{"id":"56b62381-3df0-4fad-8d48-af3a3c71f7e0","arxiv_id":"2511.03468","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Cavity pole trajectories that swap positions in the complex frequency plane sharply mark the onset of strong light–matter coupling, with coupling rates computed from overlap integrals.","lead":"Resonances of a photonic cavity, tracked at their true complex frequencies as the cavity geometry changes, swap positions exactly when light–matter coupling becomes strong — a clean, spectrum-independent signature. The same framework yields per-resonance coupling rates directly from overlap integrals, replacing fits to transmission or absorption data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Higher-order cavity modes may bias the single-RS effective Hamiltonian and inverse-eigenproblem coupling rates; the paper's own spectra show these modes are non-negligible.","rationale":"The reader's weakest assumption—the single-RS Green's function approximation—is indeed the most load-bearing element of the paper's quantitative apparatus. The paper's central qualitative observation (RS trajectories switching from crossing to swapping at strong coupling) is demonstrated with exact nonlinear eigenproblem solutions in Figs. 3 and 4, so it is on firmer ground. But the more novel quantitative claims—the effective Hamiltonian of Eq. (10), the overlap-integral coupling rates, and the inverse-eigenproblem extraction of SI §D—all inherit the single-mode truncation. The paper's own evidence that higher-order modes contribute to observable spectra and form accumulation points makes this more than a formal caveat. A concrete test comparing inverse-eigenproblem extraction against overlap-integral values in a regime where higher-order modes are resonant would settle whether the bias is material. I therefore maintain the reader's CONDITIONAL verdict, with the condition that this test be performed and reported. The paper deserves credit for making the derivation transparent, providing code, and showing the right kind of validation (Fig. 6a), but the quantitative accuracy claim needs a quantitative error metric.","tokens_in":23360,"tokens_out":4187,"duration_ms":40778,"concrete_test":"Use the provided repository to compute the full nonlinear-eigenproblem RS spectrum for the SURMOF-filled planar cavity at a thickness d where a higher-order FP mode is nearly degenerate with one of the material poles (e.g., near the third peak in Fig. 2(a4)). Fit the inverse eigenproblem of SI §D to only the four selected eigenvalues (as in Fig. 7) and compare the extracted κ_p and K against the overlap-integral values from Eq. (11) and against the eigenvalues of the full 4×4 Hamiltonian. Repeat with the higher-order modes artificially suppressed (e.g., by increasing the material pole separation or using a single-mode cavity). If the extracted κ_p changes by more than a few percent when higher-order modes are included or excluded, the single-RS approximation is not sufficient for unbiased coupling-rate extraction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims—direct extraction of coupling rates from overlap integrals via Eq. (10), and the inverse-eigenproblem recovery of SI §D—rest on the single-resonant-state approximation of the Green's function, G_c ≈ c E_c⊗E_c/(ω−ω̃_c) (SI Eq. (29)). All higher-order cavity modes are discarded. Yet the paper itself demonstrates that higher-order modes are not negligible: in Fig. 2(a4) the central transmission peak is attributed to higher-order Fabry-Perot modes, and in Fig. 3(b) the RSs form accumulation points at the material pole. In the multi-resonance example of Figs. 5–7, higher-order FP modes hybridize with the same three material poles, producing additional polariton branches between the fundamental branches (Fig. 5(c)). If these modes contribute significantly to the four eigenvalues used in the inverse eigenproblem (SI §D, Eq. (43)), the characteristic polynomial is not that of the 4×4 Hamiltonian, and the extracted κ_p and diagonal shift K are biased. The same single-mode Green's function underlies the overlap-integral κ_p in Eq. (11), so both routes share this vulnerability. The paper provides no quantitative error metric for the 'excellent agreement' in Fig. 6(a), nor a direct comparison between the overlap-derived and inverse-eigenproblem κ_p values.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Using resonant states (quasinormal modes) at complex frequencies, the paper argues for a criterion separating weak and strong light-matter coupling: when a geometric parameter is swept, the associated RS trajectories either pass through each other (weak coupling) or swap identities (strong coupling), with an exceptional point at the transition. Under a single-dominant-photonic-mode assumption, the authors derive from Maxwell's equations an effective Hamiltonian (Eq. (10)) coupling one cavity RS to an arbitrary number of Lorentz material poles, with overlap-integral couplings κ_p = sqrt(g_p Ω̃_p) and a diagonal shift −Σ g_p. They demonstrate the criterion on a planar Fabry-Perot cavity with one and three material poles and on a core-shell nanosphere, extract coupling rates both from overlap integrals and from an inverse eigenproblem, and compare the single-mode resonant-state expansion to a direct nonlinear eigenproblem solution (Fig. 6(a)). Code and data are provided.","tokens_in":23540,"tokens_out":6947,"duration_ms":64649,"significance":"If the central quantitative claims hold, the paper offers a parameter-free, first-principles route to coupling rates in open cavities, directly from normalized RS fields, and it identifies a physical correction (the diagonal shift) absent from standard coupled-oscillator models. The trajectory criterion is intuitive and is checked against direct nonlinear eigenproblem solutions; the availability of code/data and the transparent SI derivation increase confidence. The main weakness is that the quantitative extraction—overlap integrals and inverse eigenproblem—relies on a single-RS Green's function approximation whose validity is asserted rather than quantified in exactly the multi-resonance, multi-mode regime the paper targets.","major_comments":[{"comment":"The derivation of the effective Hamiltonian and the coupling rates κ_p = sqrt(g_p Ω̃_p) assumes G_c ≈ c E_c⊗E_c/(ω−ω̃_c), i.e., a single cavity RS. This is load-bearing for the paper's main quantitative claims, but the paper itself demonstrates that higher-order cavity RSs are not negligible: they dominate the central transmission peak in Fig. 2(a4), form accumulation points at the material pole in Fig. 3(b), and produce additional polariton branches between the fundamental branches in Fig. 5(c). In such a regime the four eigenvalues used to validate the model may not be governed by the truncated 4×4 Hamiltonian, and the overlap-integral κ_p can be biased by missing modes. Please provide a quantitative error metric for the \"excellent agreement\" in Fig. 6(a) (e.g., maximum/mean |Δω̃| over the plotted thickness range), compare the overlap-derived κ_p with those recovered by the inverse eig","section":"SI §C, Eq. (29); Eqs. (10)–(11), Fig. 6(a)"},{"comment":"The inverse-eigenproblem extraction treats the products κ̂_p κ_p and the shift K as unknowns determined by matching the characteristic polynomial of the 4×4 Hamiltonian to exactly four known RS eigenvalues. This is only valid if those four eigenvalues correspond to the single cavity RS plus the three material poles and no other RS contributes to the secular equation. Given the additional polariton branches visible in Fig. 5(c), the selection rule for the four eigenvalues is not specified, and any contamination from higher-order FP modes would bias κ_p and K. Please state how the eigenvalues are selected and include a robustness check, e.g., solving the inverse problem from subsets of eigenvalues or from noisy synthetic data.","section":"SI §D, Eqs. (41)–(45); Fig. 5(c)"},{"comment":"The three-pole Lorentz permittivity is fitted on the real frequency axis (Fig. 5(a)) and then continued analytically into the complex plane; all trajectory features—material poles, zeros, accumulation points—and the resulting strong-coupling assignments depend on the locations of those complex poles. A real-axis fit does not uniquely determine the complex-plane continuation. The direct nonlinear eigenproblem used for validation employs the same pole model, so it cannot test this assumption. Please demonstrate robustness by comparing against an independent rational approximation of the same real-axis permittivity (e.g., an AAA-based fit along the lines of Refs. [54]–[56]) or by perturbing the fitted pole parameters within their uncertainty, and state whether the strong-coupling classification changes.","section":"Eq. (4) and SI §B"}],"minor_comments":[{"comment":"The text states that κ_eff^p is compared at a thickness d_p where the real parts of the bare cavity mode and the p-th material resonance match, but the colored vertical lines in Fig. 6(b) are not explained in the caption. Please define d_p explicitly in the figure caption.","section":"Fig. 6(b) and Eq. (14)"},{"comment":"The table mixes geometry parameters, the SURMOF material model, and the silver model in a single column set. Separating the geometry and material-model columns would improve readability and reduce the chance of misassignment.","section":"Table I"},{"comment":"The sign convention relating A_j, B_j, and σ_p is easy to confuse; a short numerical check for one Lorentz pole (e.g., the resulting ε_r at a real frequency) would help the reader verify the convention.","section":"Eqs. (15)–(20)"},{"comment":"The branch colors are described in the text but not in the captions. Adding a legend or explicit color labels to the figures would make the trajectory discussion much easier to follow.","section":"Figure 3 and 4 captions"}],"recommendation":"major_revision","confidential_remarks":"The trajectory-based criterion part is solid and well validated; the main issue is the quantitative support for the single-RS truncation underlying the coupling-rate extraction. This should be fixable with additional numerical analysis. No concerns about citation ethics or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a serious referee. The main new thing isn't the complex-plane trajectory idea per se—the exchange at an exceptional point is known—but the derivation of a single-mode effective Hamiltonian whose couplings are overlap integrals, and a plausible demonstration that the trajectory crossing/swap is a robust weak/strong classifier in open cavities. The validation against the nonlinear eigenproblem (Fig. 6a) is the right check, and it holds.\n\nWhat the paper does well: the SI is transparent, the diagonal self-shift −Σg_p is a real correction absent from most coupled-oscillator fits, and the SURMOF example is a realistic test. They also ship code and data, which is real evidence.\n\nSoft spots. First, the single-resonant-state approximation of the Green's function is load-bearing, and the paper itself shows higher-order modes are non-negligible—they produce the 'third peak' in Fig. 2(a4) and the accumulation points in Fig. 3. The inverse-eigenproblem extraction in SI §D assumes exactly the truncated 4×4 characteristic polynomial. If higher-order modes materially perturb the four eigenvalues used, the extracted κ_p and K are biased. I think this is a legitimate concern, but the Fig. 6a comparison suggests the truncation error is small for the fundamental branches; still, the authors should quantify it, e.g., by adding a residual or directly comparing overlap-derived and inverse-eigenproblem κ_p. Second, there's no quantitative error metric for 'excellent agreement'—an easy fix. Third, Table I has an eye-catching coincidence: the silver Lorentz resonance frequency equals the molecular pole frequency (2π×412.94 THz) in the single-pole examples. That looks like a typo, and the authors should confirm the intended silver model. Fourth, the 'for the first time' / 'unambiguous' framing is a bit strong given the approximations. And the novelty relative to the dispersive resonant-state expansion work (Refs 22–23) is asserted rather than clearly delineated.\n\nBottom line: the core physics is coherent and the derivation is honest. Anyone modeling polaritons in dispersive open cavities will get value from the overlap-integral Hamiltonian. I'd send it to peer review rather than desk reject, and I'd ask the referee to check Table I and quantify the single-mode truncation error.","headline":"Solid methods paper: the complex-plane trajectory criterion and overlap-integral Hamiltonian are useful, validation is decent, but the single-mode truncation and a suspicious Table I deserve referee attention.","tokens_in":24205,"tokens_out":4809,"would_cite":true,"duration_ms":40143,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Resonant states in the complex frequency plane reveal when light-matter coupling becomes strong: rather than merely perturbing each other, the states swap positions at the transition, while the paper also derives an effective Hamiltonian th","keywords":["resonant states","strong coupling","nanophotonic cavities","complex frequency plane","quasinormal modes","exceptional points","effective Hamiltonian","overlap integrals"],"falsifier":"Take the planar cavity with the three-pole SURMOF material and solve the full nonlinear Maxwell eigenproblem at cavity thicknesses where higher-order modes create the third spectral peak; if the exact complex eigenfrequencies deviate significantly from the effective-Hamiltonian predictions (Eq. 10) in that region, the single-resonant-state assumption, and hence the coupling rates extracted from it, is falsified.","tokens_in":23113,"feed_emoji":"💡","tokens_out":5766,"duration_ms":60089,"temperature":0.7,"pith_summary":"This paper argues that the distinction between weak and strong light-matter coupling in open nanophotonic cavities is carried not by real-frequency spectra but by the motion of the cavity's resonant states — its complex-frequency eigenmodes. As a geometric parameter such as cavity thickness or core radius is swept, the hybridized states pass each other with only weak interaction in the weak-coupling regime, but at the onset of strong coupling they abruptly swap positions in the complex frequency plane, an unambiguous signature tied to an exceptional point. The authors then derive, from Maxwell's equations, an effective Hamiltonian for a single dominant photonic mode coupled to any number of material resonances, with coupling rates given by overlap integrals and with an additional diagonal shift of the bare photonic eigenfrequency that is absent from conventional coupled-oscillator models. They validate the approach on planar and spherical silver cavities filled with a molecular material whose permittivity was extracted from quantum-chemical simulations, showing that individual material resonances can be disentangled even when several overlap spectrally. If correct, this provides a parameter-free, fit-free route to coupling rates and a robust criterion for strong coupling without relying on ambiguous spectral features.","feed_headline":"Resonant states swap places at the onset of strong coupling","feed_subtitle":"Complex-frequency trajectories separate weak from strong coupling without ambiguous spectral fits.","key_machinery":"The central object is the resonant state (quasinormal mode): an eigenmode of the source-free Maxwell equations at a complex frequency whose real part gives the resonance frequency and imaginary part the damping rate. The argument is carried by tracing these eigenfrequencies through the complex plane as a system parameter changes — weak coupling shows passing, strong coupling shows swapping via an exceptional point. The quantitative engine is the single-mode effective Hamiltonian (Eq. 10) built from a resonant-state Green's function approximation: coupling rates are overlap integrals κ_p = sqrt(g_p Ω̃_p), with g_p = ∫ E_c·iσ_p E_c dV, and the bare cavity frequency is shifted by −Σ g_p. This r","core_discovery":"The central discovery is that resonant state trajectories in the complex frequency plane undergo a qualitative change exactly at the onset of strong coupling: instead of passing each other with only perturbative interactions, the two resonant states swap positions, going through an exceptional point where the square-root splitting in the coupled-oscillator discriminant vanishes. Under a single-dominant-photonic-mode assumption, the paper derives an effective Hamiltonian (Eq. 10) from the source-free Maxwell equations, in which the off-diagonal coupling between the photonic mode and the p-th material resonance is κ_p = sqrt(g_p Ω̃_p), with g_p = ∫ E_c · iσ_p E_c dV an overlap integral over th","pith_inferences":["If the trajectory-swap signature is generic beyond the single-mode approximation, it could serve as a model-free experimental test of strong coupling: one would directly retrieve complex eigenfrequencies from time-domain ringdown or scattering-matrix pole measurements and look for the exchange rather than fitting spectra.","The single-mode effective Hamiltonian excludes higher-order cavity modes, yet the paper itself identifies higher-order modes as the origin of a distinctive third spectral peak; extending the Hamiltonian to include several photonic modes would naturally add further diagonal shifts and couplings, possibly explaining the relative intensities of such multi-peak polariton spectra.","The per-resonance criterion κ_p^eff ≳ γ_c/√2 suggests a natural experimental check: for a given cavity, one could compare the predicted observable-strong-coupling threshold for each material resonance with the appearance of resolvable polariton branches in absorption, photoluminescence, and scattering of the same structure.","Because the pole trajectories depend on the analytic continuation of the fitted Lorentz permittivity into the complex plane, the accuracy of the predicted exceptional-point positions could be tested by using independently determined permittivity data in the complex frequency region, rather than only real-axis fits."],"forward_implications":["The complex-plane trajectory-swap signature provides a criterion for strong coupling that does not rely on fitting peaks or dips in real-frequency spectra, avoiding ambiguities such as Fano-interference-induced splittings that mimic strong coupling.","The effective Hamiltonian gives direct, fit-free access to individual coupling rates κ_p from overlap integrals, allowing the contribution of each material resonance to be disentangled even when several resonances overlap and the overall splitting is a superposition.","The predicted diagonal shift −Σ g_p implies that hybridization renormalizes the bare photonic eigenfrequency itself — reading the cavity frequency from the uncoupled spectrum would miss this correction, which is absent in standard coupled-oscillator models.","The inverse eigenproblem enables the extraction of coupling rates from measured or computed complex eigenfrequencies alone, for example from pole positions of the scattering matrix, without requiring detailed field distributions.","Generalizing the observable-strong-coupling criterion to κ_p^eff > sqrt(γ_c^2 + γ_p^2)/√2 per material resonance provides a robust way to judge whether a specific resonance is strongly coupled even when other, weakly coupled resonances are present."],"fun_headline_variants":["Resonant states trade places at strong coupling onset","Complex plane swap marks strong light-matter coupling","Strong coupling: when resonant states exchange spots","Exceptional point signals strong coupling in cavities","Swap of resonances: unambiguous strong-coupling marker"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central results assume both that the material permittivity can be safely analytically continued into the complex plane through its fitted Lorentz poles and, for the coupling-rate extraction, that a single bare cavity mode dominates the Green's function; if either fails, the predicted trajectories or the extracted rates lose their grounding.","fun_headline_variants_meta":{"raw":{"variants":["Resonant states trade places at strong coupling onset","Complex plane swap marks strong light-matter coupling","Strong coupling: when resonant states exchange spots","Exceptional point signals strong coupling in cavities","Swap of resonances: unambiguous strong-coupling marker"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1492,"prompt_tokens":742,"completion_tokens":750,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":679}},"tokens_in":486,"tokens_out":750,"duration_ms":8251,"temperature":1.0,"reasoning_tokens":679,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:57:26.906101+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the planar cavity with the three-pole SURMOF material and solve the full nonlinear Maxwell eigenproblem at cavity thicknesses where higher-order modes create the third spectral peak; if the exact complex eigenfrequencies deviate significantly from the effective-Hamiltonian predictions (Eq. 10) in that region, the single-resonant-state assumption, and hence the coupling rates extracted from it, is falsified.","supporting_citations":[],"review_version":1}