Merging the characteristics of an exceptional point and a quasi-bound state in the continuum in nanophotonic cavities

Jun He; Lin Ma; Xiao-Jing Du; Xi-Hua Guan; Yue You; Zhong-Jian Yang

REVIEW 4 major objections 5 minor 2 references

Merging the characteristics of an exceptional point and a quasi-bound state in the continuum in nanophotonic cavities

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A single excitation-phase choice can merge an exceptional point and a quasi-bound state in the continuum at the same frequency in two-mode nanophotonic cavities.

desk verdict Theta never enters the Hamiltonian, so the EP-QBIC claim is unsubstantiated; the real content is a solid, useful demonstration of phase-controlled Fano line narrowing in stacked nanorods. read the letter →

arxiv 2512.07494 v3 pith:ONUCZAGT submitted 2025-12-08 physics.optics

Xiao-Jing Du , Xi-Hua Guan , Yue You , Lin Ma , Jun He , Zhong-Jian Yang This is my paper · ORCID

classification physics.optics

keywords exceptionalpointboundstateinthecontinuumquasi-BICexcitationphasecoupledmodetheorynanophotoniccavityFanoresonanceplasmonicnanoroddimer

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

This paper tries to establish that the usual impossibility of placing an exceptional point and a quasi-bound state in the continuum at the same spectral position in a two-mode non-Hermitian cavity is a limitation of the eigenvalue framework, not of the underlying physics. Adding an excitation-phase delay between the driving of the two modes reshapes the interference between their radiation channels without altering the intrinsic eigenmodes, and this extra knob permits the exceptional-point condition and the quasi-BIC condition to be met simultaneously. Using coupled-mode theory and full-wave simulations of stacked gold-silicon nanorod dimers, the authors demonstrate a single narrow resonance with Q around 125 and an order-of-magnitude quality-factor enhancement, and they report a purely plasmonic case whose Q exceeds the intrinsic material-loss limit. If the claim holds, the excitation phase becomes a practical control for high-Q resonances at eigenmode degeneracies.

What carries the argument

The central object is the excitation phase θ in the source vector of the coupled-mode equations, working together with the non-Hermitian 2×2 Hamiltonian whose eigenvalues define the exceptional point (a degeneracy where two eigenvalues and their eigenvectors coalesce). Because θ does not appear in the Hamiltonian, it cannot change eigenmode frequencies or linewidths; instead it controls the relative phase with which the two coupled modes are driven, i.e., the interference between radiation channels. This lets the extinction spectrum be tuned from a Fano double-peak (conventional QBIC) to a single-peak EP-QBIC while the intrinsic mode structure stays fixed, and it provides a direct mapping be

What would settle it

Measure the extinction spectrum of the stacked dimer while sweeping the excitation phase from θ=0 to θ=π/2 with the coupling phase fixed. If the narrow peak keeps the same frequency and linewidth while the broad peak disappears, the phase is only deselecting a mode rather than creating a degeneracy. A second check: at the claimed EP, a small detuning of φ or κ should split the resonance with a square-root dependence, the signature of an exceptional point.

Watch

Extended reading notes

Core claim

The paper claims that in a two-mode non-Hermitian cavity, the failure to merge an exceptional point and a quasi-BIC at one frequency is not fundamental but stems from leaving the driving field out of the picture. In the coupled-mode equations, the excitation phase enters only the source term and not the Hamiltonian, so it can be varied to change which superposition of the two eigenmodes the incident field excites, without moving the eigenfrequencies or linewidths. With a suitable coupling strength and coupling phase, the right excitation phase produces a single-peak, narrow-linewidth response (Q≈125) in which the two new modes share both frequency and linewidth — the exceptional point — and

Load-bearing premise

The load-bearing premise is that the single narrow peak obtained at the chosen excitation phase is genuinely an exceptional point of the two-mode Hamiltonian, and not a conventional Fano dark-mode line in which the broad mode is simply not excited; if that identification is wrong, the central claim collapses.

Editorial extensions

If this is right

At the EP-QBIC condition, the cavity response is a single narrow resonance with Q≈125, more than an order of magnitude above the uncoupled nanorod modes.
The excitation phase is a necessary control: at θ=0 no choice of coupling strength or coupling phase produces an EP-QBIC, and tuning θ sweeps the spectrum between conventional QBIC and EP-QBIC.
Because the EP region lies wholly inside the QBIC region in the θ-κ plane, the merging is a systematic feature of the two-mode model rather than an accidental crossing.
Increasing the coupling strength drives both conventional QBIC and EP-QBIC toward ideal BICs, so the model supports an EP-BIC as a limiting case.
In a purely plasmonic dimer, the phase-controlled QBIC raises Q by over 15 times, exceeding the intrinsic material-loss limit of roughly 12.

Reading between the lines

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

A practical next step would be to make the excitation phase dynamically adjustable with a shaped or tilted wavefront; the same cavity could then toggle between a broad Fano response and a narrow high-Q resonance at fixed geometry.
Because θ is absent from the Hamiltonian, the merged resonance is the selected excitation of an existing degenerate eigenmode rather than a newly created mode — this suggests the phase-control strategy should transfer to any two-mode non-Hermitian radiator, including metasurfaces, waveguide-coupled rings, or acoustic analogs, wherever a relative drive phase can be imposed.
A direct experimental falsification of the 'merging' interpretation is to perturb φ or κ slightly around the claimed EP point and look for the characteristic square-root splitting of the resonance; observing linear splitting would indicate a diabolic-point-like degeneracy instead.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the central claim—that tuning an excitation phase θ merges an exceptional point and a quasi-BIC—does not survive contact with its own equations. θ appears only in the source vector of Eq. (1); the Hamiltonian in Eq. (3) is θ-independent. The single-peak, high-Q spectrum at θ=π/2 is just the Fano dark-mode case where the broad eigenmode is not excited. That is a useful trick, but it is not an EP.

What is genuinely new: the parameter maps in Fig. 1d and Fig. 2 connecting coupling strength, coupling phase, and excitation phase to QBIC conditions, plus the FDTD demonstration on stacked Au-Si nanorods where the gap controls coupling and the linewidth narrows. The CMT-to-FDTD comparison in Fig. 3d, with all parameters except coupling taken from the individual-mode simulations, is a reasonable validation and shows an honest discrepancy at small gaps.

The soft spots are load-bearing. First, the EP-QBIC label. In Fig. 1c, φ and κ are fixed, so the eigenmodes of H are fixed; θ only changes the weights of the two Lorentzians in the extinction. That is textbook Fano interference, not coalescence. In Fig. 2, the authors set φ=θ, which does make θ enter H, but then the abstract's claim that the intrinsic eigenmodes are "not modified" is wrong—they are modifying the coupling phase. Second, the QBIC region in Fig. 2c is defined as γ₋ < γ_L where γ_L is the linewidth of the low-frequency original mode. In their example (Au and Si), that is the high-loss mode, so the inequality is easy to satisfy; the EP region landing inside this 'QBIC' region is trivial. Third, the pure-plasmonic claim of exceeding the material-loss limit is not rigorously supported; the 15× Q enhancement and the comparison to a 'limit' would need a careful derivation of what that limit is for a coupled dimer.

The paper is worth a serious referee because the engineering demonstration is plausible and the CMT machinery is mostly sound, but the interpretation is wrong. A referee should ask for a rewrite that either drops the EP language or clearly separates the Hamiltonian-level EP (which exists when φ, κ are tuned) from the excitation-phase effect. In its current form, I would not cite it for the EP-QBIC claim, but I would be interested in the phase-controlled line-narrowing data. Recommendation: send to peer review, expect major revision or rejection.

Referee Report

4 major / 5 minor

Summary. The paper claims that introducing an excitation-phase degree of freedom in a two-mode non-Hermitian nanophotonic cavity allows an exceptional point (EP) and a quasi-bound state in the continuum (QBIC) to merge at the same spectral position, even though conventional eigenvalue analyses supposedly forbid such coexistence. The authors develop a coupled-mode theory (CMT) with a source term containing an excitation phase θ, show that a single narrow extinction peak appears for θ=π/2 at fixed coupling phase and strength, label this as an EP–QBIC state, and support the claim with FDTD simulations of Au–Si and Au–Au stacked nanorod dimers. They also report Q-factor enhancements of over an order of magnitude and a plasmonic case exceeding the material-loss limit.

Significance. If the central claim were correct, it would offer a conceptually new route to high-Q resonances at eigenvalue degeneracies, with potential impact in sensing, lasing, and nonlinear photonics. The manuscript includes both CMT analytics and FDTD simulations, and the numerical effort is substantial. However, the claim rests on a misidentification: the excitation phase θ does not enter the Hamiltonian, so it cannot create an exceptional point. The observed single-peak response at θ=π/2 is a standard dark-mode/Fano excitation effect, not eigenvalue coalescence. The QBIC classification is also weakened by an inconsistent definition of γ_L. These issues are load-bearing and render the central result unsupported.

major comments (4)

[Methods, Eqs. (1)–(3) and Fig. 1(c)] The excitation phase θ appears only in the source vector on the right-hand side of Eq. (1), while the Hamiltonian H in Eq. (3) is independent of θ. Consequently, the eigenfrequencies and eigenmodes are fixed for given ω1,ω2,γ1,γ2,κ,φ. The spectral change from a Fano doublet at θ=0 to a single narrow peak at θ=π/2 in Fig. 1(c) is therefore a change in the excitation overlap—one eigenmode becomes dark—not a coalescence of eigenvalues. The single narrow peak at θ=π/2 is the same high-Q eigenmode present at θ=0. Calling this an EP requires either an unstated change in φ (as in Fig. 2, where φ=θ) or a nonstandard definition of EP. The abstract's claim that the mechanism works 'without modifying the intrinsic eigenmodes' is inconsistent with the Fig. 2 procedure, which tunes φ=θ and thus changes H. The central identification of an EP–QBIC state is not established by the CMT as presented.
[Results, Fig. 2(c) and surrounding text] The QBIC criterion is defined as γ− ≤ γ_L, where γ_L is stated to be the linewidth of the original low-frequency mode. In the example of Fig. 1, the low-frequency mode is the broad, high-loss mode (γ≈0.556 eV), while the high-frequency mode is the narrow, low-loss mode (γ≈0.0102 eV). The text nevertheless refers to γ_L as the 'original low-loss one.' With γ_L taken as the broad-mode linewidth, the condition γ− < γ_L is satisfied whenever the coupled low-frequency mode is narrower than the original broad mode, which is nearly always true and trivially includes the EP region, whose linewidth is (γ_H+γ_L)/2 ≈ 0.28 eV < 0.556 eV. The EP–QBIC 'merging' shown in Fig. 2(c,f) is therefore an artifact of an overly permissive and inconsistently defined QBIC criterion, not a nontrivial coincidence.
[Fig. 3(d) and 'Simulated and analytical extinction spectra'] The FDTD validation is partly circular. All CMT parameters except the coupling strength κ are extracted from isolated-rod simulations, and κ is the only free parameter. The simulated single peak in the hybrid dimer is then labeled EP–QBIC using the same CMT criteria whose validity is under question. Since a single narrow peak can arise from a dark-mode Fano configuration without any eigenvalue degeneracy, the FDTD spectra of Fig. 3(b,c) do not independently confirm the existence of an EP. The agreement shown in Fig. 3(d) is a trend in linewidth versus gap, not a test of eigenmode coalescence. A direct eigenvalue analysis or an unambiguous signature of the EP (e.g., eigenvalue Riemann-surface topology or eigenvector coalescence) is lacking.
[Abstract and Introduction, 'eigenvalue framework' claim] The paper asserts that the impossibility of EP–QBIC coexistence is a limitation of the eigenvalue framework and that an excitation-phase degree of freedom overcomes it. However, when φ=θ is used (as in Fig. 2), θ does enter the Hamiltonian through the coupling phase and hence modifies the eigenvalues. The eigenvalue framework is fully capable of describing the resulting EP; the source phase is not needed for the eigenvalue degeneracy. The philosophical claim that the eigenvalue framework itself is the obstacle is therefore not supported by the model.

minor comments (5)

[Affiliations] The affiliation line repeats 'University, Changsha 410083, China' twice; this should be corrected.
[Fig. 1(d) and text] The statement that 'the phase delay θ induced by plane wave excitation cannot exceed the spatial phase delay φ between two modes' is asserted without justification or a formal definition of the phase relationship. Clarify the physical origin of this inequality.
[Fig. 4(c)] The 'volcanic eruption' line shape and its interpretation in terms of a diabolic-point-like state need a more precise definition. Is the frequency degeneracy with unequal linewidths a diabolic point in the usual sense, and what is its relation to the EP–QBIC discussion?
[Fig. 4(b) and associated text] The claim that the Q factor 'surpasses the theoretical limit imposed by intrinsic material loss' should be clarified: for a single plasmon resonance the maximum Q is limited by Im(ε)/Re(ε), but a coupled-mode scenario can produce a narrower dark mode because radiative loss is suppressed while material loss remains. Specify the precise reference limit and how it is computed.
[Supplementary Fig. S2] The lower panel is said to show 'EP-QBIC gradually approaching EP-BIC,' but the text at the end of the Results section states that approaching BIC in practical systems is challenging. This is consistent, but the Supplementary caption should be checked for a typo: 'EP–QBIC' appears twice in the lower panel description.

Circularity Check

3 steps flagged · score 6.0 of 10

The central EP–QBIC result is imposed by a response-based definition and a QBIC threshold chosen so the EP automatically falls inside; the excitation phase alone never changes the Hamiltonian.

renaming known result [Results and Discussion, Fig. 1(c) paragraph (after Eqs. (1)–(6))]
"The case with the excitation phase 𝜃 = 0 exhibits a Fano-resonance lineshape... analogous to the coherent interference between bright and dark modes in common Fano-resonant nanophotonic systems... When the excitation phase is adjusted to 𝜃 = 2𝜋 4⁄, the coupled system exhibits a single-peak, narrow-linewidth resonance characteristic. This result indicates the emergence of two new modes at the same frequency, accompanied by a high-Q resonance (Q ≈ 125), signifying the formation of an EP–QBIC state."

θ appears only on the RHS source vector of Eq. (1); the Hamiltonian in Eq. (3) is independent of θ when φ and κ are fixed. The eigenmodes are therefore identical for θ=0 and θ=π/2. The single narrow peak at θ=π/2 is the same eigenmode pair with the broad mode suppressed by the drive vector — the standard Fano/dark-mode effect described in the same paragraph for θ=0. Calling it EP–QBIC defines the state by the response lineshape rather than by eigenvalue coalescence, so the claimed prediction reduces to a known dark-mode result relabeled as an EP.
self definitional [Results and Discussion, Fig. 2(c–f) mapping paragraph]
"Since the formation of a QBIC requires that the linewidth of one coupled mode be smaller than the original low-loss one (𝛾𝐿), we calculated the 𝛾− − 𝛾𝐿 as shown in Fig. 2c. Then, the QBIC region can be easily identified within the area of 𝛾− − 𝛾𝐿 < 0... The EP region falls entirely within the QBIC domain, indicating the generation of EP–QBIC states."

At an EP the two coupled linewidths coalesce, γ+ = γ− = (γ1+γ2)/2, which is larger than the smaller of γ1 and γ2. Therefore γ− < γL can hold only if γL is the loss of the low-frequency (lossier) original mode, not the 'low-loss one.' With that reading, the inequality is an arithmetic consequence of averaging, not a narrow-linewidth QBIC condition. The statement that the EP region lies inside the QBIC domain is thus true by the choice of reference loss; the coexistence is put into the definition rather than derived.

1 more flagged steps

other [Results and Discussion, beginning of Fig. 2 section]
"To further investigate the formation of EP–QBIC within such a two-mode coupled system, we systematically studied the influence of coupling strength 𝜅 and phase 𝜃 on the optical response of the system. 𝜑 is set to be 𝜃 = 𝜑 for simplicity."

The abstract claims the excitation phase acts 'without modifying the intrinsic eigenmodes.' But setting φ=θ makes the tuning parameter also enter H through k = κe^(−iφ), so the plotted EP regions are ordinary eigenvalue degeneracies controlled by the pre-existing coupling phase. The independent 'excitation-phase degree of freedom' has been identified with an old Hamiltonian parameter, so the claimed escape from the eigenvalue framework reduces to tuning and relabeling that parameter.

full rationale

The CMT algebra itself is not circular: given ω1, ω2, γ1, γ2, κ, φ and the source vector, the extinction follows from Eq. (6), and no fitted quantity is renamed as a prediction in that calculation. The FDTD simulations are independent of the CMT fit apart from standard extraction of mode parameters and one coupling-strength parameter. No load-bearing self-citation chain or imported uniqueness theorem is used. However, the interpretive layer that constitutes the paper's central claim is circular in two places. First, the 'excitation phase' θ appears only in the source term, so at fixed φ and κ it cannot move the eigenfrequencies or linewidths; the θ=π/2 single peak is the standard Fano dark-mode response of the same eigenmode pair, relabeled 'EP–QBIC' because a single narrow peak is declared to signify it. Second, the QBIC domain in Fig. 2 is defined by γ− < γL, with γL taken as the loss of the low-frequency (lossier) original mode; at an EP γ− is the arithmetic mean of the two original losses, so the EP automatically satisfies the criterion. The 'merging' is therefore put in by the threshold choice. The additional move φ=θ in Fig. 2 removes the promised independence of the new degree of freedom: the EP maps are ordinary eigenvalue degeneracies controlled by the coupling phase already present in H. These are definitional/renaming reductions of the central claim, not fitted-input substitutions, so the appropriate score is partial circularity rather than full equivalence.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model rests on standard two-mode CMT plus two nonstandard interpretive steps: (i) the identification of γ−<γ_L with QBIC even when γ_L is the larger original loss, and (ii) the identification of a single narrow response peak at θ=π/2 with an EP despite θ not affecting H. The FDTD comparison additionally requires a coupling strength that is effectively fitted. No new physical entities are introduced.

free parameters (3)

illustrative coupling ratio κ/γ_L = 0.94 = 0.94 γ_L
Chosen for Fig. 1(c); the EP–QBIC and C–QBIC regions in the parameter maps depend on this value.
illustrative coupling phase φ = π/2 (Fig. 1), φ = θ (Fig. 2) = π/2; φ=θ
Free phase choices; the central demonstration is possible only for these phase relations.
coupling strength κ for the Au-Si dimer CMT model = not stated
The only CMT parameter not extracted from isolated-rod simulations; effectively fit so the analytical spectra match the coupled FDTD results in Fig. 3(d).

assumptions (4)

domain assumption Two-mode CMT (Eqs. 1–6) is adequate for the stacked nanorod dimers.
Assumes only two relevant modes, linear coupling, Markovian loss; the FDTD agreement is used as support, but the model omits the known Au plasmon redshift at small gaps.
ad hoc to paper QBIC condition γ− < γ_L defines a quasi-BIC.
γ_L is defined as the loss of the low-frequency mode; Fig. S1 suggests that mode has the larger loss, so the condition is easily satisfied and the EP region lies inside it trivially.
ad hoc to paper Single narrow peak at θ = π/2 indicates an exceptional point.
θ is not in H, so no eigenvalue coalescence is implied; this interpretation is load-bearing for the title claim.
domain assumption Total loss of a passive resonator is the sum of radiation and absorption, so Q ≤ ω/γ_abs.
Standard passivity constraint used to define the 'theoretical limit' (Q≈12); the paper's claim of Q=20 contradicts this unless Q refers to a Fano linewidth rather than a modal Q.

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Cite this review

Pith. "Pith review of Merging the characteristics of an exceptional point and a quasi-bound state in the continuum in nanophotonic cavities." pith.science (2026). https://pith.science/paper/ONUCZAGT

@misc{pith2026251207494,
  author       = {Pith},
  title        = {Pith review of: Merging the characteristics of an exceptional point and a quasi-bound state in the continuum in nanophotonic cavities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ONUCZAGT}},
  note         = {Machine review of arXiv:2512.07494}
}

read the original abstract

In conventional eigenvalue analyses of non Hermitian two mode systems, mode coupling cannot produce the simultaneous occurrence of an exceptional point (EP) and a quasi bound state in the continuum (QBIC) at the same spectral position. This work shows that this limitation originates from the eigenvalue framework itself. By introducing an excitation phase degree of freedom, the interference between radiation channels can be reshaped without modifying the intrinsic eigenmodes of the system, thereby overcoming this constraint. Based on coupled mode theory, we demonstrate that the excitation phase enables the merging of an EP and a QBIC (and even a BIC) in nanophotonic cavities, and we validate this mechanism through full wave simulations of practical stacked structures. In the EP-QBIC regime, the mode quality (Q) factor is enhanced by more than one order of magnitude. We further systematically analyze the formation conditions of EP-QBIC states and conventional QBICs. Moreover, in a purely plasmonic structure, introducing an excitation phase leads to a more than 15 fold increase in the Q factor due to QBIC formation-surpassing the theoretical limit imposed by intrinsic material loss.

Figures

Figures reproduced from arXiv: 2512.07494 by the authors.

**Figure 2.** EP–QBIC and C–QBIC states with varying the coupling strength 𝜿 and phase 𝜽. (a, b) Linewidths 𝛾+ and 𝛾− of the coupled modes. The dashed line shows the region of 𝛾+ = 𝛾−. (c) Mapping of 𝛾− − 𝛾𝐿 . The dashed white-line area shows the region of 𝛾− ≤ 𝛾𝐿 . (d, e) Resonant frequencies 𝜔+ and 𝜔− of the coupled modes. The dashed-line area shows the region of 𝜔+ = 𝜔− . (f) Frequency differences 𝜔+ − 𝜔− . The dashed purple-l… view at source ↗

**Figure 3.** Generation of EP–QBIC states in practical stacked hybrid cavities. (a) Schematic of the hybrid dimer system consisting of a Si nanorod and an Au nanorod. The length, width, and height of the Si rod are 340, 230 and 450 nm, respectively. The dimensions of the gold rod are 400, 230 and 150 nm. (b) Simulated extinction spectra of individual Si and Au nanorods as well as the hybrid structure with a surface gap of 10 nm.… view at source ↗

**Figure 4.** The case of a pure plasmonic system. (a) Schematic illustration of a stacked plasmonic nanorod dimer. The two Au nanorods possess identical geometrical dimensions, with a length, width, and height of 308 nm, 120 nm, and 110 nm, respectively. (b) Simulated and analytical extinction spectra for dimers with gap distances varying from 500 to 30 nm. (c) Extinction spectra for a dimer with a gap of 30 nm and the monomer s… view at source ↗

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Works this paper leans on

2 extracted references

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

D., Handbook of optical constants of solids

(1) Palik, E. D., Handbook of optical constants of solids. Academic press: 1998; V ol

1998

[2] [3]

D.; Fan, J

(2) Liu, S. D.; Fan, J. L.; Wang, W. J.; Chen, J. D.; Chen, Z. H. Resonance Coupling between Molecular Excitons and Nonradiating Anapole Modes in Silicon Nanodisk-J-Aggregate Heterostructures. ACS Photonics 2018, 5 (4), 1628-1639. (3) Yang, Z. J.; Deng, Y . H.; Yu, Y .; He, J. Magnetic toroidal dipole response in individual all -dielectric nanodisk cluste...

2018