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arxiv: 2604.12605 · v1 · submitted 2026-04-14 · ❄️ cond-mat.mes-hall · physics.optics· quant-ph

Quantum dynamics of coupled quasinormal modes and quantum emitters interacting via finite-delay propagating photons

Robert Meiners Fuchs , Juanjuan Ren , Sebastian Franke , Stephen Hughes , Marten Richter This is my paper

Pith reviewed 2026-05-10 15:41 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.opticsquant-ph
keywords quasinormal modesretarded interactionsquantum emittersphoton bathopen quantum systemscavity quantum electrodynamics
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The pith

Retarded interactions between distant lossy cavities and quantum emitters reduce exactly to system-bath correlation functions.

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

The paper develops a time-dependent theory for how spatially separated lossy cavities interact through photons that take finite time to travel between them. It demonstrates that these delayed dynamics are fully captured by correlation functions between the cavity quasinormal modes and the photon bath, with emission from one cavity serving as the input field to another. Quantum emitters placed inside cavities or in the surrounding medium couple to both the modes and the bath, producing two distinct channels of interaction. A reader would care because the approach supplies a practical, reduced description for quantum dynamics in open nanophotonic systems without needing to propagate the full electromagnetic field explicitly.

Core claim

The retarded inter-cavity dynamics are fully described by system-bath correlation functions in which the emission from one cavity appears as the input field for another. The bath of traveling photons is represented by non-bosonic operators that remain orthogonal to the open-cavity quasinormal modes. When quantum emitters are included, both bath-mediated and quasinormal-mode-mediated interactions between emitters arise naturally within the same framework.

What carries the argument

System-bath correlation functions that encode the finite propagation delay between quantized quasinormal modes of separate cavities.

If this is right

  • Emission from one cavity acts as the exact driving field for a second cavity after the light-travel time.
  • Emitter interactions occur through two parallel channels: direct bath mediation and indirect quasinormal-mode mediation.
  • The same correlation-function structure applies to arbitrary cavity shapes and separations in any homogeneous medium.
  • Time-dependent quantum dynamics can be simulated without discretizing the entire propagation path between cavities.

Where Pith is reading between the lines

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

  • The method could lower the computational cost of simulating extended quantum photonic networks by eliminating explicit propagation degrees of freedom.
  • The correlation-function description may map onto existing input-output formalisms once finite delays are retained.
  • Natural extensions include adding nonlinear cavity or emitter responses while preserving the same bath structure.
  • Experimental signatures in nanophotonic circuits would appear as precise time-delayed correlations between detected photons from different cavities.

Load-bearing premise

The traveling photon bath can be modeled with operators that are orthogonal to the quasinormal modes of the open cavities.

What would settle it

A direct calculation or measurement of the time-dependent field correlation between two cavities that deviates from the predicted system-bath correlation functions would falsify the completeness of the description.

Figures

Figures reproduced from arXiv: 2604.12605 by Juanjuan Ren, Marten Richter, Robert Meiners Fuchs, Sebastian Franke, Stephen Hughes.

Figure 1
Figure 1. Figure 1: Schemes of the different contributions to the Hamil [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sketch of the area of direct influence for the in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: We obtain the retarded overlap matrix Siµ←jη from Eq. (41) by eliminating the propagation from the inter-cavity overlap integral to obtain an effective, frequency-independent overlap. QNM-QNM coupling, we formulate the correlation us￾ing effective coupling elements (see Appendix B for the derivation), χ (−) iµ←jν = X µ′ν′  S −1/2  iµiµ′ i(˜ωiµ′ − ω˜ ∗ jν′ )Siµ′←jν′  S −1/2  jν′ jν , (40) with the retar… view at source ↗
Figure 5
Figure 5. Figure 5: QNM correlation function for two coupled metal dimers 1 and 2 (cf. Fig. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Bath-mediated coupling between the QNM and [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Bath-mediated coupling between the QNM and [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Diagonal terms of TLS correlation function from [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Off-diagonal terms of TLS correlation function from [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Schematic for coupled dimers. r1 = 10 nm, L1 = 80 nm, d1 = 10 nm. r2 = 10 nm, L2 = 90 nm, d2 = 20 nm. The surface-to-surface gap distance between two dimers is dgap. Schematic for (b) dimer 1 alone and (c) dimer 2 alone [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) Red (blue) solid curve and red (blue) squares show the Purcell factors at the gap center of the dimer 1 (dimer [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Schematic diagram of integral over near field surface [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
read the original abstract

A time-dependent theory for the interactions between spatially separated lossy cavities in a homogeneous background medium using quantized quasinormal modes (QNMs) is presented. The cavities interact via a bath of traveling photons, described by non-bosonic operators that are orthogonal to the open-cavity QNMs. The retarded (i.e., time-delayed) inter-cavity dynamics are fully described by system-bath correlation functions, in which the emission from one cavity appears as the input field for another. Coupling between quantum emitters (described as two-level systems), placed inside a cavity or embedded in an external medium, and the electromagnetic field (cavity modes and bath photons) is included in the theory, which gives rise to both bath-mediated and QNM-mediated interactions between the emitters.

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 develops a time-dependent quantum theory for interactions between spatially separated lossy cavities using quantized quasinormal modes (QNMs). Cavities couple through a bath of traveling photons modeled by non-bosonic operators orthogonal to the open-cavity QNMs. Retarded inter-cavity dynamics are encoded solely in system-bath correlation functions, with emission from one cavity acting as input to another. Coupling to quantum emitters (two-level systems) inside cavities or in the medium is included, producing both bath-mediated and QNM-mediated emitter interactions.

Significance. If the non-bosonic bath algebra is shown to be consistent, the framework would offer a useful extension of QNM quantization to multi-cavity systems with explicit finite-delay propagation. It could enable more accurate modeling of retardation effects in nanophotonic quantum optics without common approximations, with direct relevance to hybrid systems involving emitters.

major comments (2)
  1. [§3.1] §3.1, Eq. (10): the orthogonality projection defining the non-bosonic bath operators is used to remove direct QNM-bath overlap so that all retarded dynamics enter via correlations, but the resulting commutation relations are not explicitly derived or verified to preserve standard input-output relations and vacuum fluctuations once emitters are coupled; this is load-bearing for the central claim that correlations alone fully capture the dynamics.
  2. [§4.2] §4.2: the separation of emitter interactions into QNM-mediated and bath-mediated channels assumes the bath model introduces no residual terms, yet without shown consistency of the non-bosonic algebra this separation risks being incomplete when finite delays and emitters are both present.
minor comments (2)
  1. [Introduction] The introduction would benefit from a clearer contrast with prior bosonic input-output treatments of retardation to highlight the specific role of the non-bosonic construction.
  2. [Figure 1] Figure captions for the schematic of the multi-cavity setup could explicitly label the delay time τ to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation of the non-bosonic bath algebra. We address each major comment below and have revised the manuscript accordingly to provide the requested explicit derivations and consistency checks.

read point-by-point responses

  1. Referee: [§3.1] §3.1, Eq. (10): the orthogonality projection defining the non-bosonic bath operators is used to remove direct QNM-bath overlap so that all retarded dynamics enter via correlations, but the resulting commutation relations are not explicitly derived or verified to preserve standard input-output relations and vacuum fluctuations once emitters are coupled; this is load-bearing for the central claim that correlations alone fully capture the dynamics.

    Authors: We agree that an explicit derivation and verification of the commutation relations is necessary to substantiate the central claim. The orthogonality projection in Eq. (10) is introduced precisely to eliminate direct QNM-bath overlap, ensuring that all inter-cavity retardation appears through the system-bath correlation functions. In the revised manuscript we have added Appendix C, which derives the commutation relations for the projected non-bosonic bath operators and shows that they satisfy the required algebra while remaining orthogonal to the QNMs. We further verify that the standard input-output relations and vacuum fluctuations are preserved when the emitters are included, because the projection removes any direct overlap terms and the bath is taken in the vacuum state; the resulting Heisenberg-Langevin equations therefore contain no additional residual commutators beyond those already encoded in the correlation functions. revision: yes

  2. Referee: [§4.2] §4.2: the separation of emitter interactions into QNM-mediated and bath-mediated channels assumes the bath model introduces no residual terms, yet without shown consistency of the non-bosonic algebra this separation risks being incomplete when finite delays and emitters are both present.

    Authors: The decomposition into QNM-mediated and bath-mediated channels follows directly from the orthogonality condition of Eq. (10), which partitions the total electromagnetic field without overlap. We acknowledge that the consistency of this separation under simultaneous finite delays and emitter coupling requires explicit confirmation. The revised manuscript now includes a short verification subsection in §4.2 together with the new Appendix C. There we substitute the projected bath operators into the emitter equations of motion and confirm that no residual terms appear; the finite-delay propagation remains fully contained in the bath correlation functions, and the non-bosonic algebra is shown to be consistent with the separation for the parameter regimes considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via modeling assumptions

full rationale

The paper constructs a time-dependent theory for coupled QNMs interacting through a bath of traveling photons using non-bosonic operators defined to be orthogonal to the open-cavity QNMs. The central claim—that retarded inter-cavity dynamics are fully captured by system-bath correlation functions—is presented as following directly from this orthogonality choice and the input-output mapping, without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations that presuppose the target result. No equations or steps in the abstract or description exhibit a prediction that is equivalent to its inputs by construction; the orthogonality is an explicit modeling assumption rather than a derived or renamed known result. The theory incorporates quantum emitters and both bath- and QNM-mediated interactions as extensions, remaining independent of any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; detailed free parameters, axioms, and entities cannot be extracted without full text.

axioms (2)
  • domain assumption Quantized quasinormal modes provide a valid basis for describing lossy open cavities.
    Standard assumption in nanophotonics and quantum optics.
  • domain assumption Photon bath operators are non-bosonic and orthogonal to cavity QNMs.
    Invoked to separate cavity and bath degrees of freedom.

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Reference graph

Works this paper leans on

104 extracted references · 104 canonical work pages · cited by 1 Pith paper

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    II, we discussed how the instantaneous (or di- rect) coupling between different QNM cavities is sup- pressed for cases with sufficient separation

    General form In Sec. II, we discussed how the instantaneous (or di- rect) coupling between different QNM cavities is sup- pressed for cases with sufficient separation. As a result, the QNM Hamiltonian [Eq. (17)] contains no direct cou- pling between QNMs of separate cavitiesi̸=j. In a time- dependent theory, propagating photons transmit energy between the...

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    (37) al- lows for a perturbative treatment of the intercavity scat- tering

    Perturbative intercavity coupling The form of the correlation function from Eq. (37) al- lows for a perturbative treatment of the intercavity scat- tering. Since instantaneous intercavity transfer is negligi- ble for well-separated cavities (cf. Sec. II C), only a finite number of intercavity scattering processes contribute sig- nificantly within a finite...

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    Coupled metal dimers To illustrate the calculation of the correlation function, we consider an example of two metal dimers in vacuum (nB = 1) serving as QNM cavities with one dominant QNM each (cf. Fig. 2). We consider the caseN≤1 as discussed above. In Fig. 5, we show the absolute value and real part of the QNM-QNM correlation function from Eq. (39) for ...

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    da ·E jν(ra, t′ −t 1) iℏ #∗ −Θ(t ′−t) X j,ην ∞X N=0 (δηµ∂t+iχ∗ iηiµ) Z t′ t dt1 h KN jν iη(t1 −t) i∗ ×

    General form A TLS that transitions from the excited state to the ground state emits a photon into the surrounding medium, described in Eq. (17) by the coupling to the bath modes. The resulting bath photon can propa- gate through space and excite another system (such as a QNM cavity) elsewhere, leading to an effective interac- tion. In contrast to the QNM...

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    Perturbative treatment of intercavity transfer For cavities with significant separation (P ij ≫0, c.f. Sec. II C), we treat the intercavity scattering pertur- batively and restrict the number of intercavity scattering processes toN≤1. Note, however, that the QNM-TLS couplingd a ·E jν(ra, t′ −t 1)/(iℏ) may also contain inter- cavity scattering terms if the...

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    We place a TLS in the gap of each dimer (see Fig

    TLS coupled to a metal dimer We again use metal dimers as QNM cavities. We place a TLS in the gap of each dimer (see Fig. 2). The TLS dipoles are polarized along thez-axis, i.e., along the sym- metry axis of the cylindrical dimers. The dipoles are also assumed to be identical withd 1 =d 2 =d ˆnz, where ˆnz is the unit vector in thezdirection andd= 1 e·nm ...

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    da ·E iµ(ra, t−t 1) iℏ (δην ∂t2 −iχjηjν)KN iµjν(t1 −t 2) #

    General form The photon emitted by a TLS can also excite another TLS (or re-excite the same TLS), leading to an effective TLS-TLS interaction mediated by propagating bath pho- tons. Using the Hamiltonian for the TLS-bath coupling from Eq. (46) and the results derived in the previous sec- tions, we derive the TLS-TLS correlation function (see Appendix C 3 ...

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    TLS coupling in the presence of metal dimer QNM cavities We again consider the case of two metal dimers serving as QNM cavities with TLSs in the dimer gaps from Fig. 2. We also takeN≤1 in Eq. (50) under the assumption of weak intercavity coupling, as in Eq. (49). For the full Green’s function, we use the expansion from Appendix D. In Fig. 8, we show the d...

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    Derivation of the QNM correlation function With the QNM-bath coupling Hamiltonian from Eq. (36) together with the initial scattering ˆCiµ(t) from Eq. (31), the correlation function from Eq. (25) becomes CQNM iµjν (t−t ′) = ∞X N=0 ∞X M=0 X k,l X η,κ X pq Z t t0 dt1 Z t′ t0 dt2KN iµkη(t−t 1) × Z ∞ 0 Z d3rgkη,p(r, ω)e−iω(t1−t0) ×tr B ˆcp(r, ω, t0)ˆc† q(r′, ω...

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    (36) and the Hamiltonian for TLS-bath coupling from Eq

    Derivation of the QNM-TLS correlation function Using the Hamiltonian of QNM-bath coupling from Eq. (36) and the Hamiltonian for TLS-bath coupling from Eq. (46), the correlation function for the bath-mediated coupling between a QNM and TLS reads [cf. Eq. (25)] CQ−T iµa (t−t ′) = X j,ν ∞X N=0 Z t t0 dt1KN iµjν(t−t 1) × X p Z d3r Z ∞ 0 dωg∗ a,p(r, ω)eiω(t′−t...

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    Derivation of the TLS correlation function Using the Hamiltonian of the TLS-bath interaction from Eq. (46), the correlation function for the bath- mediated interaction between two TLSsaandbreads, CTLS ab (t−t ′) = X pq Z d3r Z ∞ 0 dωga,p(r, ω)e−iω(t−t0) ×tr B ˆcp(r, ω, t0)ˆc† q(r′, ω′, t0)ρB × Z d3r′ Z ∞ 0 dω′g∗ b,q(r′, ω′)eiω′(t′−t0) + X i,µ Z t t0 dt1 d...

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