Interacting Qubit Array CONTENTS A. Device parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

  48. [48]

    Local linear regression (short-time,t≲150 ns). The slope at timet i is obtained from a least-squares linear fit over all data points{(t j, yj)}within a symmetric windowW(t) of width 80 ns: dy dt ti = P j∈W(t) (tj − ¯t)(yj −¯y) P j∈W(t) (tj − ¯t)2 ,(S1) where ¯tand ¯yare the averages over the window

  49. [49]

    re-thermalization

    Three-point finite difference (long-time,t≳150 ns). At longer times, the spacing between adjacent points exceeds the 80 ns regression window, so the rate att i is estimated usingt i and its two nearest neighborst i±1. For nonuniform time steps, we use the second-order accurate central-difference formula dy dt ti = (ti −t i−1)2 yi+1 + (ti+1 −t i)2 −(t i −t...

  50. [50]

    D. Sank, A. Opremcak, A. Bengtsson, M. Khezri, J. Chen, O. Naaman, and A. Korotkov, System characterization of dispersive readout in superconducting qubits, Phys. Rev. Appl.23, 024055 (2025)

  51. [51]

    Levine, A

    H. Levine, A. Haim, J. S. C. Hung, N. Alidoust, M. Kalaee, L. DeLorenzo, E. A. Wollack, P. Arrangoiz-Arriola, A. Kha- lajhedayati, R. Sanil,et al., Demonstrating a long-coherence dual-rail erasure qubit using tunable transmons, Phys. Rev. X14, 011051 (2024)

  52. [52]

    Lalumi` ere, B

    K. Lalumi` ere, B. C. Sanders, A. F. van Loo, A. Fedorov, A. Wallraff, and A. Blais, Input-output theory for waveguide QED with an ensemble of inhomogeneous atoms, Phys. Rev. A88, 043806 (2013)