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REVIEW 3 major objections 5 minor 97 references

Exceptional, but Separate: Precursors to Spontaneous Symmetry Breaking

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Pith's one-line read In three Kerr-resonator systems, spontaneous symmetry breaking and exceptional points occur at different parameter points, but every route to breaking still crosses an exceptional point first.

desk verdict Clean homogeneous-state result proving Jacobian EPs precede SSB, but the paper overclaims the scope to patterned LLE states. read the letter →

arxiv 2505.02691 v1 pith:WLHNBUHB submitted 2025-05-05 physics.optics math-phmath.MP

Lewis Hill , Julius T. Gohsrich , Alekhya Ghosh , Jacob Fauman , Pascal Del'Haye , Flore K. Kunst This is my paper · ORCID
classification physics.opticsmath-phmath.MP
keywords spontaneoussymmetrybreakingexceptionalpointsKerrresonatorsLugiato-Lefeverequationnon-Hermitianphotonicsquasi-chiralopticalbistabilityJacobianstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies three common Kerr-resonator configurations — two co-propagating polarizations in a ring, two counter-propagating pumps in a ring, and a Fabry-Pérot cavity — and asks whether spontaneous symmetry breaking (SSB) coincides with exceptional points (EPs) in the system's Jacobian. After reducing all three to the same homogeneous-state equation, it shows that the algebraic conditions for a Jacobian EP and for the SSB instability are different, so the two phenomena generally sit at different locations in parameter space. For the homogeneous, round-trip-uniform states analyzed, it then proves that every path from a stable symmetric state to SSB must cross a Jacobian EP, making such EPs necessary precursors even though they are not the bifurcation points. The result matters because much of non-Hermitian photonics assumes EP and SSB are tied together; the paper argues that assumption must be made carefully, and that the relevant class of EP has to be identified.

What carries the argument

The load-bearing object is the Jacobian matrix $J$ of the linearized coupled field equations, evaluated on the symmetric stationary state. Writing the shifted traceless Jacobian $\tilde J=J+\tfrac14$, the four eigenvalues take the two-parameter form $\lambda_{\pm_1\pm_2}=-1\pm_1\sqrt{\eta\pm_2 2\sqrt{\eta^2-\nu}}$, with $\eta=\operatorname{tr}(\tilde J^2)/4$ and $\nu=\det(\tilde J)$. This two-invariant structure separates the conditions: eigenvalue degeneracies (EPs) live on $\nu=\eta^2$ or $\nu=0$ with $\eta\neq 0$, while the onset of instability obeys Eq. (20). The same matrix carries $\mathcal{PT}$ and quasi-chiral symmetry, and the symmetry-phase argument is what forces a trajectory from the stable region to the unstable region to pass through an EP.

What would settle it

Run a full numerical integration of Eqs. (5)-(7) without the homogeneous-state restriction while scanning detuning and input intensity; if any symmetry-broken branch is reached from a stable symmetric state without the trajectory crossing $\nu=\eta^2$ or $\nu=0$ in the homogeneous Jacobian, the necessary-precursor claim is false.

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Extended reading notes

Core claim

The central claim is that for the generalized coupled field equation governing homogeneous states of all three resonators, Jacobian EPs and SSB bifurcations obey distinct algebraic conditions. A dual EP2 — two simultaneous second-order EPs — occurs when $\nu=\eta^2$, and a single EP2 occurs when $\nu=0$ with $\eta\neq 0$; the SSB instability boundary is $\eta\geq 2-\sqrt{\nu}$ for $\nu\geq 1$ and $\eta\geq (1+\nu)/2$ for $\nu\leq 1$, where $\eta$ and $\nu$ are trace and determinant invariants of the shifted Jacobian. Because these curves differ, SSB and Jacobian EPs are generally dislocated in parameter space. Yet every parameter scan considered begins and ends in the quasi-chiral-symmetry-unbroken phase, so reaching the unstable region in which SSB occurs forces a crossing of a Jacobian EP. The paper therefore concludes that Jacobian EPs are necessary precursors to SSB, not coincident markers of it.

Load-bearing premise

The analysis covers only states that do not vary along the cavity round-trip time; if symmetry breaking in these resonators instead happens through moving or patterned states, the dislocation and precursor claims could fail.

Editorial extensions

If this is right

  • In the three Kerr-resonator platforms, locating an EP in the spectrum does not by itself locate the SSB threshold; devices tuned to an EP to trigger symmetry breaking may be tuned to the wrong parameter point.
  • The necessary-precursor result provides a diagnostic: before SSB can appear from a stable symmetric state, the trajectory in the $(\eta,\nu)$ plane must cross a Jacobian EP, so monitoring the Jacobian's symmetry phase can flag impending SSB.
  • The distinction between dual and single EP2s matters: in the large-detuning and large-intensity limits for $B<3A$ the system approaches the dual EP2 line, while for $B>3A$ the high-intensity symmetry-broken branch stays in the qCS-unbroken phase.
  • Because the same separation appears across three experimentally distinct resonator geometries, the authors argue the dislocation is likely a general principle rather than an accident of one platform.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to check whether the precursor relation survives for non-uniform states such as temporal solitons, breathers, and faticons, whose homogeneous-state reduction this paper deliberately leaves out.
  • The algebraic separation suggests an experimental route: measure the Jacobian eigen-spectrum with weak probe perturbations while sweeping detuning, and compare the EP location with the SSB bifurcation; a mismatch would directly confirm dislocation.
  • The symmetry-phase logic may extend beyond optics to any non-Hermitian system whose linearization has the same two-invariant trace-determinant structure, although the paper itself does not claim that extension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the relation between spontaneous symmetry breaking (SSB) and exceptional points (EPs) in three Kerr-resonator models. Starting from the coupled Lugiato-Lefever equations (5)-(7), it imposes the homogeneous-state reduction ∂²τE1,2=0 and ⟨|E|²⟩→|E|², obtaining a single two-component ODE (8). It then derives the Jacobian, its eigenvalues (16), the EP conditions (19), and the SSB/instability boundary (20) in terms of two invariants η and ν. The central claims are that SSB and Jacobian EPs are generically located at different points in parameter space, but that crossing a Jacobian EP is a necessary precursor to SSB, because parameter scans start and end in the qCS-unbroken phase of the Jacobian and therefore must cross an EP line to reach an unstable regime.

Significance. If correct within its stated homogeneous-state domain, the paper gives a clean, analytic demonstration that Jacobian EPs and SSB bifurcations do not have to coincide, together with a non-trivial necessary-condition relation between them. The construction of the η-ν plane and the use of symmetry phases to organize both the EP and stability conditions is elegant, and the central derivation contains no fitted parameters: Eqs. (19) and (20) are derived algebraically from the model. The cautionary message about not equating EPs with SSB is valuable. However, as detailed below, the displayed eigenvalue formula contains an internal inconsistency, the precursor claim as stated in the abstract and conclusion goes beyond the homogeneous-state analysis, and one auxiliary asymptotic result in the Methods is asserted without derivation. These issues are local and addressable, so the paper is promising but needs revision.

major comments (3)
  1. [Eq. (16); Methods B.2] The eigenvalue formula (16) is inconsistent with the subsequent derivation. Equation (16) and the display before Methods B.2 write λ±1±2 = −1 ± √(η ± 2S) with S = √(η²−ν), but the stability analysis in Methods B.2 uses λ = −1 + √(η+S) (for example, it states that in Regions I and IV, where η+S≥0, Eq. (28) becomes √(η+S)≥1). For a traceless 4×4 matrix with invariants η=tr(J̃²)/4 and ν=det(J̃), the characteristic polynomial has roots ±√(η ± √(η²−ν)), without the factor 2 in the inner radical. If Eq. (16) were taken literally, the outer square roots would vanish at ν=3η²/4 rather than at ν=0, which would change the single-EP2 condition in Eq. (19). The EP conditions (19) and the stability boundary (20) appear to be based on the correct form, but the manuscript as written is internally contradictory; Eq. (16) and the corresponding display in Methods B.2 should be corrected and their consistency with Eq. (19) checked explicitly.
  2. [§I.B, §II.C, Conclusion] The necessary-precursor claim is proven only for the homogeneous reduction. Eq. (8) is obtained from Eqs. (5)-(7) by imposing ∂²τE1,2=0 and replacing all fast-time averages by local intensities, and the Jacobian (14) is built from this ODE. Section I.B explicitly notes that the full coupled LLEs also support bright and dark solitons, breathers, faticons, and soliton chains. The proof in §II.C that parameter scans begin and end in the qCS-unbroken phase uses Eq. (21) and the homogeneous-state relation d=4B²(I−P)P, so it establishes at most that the k=0 Jacobian EP precedes the k=0 SSB instability of the homogeneous branch. The Conclusion, however, states without qualification that "the emergence of Jacobian EPs is a necessary precursor to the onset of SSB" for the three LLE platforms. For patterned or finite-wavevector instabilities, the mode-resolved Jacobian involves the replacement θ→θ−ηk² and modified averaged XPM terms, so the EP and stability boundaries become k-dependent; the present analysis does not cover them. The authors should either restrict the abstract and conclusion to homogeneous-state SSB, or extend the analysis to the k-resolved Jacobian and show that the precursor property survives.
  3. [Methods B.3, Eq. (31)] The treatment of the B>3A case relies on an asymptotic expansion that is not derived. Eq. (31) gives the leading-order form of d(θ,P1,P2) for P1,P2→∞, and the text infers from its positivity that the intensity scan remains in the qCS-unbroken phase and does not approach the dual EP2 line. The expansion is stated without derivation and under the assumption that P1 and P2 grow with the same asymptotic scaling; the validity of that scaling is not justified. Because the B>3A regime is explicitly invoked to extend the precursor argument beyond the B<3A case analyzed in the main text, this omission leaves a gap in the proof. Please provide a derivation of Eq. (31) and a justification of the scaling assumption, or explicitly label the B>3A discussion as heuristic.
minor comments (5)
  1. [Abstract and §III] The phrase "recurring behavior across disparate platforms" overstates what is demonstrated: the three models are reduced to the same homogeneous ODE, Eq. (8), so the analysis of their common reduction does not by itself establish platform-specific generality. Suggest rewording to reflect that the three systems share the same homogeneous-state dynamics.
  2. [§I.B] The mapping from Eq. (7) to Eq. (8) via the substitution (3A,3B)→(A,B) is stated in one clause; a short derivation or explicit definition of the rescaled A and B would help readers verify that no cross-averaged terms survive in the homogeneous limit.
  3. [Eq. (31)] The displayed expression for d in Eq. (31) is difficult to parse as typeset; please add explicit multiplication signs and parentheses, for example around the terms involving P1²P2² and P1P2(P1²+P2²), so that the algebraic expression is unambiguous.
  4. [Fig. 4 caption] The caption states that "the gray dashed area denotes instability," while the text describes the gray line as the stability boundary; clarifying whether the dashed side or the enclosed area is unstable would remove ambiguity.
  5. [§II.C] The text moves between "onset of instability" and "onset of SSB." Given that Eq. (20) is the instability boundary, the equivalence of these notions for the homogeneous solutions considered here should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EP conditions and SSB stability boundary are derived independently from the same model, with no fitted inputs and no load-bearing self-citation.

full rationale

The derivation chain is self-contained. The Jacobian is obtained by linearizing Eq. (8), and its eigenvalues are expressed in Eq. (16) purely in terms of the invariants eta and nu defined in Eq. (17). The EP conditions in Eq. (19) are the algebraic degeneracy conditions of Eq. (16), while the SSB/stability boundary in Eq. (20) is obtained by imposing Re(lambda_++) >= 0. These are distinct algebraic conditions, and neither is fitted to the other. There is no parameter fitting, no quantity called a prediction that is actually an input, and no uniqueness theorem invoked to force a choice. The self-citations (Refs. 17 and 23) provide background, a formalism, and the B < 3A classification of bounded symmetry-broken regions, but the central claim is re-derived from the paper's own equations and is not reduced to those citations; the B < 3A result is also extended independently in the Methods for B > 3A. The paper explicitly restricts the analysis to homogeneous states before Eq. (8), which limits the scope of the precursor statement relative to the full LLE systems supporting solitons, breathers, and faticons, but this is a stated scope limitation rather than a circular step. No circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The analysis rests on the homogeneous-state reduction, the standard LLE modeling assumptions, and the imported eigenvalue formalism from Ref 87. No free parameters are fitted. No new physical entities are introduced; 'single EP2' and 'dual EP2' are classifications of degeneracies, not new entities.

assumptions (4)
  • domain assumption Kerr material response is lossless, isotropic, and dispersionless, with only χ(1) and χ(3) susceptibilities (Eq 1-2).
    Underlies the LLE models (Eq 3-7); if higher-order or anisotropic nonlinearities enter, the Jacobian structure and EP conditions change.
  • ad hoc to paper Focus on homogeneous (fast-time constant) states, reducing Eqs 5-7 to the common form Eq 8.
    The central precursor claim is proven only in this reduced setting; the paper does not analyze patterned states.
  • standard math The eigenvalue parameterization of the Jacobian in Eq (16) follows Ref 87 (Montag and Kunst); the symmetry phases (PT, qCS) are taken as standard.
    Background formalism imported from prior work; not re-proven in this paper.
  • domain assumption For the generic precursor argument, the parameter scan must start and end in the qCS-unbroken phase; shown for B<3A in the limits, and argued for B>3A via Eq (31).
    If a scan starts outside qCS-unbroken, the necessity of crossing an EP before SSB is not established.

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Pith. "Pith review of Exceptional, but Separate: Precursors to Spontaneous Symmetry Breaking." pith.science (2026). https://pith.science/paper/WLHNBUHB

@misc{pith2026250502691,
  author       = {Pith},
  title        = {Pith review of: Exceptional, but Separate: Precursors to Spontaneous Symmetry Breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLHNBUHB}},
  note         = {Machine review of arXiv:2505.02691}
}
read the original abstract

Spontaneous symmetry breaking (SSB) and exceptional points (EPs) are often assumed to be inherently linked. Here we investigate the intricate relationship between SSB and specific classes of EPs across three distinct, real-world scenarios in nonlinear optics. In these systems, the two phenomena do not coincide for all classes of EPs; they can occur at dislocated points in parameter space. This recurring behavior across disparate platforms implies that such decoupling is not unique to these optical systems, but likely reflects a more general principle. Our results highlight the need for careful analysis of assumed correlations between SSB and EPs in both theoretical and applied contexts. They deepen our understanding of nonlinear dynamics in optical systems and prompt a broader reconsideration of contexts where EPs and SSB are thought to be interdependent.

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Reference graph

Works this paper leans on

97 extracted references · 67 canonical work pages

  1. [1]

    NTopQuant

    The Lugiato-Lefever Equation One of the most successful models for Kerr ring res- onators is the Lugiato-Lefever equation (LLE) [57, 58], which in its purely temporal [59] and normalized form is given by ∂E ∂t =Ein−E− iθE− iη∂2E ∂τ 2 + i|E|2E. (3) The LLE describes the evolution of the complex envelope of the intracavity electric field,E, over a slow time...

  2. [2]

    Symmetry Phases of the Jacobian To identify the symmetry phases, we analyze the re- gions in the η-ν-plane shown in Fig. 4: I. ν >0 andη >√ν: S is real and η >S, so all λ±1±2 are real. The Jacobian is in thePT -unbroken phase. II. ν >0 and−√ν < η <√ν: S is purely imaginary, making η±S complex. Hence, all eigenvalues are complex and bothPT and qCS symmetri...

  3. [3]

    4 to derive Eq

    Stability from the Jacobian We now analyze stability using the same regions in the previous section and Fig. 4 to derive Eq. (20). As λ++ always has the largest real part, we consider the condition ℜ(λ++)≥ 0, which is equivalent to ℜ p η +S ≥ 1. (28) In Region III (qCS-unbroken), all eigenvalues satisfy ℜ(λ) =−1, so this region is entirely stable. In Regi...

  4. [4]

    For intensity scans with B >3A, however, symmetry is not restored at high input power, and the symmetric solu- tion (Eq

    Limits of the parameter scans In the main text, we focus on the case where the symmetry-broken region is bounded, i.e., when B <3A. For intensity scans with B >3A, however, symmetry is not restored at high input power, and the symmetric solu- tion (Eq. (21)) no longer describes the limit I→∞ [23]. In this regime, the system remains in a symmetry-broken st...

  5. [5]

    Wang, Y.-H

    H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Vahala, Nature Communications 11, 1610 (2020)

    2020

  6. [6]

    Wang, G.-h

    X.-g. Wang, G.-h. Guo, and J. Berakdar, Nature Com- munications 11, 5663 (2020)

    2020

  7. [7]

    Bergman, R

    A. Bergman, R. Duggan, K. Sharma, M. Tur, A. Zadok, and A. Al` u, Nature Communications12, 486 (2021)

    2021

  8. [8]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics 93, 015005 (2021)

    2021

Show all 97 references
  1. [9]

    Kato, Perturbation Theory for Linear Operators (Springer, 1995)

    T. Kato, Perturbation Theory for Linear Operators (Springer, 1995)

    1995

  2. [10]

    Berry, Czechoslovak Journal of Physics 54, 1039 (2004)

    M. Berry, Czechoslovak Journal of Physics 54, 1039 (2004)

    2004

  3. [11]

    W. D. Heiss, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012)

    2012

  4. [12]

    A. E. Kaplan and P. Meystre, Optics Letters 6, 590 (1981)

    1981

  5. [13]

    A. E. Kaplan and P. Meystre, Optics Communications 40, 229 (1982)

    1982

  6. [14]

    E. M. Wright, P. Meystre, W. J. Firth, and A. E. Kaplan, Physical Review A 32, 2857 (1985)

    1985

  7. [15]

    J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Optics Communications 111, 623 (1994)

    1994

  8. [16]

    Xu and S

    Y. Xu and S. Coen, Optics Letters 39, 3492 (2014)

    2014

  9. [17]

    J. Li, X. Zhan, C. Ding, D. Zhang, and Y. Wu, Physical Review A 92, 043830 (2015)

    2015

  10. [18]

    Rossi, R

    J. Rossi, R. Carretero-Gonz´ alez, P. G. Kevrekidis, and M. Haragus, Journal of Physics A: Mathematical and Theoretical 49, 455201 (2016)

    2016

  11. [19]

    Del Bino, J

    L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, Scientific Reports 7, 43142 (2017)

    2017

  12. [20]

    Q.-T. Cao, H. Wang, C.-H. Dong, H. Jing, R.-S. Liu, X. Chen, L. Ge, Q. Gong, and Y.-F. Xiao, Physical Re- view Letters 118, 033901 (2017)

    2017

  13. [21]

    M. T. M. Woodley, J. M. Silver, L. Hill, F. Copie, L. Del Bino, S. Zhang, G.-L. Oppo, and P. Del’Haye, Physical Review A 98, 053863 (2018)

    2018

  14. [22]

    Hendry, W

    I. Hendry, W. Chen, Y. Wang, B. Garbin, J. Javaloyes, G.-L. Oppo, S. Coen, S. G. Murdoch, and M. Erkintalo, Physical Review A 97, 053834 (2018)

    2018

  15. [23]

    L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Steb- bings, X. Zhao, and P. Del’Haye, Optica 5, 279 (2018)

    2018

  16. [24]

    Copie, M

    F. Copie, M. T. M. Woodley, L. Del Bino, J. M. Silver, S. Zhang, and P. Del’Haye, Physical Review Letters122, 013905 (2019)

    2019

  17. [25]

    C. Wu, J. Fan, G. Chen, and S. Jia, Optics Express 27, 28133 (2019)

    2019

  18. [26]

    N. D. Cuong, B. D. Thuan, D. X. Khoa, C. L. Van, M. Trippenbach, and D. T. Thuy, Vinh University Jour- nal of Science 48, 39 (2019)

    2019

  19. [27]

    Hill, G.-L

    L. Hill, G.-L. Oppo, M. T. M. Woodley, and P. Del’Haye, Physical Review A 101, 013823 (2020)

    2020

  20. [28]

    Garbin, J

    B. Garbin, J. Fatome, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and S. Coen, Physical Review Research 2, 023244 (2020)

    2020

  21. [29]

    Dolinina and A

    D. Dolinina and A. Yulin, Optics Letters45, 3781 (2020)

    2020

  22. [30]

    Moroney, L

    N. Moroney, L. D. Bino, M. T. M. Woodley, G. N. Ghalanos, J. M. Silver, A. Ø. Svela, S. Zhang, and P. Del’Haye, Journal of Lightwave Technology 38, 1414 (2020)

    2020

  23. [31]

    G. N. Ghalanos, J. M. Silver, L. Del Bino, N. Mo- roney, S. Zhang, M. T. M. Woodley, A. Ø. Svela, and P. Del’Haye, Physical Review Letters124, 223901 (2020)

    2020

  24. [32]

    M. T. M. Woodley, L. Hill, L. Del Bino, G.-L. Oppo, and P. Del’Haye, Physical Review Letters 126, 043901 (2021)

    2021

  25. [33]

    J. M. Silver, K. T. V. Grattan, and P. Del’Haye, Physical Review A 104, 043511 (2021)

    2021

  26. [34]

    G. Xu, A. U. Nielsen, B. Garbin, L. Hill, G.-L. Oppo, J. Fatome, S. G. Murdoch, S. Coen, and M. Erkintalo, Nature Communications 12, 4023 (2021)

    2021

  27. [35]

    Garbin, J

    B. Garbin, J. Fatome, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and S. Coen, Physical Review Letters 126, 023904 (2021)

    2021

  28. [36]

    Fatome, G

    J. Fatome, G. Xu, B. Garbin, N. Berti, G.-L. Oppo, S. G. Murdoch, M. Erkintalo, and S. Coen, (2021), arXiv:2106.07642

    2021 arXiv

  29. [37]

    Li, Q.-T

    Y.-Y. Li, Q.-T. Cao, J.-h. Chen, X.-C. Yu, and Y.-F. Xiao, Physical Review Applied 16, 044016 (2021)

    2021

  30. [38]

    J. M. Silver, L. D. Bino, M. T. M. Woodley, G. N. Gha- lanos, A. Ø. Svela, N. Moroney, S. Zhang, K. T. V. Grat- tan, and P. Del’Haye, Optica 8, 1219 (2021)

    2021

  31. [39]

    G. Xu, L. Hill, J. Fatome, G.-L. Oppo, M. Erkintalo, S. G. Murdoch, and S. Coen, Optics Letters 47, 1486 (2022). 10

    2022

  32. [40]

    G. N. Campbell, S. Zhang, L. Del Bino, P. Del’Haye, and G.-L. Oppo, Physical Review A 106, 043507 (2022)

    2022

  33. [41]

    Mai and K

    J. Mai and K. W. Cheah, Optics Express 30, 46357 (2022)

    2022

  34. [42]

    J. Mai, Y. Chen, G. Li, and K. W. Cheah, Optics Express 30, 40053 (2022)

    2022

  35. [43]

    Moroney, L

    N. Moroney, L. Del Bino, S. Zhang, M. T. M. Wood- ley, L. Hill, T. Wildi, V. J. Wittwer, T. S¨ udmeyer, G.-L. Oppo, M. R. Vanner, V. Brasch, T. Herr, and P. Del’Haye, Nature Communications 13, 398 (2022)

    2022

  36. [44]

    Hill, G.-L

    L. Hill, G.-L. Oppo, and P. Del’Haye, Communications Physics 6, 1 (2023)

    2023

  37. [45]

    Ghosh, L

    A. Ghosh, L. Hill, G.-L. Oppo, and P. Del’Haye, Physical Review Research 5, L042012 (2023)

    2023

  38. [46]

    R. D. D. Bitha, A. Giraldo, N. G. R. Broderick, and B. Krauskopf, Physical Review E 108, 064204 (2023)

    2023

  39. [47]

    W. Wang, Z. Fang, T. Xian, M. Zhang, Y. Zhao, and L. Zhan, Physical Review Applied 20, 024060 (2023)

    2023

  40. [48]

    Quinn, G

    L. Quinn, G. Xu, Y. Xu, Z. Li, J. Fatome, S. G. Mur- doch, S. Coen, and M. Erkintalo, Optics Letters 48, 3741 (2023)

    2023

  41. [49]

    G. N. Campbell, L. Hill, P. Del’Haye, and G.-L. Oppo, in CLEO/Europe-EQEC (2023) pp. 1–1

    2023

  42. [50]

    J. Mai, X. Huang, X. Guo, H. Fan, and K. W. Cheah, Communications Physics 7, 1 (2024)

    2024

  43. [51]

    Lucas, G

    E. Lucas, G. Xu, P. Wang, G.-L. Oppo, L. Hill, P. Del’Haye, B. Kibler, Y. Xu, S. G. Murdoch, M. Erkin- talo, S. Coen, and J. Fatome, (2024), arXiv:2412.05116

    2024 arXiv

  44. [52]

    G. N. Campbell, L. Hill, P. Del’Haye, and G.-L. Oppo, Optics Express 32, 37691 (2024)

    2024

  45. [53]

    Ghosh, A

    A. Ghosh, A. Pal, L. Hill, G. N. Campbell, T. Bi, Y. Zhang, A. Alabbadi, S. Zhang, and P. Del’Haye, Pho- tonics Research 12, 2376 (2024)

    2024

  46. [54]

    Hill, E.-M

    L. Hill, E.-M. Hirmer, G. Campbell, T. Bi, A. Ghosh, P. Del’Haye, and G.-L. Oppo, Communications Physics 7, 1 (2024)

    2024

  47. [55]

    S. Coen, B. Garbin, G. Xu, L. Quinn, N. Goldman, G.- L. Oppo, M. Erkintalo, S. G. Murdoch, and J. Fatome, Nature Communications 15, 1398 (2024)

    2024

  48. [56]

    A. Pal, A. Ghosh, S. Zhang, L. Hill, H. Yan, H. Zhang, T. Bi, A. Alabbadi, and P. Del’Haye, Photonics Research 12, 2733 (2024)

    2024

  49. [57]

    Rah and K

    Y. Rah and K. Yu, Physical Review Research 6, 013234 (2024)

    2024

  50. [58]

    Ghosh, A

    A. Ghosh, A. Pal, S. Zhang, L. Hill, T. Bi, and P. Del’Haye, (2024), arXiv:2407.16625

    2024 arXiv

  51. [59]

    Zhang, S

    Y. Zhang, S. Zhang, A. Ghosh, A. Pal, G. N. Gha- lanos, T. Bi, H. Yan, H. Zhang, Y. Zhuang, L. Hill, and P. Del’Haye, Photonics Research 13, 360 (2025)

    2025

  52. [60]

    G. N. Campbell, L. Hill, P. Del’Haye, and G.-L. Oppo, Physical Review A 108, 033505 (2023)

    2023

  53. [61]

    L. A. Lugiato and R. Lefever, Physical Review Letters 58, 2209 (1987)

    1987

  54. [62]

    L. A. Lugiato, F. Prati, M. L. Gorodetsky, and T. J. Kippenberg, Phil. Trans. R. Soc. A376, 20180113 (2018)

    2018

  55. [63]

    Haelterman, S

    M. Haelterman, S. Trillo, and S. Wabnitz, Optics Com- munications 91, 401 (1992)

    1992

  56. [64]

    D. C. Cole, A. Gatti, S. B. Papp, F. Prati, and L. Lugiato, Physical Review A 98, 013831 (2018)

    2018

  57. [65]

    D. V. Skryabin, OSA Continuum 3, 1364 (2020)

    2020

  58. [66]

    Bernstein, Reviews of Modern Physics 46, 7 (1974)

    J. Bernstein, Reviews of Modern Physics 46, 7 (1974)

    1974

  59. [67]

    Baskaran and P

    G. Baskaran and P. W. Anderson, Physical Review B 37, 580 (1988)

    1988

  60. [68]

    Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Physical Review Letters 106, 213901 (2011)

    2011

  61. [69]

    Fleury, D

    R. Fleury, D. L. Sounas, and A. Al` u, Physical Review Letters 113, 023903 (2014)

    2014

  62. [70]

    Goldzak, A

    T. Goldzak, A. A. Mailybaev, and N. Moiseyev, Physical Review Letters 120, 013901 (2018)

    2018

  63. [71]

    Miri and A

    M.-A. Miri and A. Al` u, Science363, eaar7709 (2019)

    2019

  64. [72]

    Haus and W

    H. Haus and W. Huang, Proceedings of the IEEE 79, 1505 (1991)

    1991

  65. [73]

    Yariv, IEEE Journal of Quantum Electronics 9, 919 (1973)

    A. Yariv, IEEE Journal of Quantum Electronics 9, 919 (1973)

    1973

  66. [74]

    J. P. Gordon and H. Kogelnik, PNAS 97, 4541 (2000)

    2000

  67. [75]

    Kogelnik, The Bell System Technical Journal 48, 2909 (1969)

    H. Kogelnik, The Bell System Technical Journal 48, 2909 (1969)

    1969

  68. [76]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Reviews of Modern Physics 86, 1391 (2014)

    2014

  69. [77]

    Fox, Quantum Optics: An Introduction (Oxford Uni- versity Press, 2006)

    M. Fox, Quantum Optics: An Introduction (Oxford Uni- versity Press, 2006)

    2006

  70. [78]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Na- ture Physics 14, 11 (2018)

    2018

  71. [79]

    K.¨Ozdemir, S

    S ¸. K.¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Nature Materials 18, 783 (2019)

    2019

  72. [80]

    Bernard and A

    D. Bernard and A. LeClair, in Statistical Field Theories , NATO Science Series (Springer, 2002) pp. 207–214

    2002

  73. [81]

    Kawabata, K

    K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys- ical Review X 9, 041015 (2019)

    2019

  74. [82]

    Delplace, T

    P. Delplace, T. Yoshida, and Y. Hatsugai, Physical Re- view Letters 127, 186602 (2021)

    2021

  75. [83]

    Sayyad and F

    S. Sayyad and F. K. Kunst, Physical Review Research 4, 023130 (2022)

    2022

  76. [84]

    J. C. Budich, J. Carlstr¨ om, F. K. Kunst, and E. J. Bergholtz, Physical Review B 99, 041406 (2019)

    2019

  77. [85]

    Yoshida, R

    T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, Physical Review B 99, 121101 (2019)

    2019

  78. [86]

    Okugawa and T

    R. Okugawa and T. Yokoyama, Physical Review B 99, 041202 (2019)

    2019

  79. [87]

    C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998)

    1998

  80. [88]

    C. M. Bender, Reports on Progress in Physics 70, 947 (2007)

    2007

  81. [89]

    Gr¨ afe, R

    M. Gr¨ afe, R. Heilmann, R. Keil, T. Eichelkraut, M. Hein- rich, S. Nolte, and A. Szameit, New Journal of Physics 15, 033008 (2013)

    2013

  82. [90]

    A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Physical Review Letters 103, 093902 (2009)

    2009

  83. [91]

    Montag and F

    A. Montag and F. K. Kunst, Physical Review Research 6, 023205 (2024)

    2024

  84. [92]

    Bai, J.-Z

    K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Physical Review Letters 130, 266901 (2023)

    2023

  85. [93]

    Bai, T.-R

    K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Physical Review Letters 132, 073802 (2024)

    2024

  86. [94]

    Yoshida, T

    T. Yoshida, T. Isobe, and Y. Hatsugai, Physical Review B 111, 064310 (2025)

    2025

  87. [95]

    L. Fang, K. Bai, C. Guo, T.-R. Liu, J.-Z. Li, and M. Xiao, (2024), arXiv:2408.03751

    2024 arXiv

  88. [96]

    F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010)

    2010

  89. [97]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables (Dover Publications, 1964)

    1964

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