Hidden nonlinear optical susceptibilities in linear polaritonic spectra

Arghadip Koner; Joel Yuen-Zhou

REVIEW 3 major objections 7 minor 49 references

Hidden nonlinear optical susceptibilities in linear polaritonic spectra

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Pith's one-line read Molecular polariton linear spectra carry a hierarchy of finite-size 1/N quantum corrections from vacuum-mediated Raman processes, visible when cavity loss is comparable to the single-molecule coupling.

desk verdict A credible 1/N expansion showing vacuum-mediated Raman sidebands as finite-size corrections to polaritonic linear spectra, but the printed continued fraction has a dagger-ordering typo and the central formula lacks a numerical check. read the letter →

arxiv 2411.09039 v2 pith:3UO3I6SY submitted 2024-11-13 quant-ph cond-mat.otherphysics.chem-phphysics.optics

Arghadip Koner , Joel Yuen-Zhou This is my paper · ORCID

classification quant-phcond-mat.otherphysics.chem-phphysics.optics

keywords molecularpolaritonsstronglight-mattercoupling1/Nexpansionvacuum-mediatedRamanscatteringlinearresponsephotonGreen'sfunctionhigh-Qcavitiesnonlinearsusceptibilities

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

Molecular polaritons — N molecules strongly coupled to one cavity photon mode — have linear spectra that are usually captured by classical linear optics. This paper argues that this classical description is only the thermodynamic-limit term of an exact 1/N expansion, and that the first quantum correction contains vacuum-mediated molecular Raman processes that produce weak sidebands around every polariton peak. Those sidebands are hidden in ordinary low-Q cavities because the cavity linewidth κ vastly exceeds the single-molecule coupling λ, but they should become visible in high-Q single-mode cavities with κ ≈ λ. If true, this explains where the expected nonlinear many-body effects of strong coupling live in linear spectroscopy and gives a concrete experimental regime in which to observe them.

What carries the argument

The central object is the block-tridiagonal representation of the single-excitation Hamiltonian, a nearest-neighbor chain of photonic and molecular blocks that the paper calls the CUT-E diagram. This chain separates fast collective couplings $V_n\propto\lambda\sqrt{N}$ — Rayleigh processes that conserve the number of ground-state phonons — from slow single-molecule couplings $v_n\propto\lambda$ — Raman processes that create or destroy one vibrational quantum. Matrix continued-fraction techniques applied to this chain produce the photon Green's function, and the timescale separation organizes the $1/N$ expansion that isolates the hidden sidebands.

What would settle it

Measure the linear absorption or transmission of a high-Q single-mode microcavity containing a known number N of identical molecules at low temperature, with κ ≈ λ and the collective coupling λ√N kept fixed while N is varied. The paper predicts sidebands offset from each zeroth-order polariton by the ground-state vibrational frequency, with peak height ∝1/N and blue shift Δ = 2λ(√N − √(N−1)). If those sidebands are absent, do not scale with N, or persist unchanged in a multimode cavity, the central claim fails.

Watch

Extended reading notes

Core claim

The paper derives the exact photon Green's function $D^R_N(\omega)$ for a single-mode cavity coupled to N vibronic molecules as a continued fraction (Eq. 6) and expands it in powers of $1/N$. The zeroth-order term $d_{N,0}$ reproduces classical linear optics: the self-energy is just the molecular linear susceptibility $\chi^{(1)}$, and in the limit $N\to\infty$ nothing else survives. The first-order term $d_{N,1}$ (Eq. 8) is the new claim: it factorizes into $d_{N,0}\cdot V_0G_{e,0}v_0\bigl(\omega-H_{\mathrm{ph},1}+i\kappa/2-V_1G_{e,1}V_1^\dagger\bigr)^{-1} v_0^\dagger G_{e,0} V_0^\dagger\cdot d_{N,0}$, which describes a Stokes Raman transition, propagation through first-order polaritons formed by the remaining $N-1$ molecules, and an anti-Stokes transition back. These vacuum-mediated processes are each penalized by $1/\sqrt{N}$, so the sidebands have height $\propto 1/N$ and blue shift $\Delta = 2\lambda(\sqrt{N}-\sqrt{N-1})$. The same continued fraction is rewritten (Eq. 9) as the photon self-energy equal to a sum of irreducible Rayleigh and Raman nonlinear susceptibilities of the molecular ensemble, making the hidden nonlinear character of the linear spectrum explicit.

Load-bearing premise

The prediction that the sidebands are visible assumes a perfectly identical set of molecules at zero temperature with only simple Markovian loss and a single cavity mode; if static disorder, thermal vibrations, or extra cavity modes blur the resonances, the 1/N features could be washed out.

Editorial extensions

If this is right

In the thermodynamic limit ($N\to\infty$) the polariton linear response becomes exactly the classical linear-optics result built from the molecular linear susceptibility, so no nonlinearity is visible.
In low-$Q$ cavities ($\kappa\gg\lambda$) the $1/N$ terms are suppressed both by the timescale separation and by limited spectral resolution, which is why transfer-matrix and effective-medium models work so well.
In high-$Q$ single-mode cavities with $\kappa\sim\lambda$, each zeroth-order polariton acquires sidebands shifted by the ground-state vibrational frequency $\omega_v$; their height scales as $1/N$ and their Rabi splitting is $2\lambda\sqrt{N-1}$.
The photon self-energy can be written as a sum over irreducible Rayleigh and Raman nonlinear susceptibilities, so a linear measurement in this regime directly carries nonlinear molecular response information.
For two molecular species, the $O(N^{-2})$ terms produce sidebands at $2\omega_{v,A}$, $2\omega_{v,B}$, and $\omega_{v,A}+\omega_{v,B}$, revealing collective Raman processes involving different molecules.

Reading between the lines

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

If the sidebands survive at the predicted $1/N$ height, a careful measurement of sideband intensity versus $N$ would give a direct estimate of the single-molecule coupling $\lambda$, which is otherwise hard to isolate in the collective strong-coupling regime.
The two-species sum-frequency sideband at $\omega_{v,A}+\omega_{v,B}$ suggests using linear transmission as a probe of cooperative Raman events between distinct molecular species, a measurement that does not require a pulsed nonlinear experiment.
The paper's own remark that multimode cavities blur the sidebands implies the observable doubles as a diagnostic: the presence of sharp sidebands certifies effective single-mode behavior, while their absence could indicate multimode or disordered broadening.
Because the first-order sidebands are genuine Raman transitions driven by vacuum fluctuations, one might expect weak nonclassical photon statistics or entanglement in the transmitted field at those frequencies; the paper leaves this as future work.

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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

The headline: this paper likely contains a real and interesting result—the linear response of molecular polaritons has finite-size 1/N corrections that show up as vacuum-mediated Raman sidebands, becoming visible when the cavity linewidth κ is comparable to the single-molecule coupling λ. That would explain why classical optics works in low-Q cavities while still leaving room for nonlinear effects at strong coupling.

What's new: the 1/N expansion of the exact polaritonic Green's function, the identification of the O(1/N) term as a sequential Stokes-anti-Stokes Raman process mediated by the vacuum field, and the diagrammatic classification in terms of irreducible nonlinear susceptibilities. The derivation of dN,0 and dN,1 from the continued fraction is clean, and the thermodynamic limit correctly reduces to the standard linear susceptibility—a good consistency check. The physical picture of the timescale separation (collective O(λ√N) Rayleigh vs single-molecule O(λ) Raman transitions) is persuasive.

The soft spots are real but addressable. First, Eq. (6) as printed has a dagger-ordering inconsistency: it writes v†0 (...) v0 inside the denominator of H_e,0, but the block structure of the Hamiltonian requires v0 (...) v†0. The supplemental derivation in Sec. S6b silently uses the correct ordering when it derives Eq. (8), so Eq. (8) is probably right, but it does not follow from the displayed Eq. (6). That typo needs fixing before the derivation is reproducible from the text. Second, the proof that the self-energy equals the sum of irreducible nonlinear susceptibilities (Eq. 9) is sketched rather than rigorous, with a "reader can check" signpost. Third, there is no numerical check of Eq. (8) against the exact continued fraction, even for a few molecules. Since the whole prediction of observable sidebands rests on that formula, an independent numerical comparison is the most important missing piece. Finally, the text contradicts itself about whether closed-form expressions exist for k ≥ 2, and the observable prediction assumes zero temperature, no disorder, and a single cavity mode; the authors acknowledge multimode blurring but do not quantify the suppression.

Bottom line: this is a solid theoretical letter with a novel, credible claim that deserves a serious referee. I would ask for a corrected Eq. (6), a numerical verification of Eq. (8), and a clearer statement about higher-order terms. With those, it would be a nice contribution for the molecular polaritonics community.

Referee Report

3 major / 7 minor

Summary. This manuscript asks why molecular polaritons' linear spectra are so well described by classical linear optics, despite the expected nonlinear character of strong coupling. Using a bosonic (Holstein–Tavis–Cummings) model restricted to the first excitation manifold, the authors express the photon Green's function as a matrix continued fraction (Eq. 6) over the block-tridiagonal 'CUT-E' chain, in which V-type couplings are collective (∝ λ√N) and v-type couplings are single-molecule Raman processes (∝ λ). They then develop a 1/N expansion of the Green's function. The zeroth order reproduces the classical linear-optics formula in terms of the ensemble linear susceptibility; the first-order term, dN,1 (Eq. 8), is interpreted as a vacuum-mediated Stokes–anti-Stokes process and predicts Raman sidebands at ±ωv around the polaritons, with height ~1/N and reduced first-order Rabi splitting 2λ√(N−1). The authors argue these sidebands are washed out in low-Q and multimode cavities but should be observable in high-Q single-mode cavities with κ ~ λ. They also classify the Dyson diagrams into reducible and irreducible contributions and claim (Eq. 9) that the photon self-energy equals the sum of irreducible odd-order nonlinear susceptibilities of the molecular ensemble, with O(N^{-2}) terms generating two-molecule Raman sidebands.

Significance. The central claim is significant and cleanly falsifiable: linear spectroscopy of high-Q single-mode molecular polaritons should reveal finite-size Raman sidebands with a specific N-scaling, providing a molecule-specific signature of nonlinear response mediated by vacuum fluctuations. If Eq. (8) holds, the paper resolves an active conceptual puzzle (polaritons as 'optical filters') by showing where the nonlinearity goes in the N→∞ limit. Strengths: the derivation is self-contained and parameter-free; the N-counting is consistent throughout; and the reduction to classical optics in the thermodynamic limit is a genuine consistency check rather than a circular assumption. The weaknesses are local but load-bearing: (i) the printed continued fraction (Eq. 6 and SI Eqs. S18, S22) uses a dagger ordering opposite to that of Eq. (5b), so Eq. (8) does not follow from the displayed equations as written; (ii) no numerical comparison of the truncated 1/N expansion with the exact Green's function is provided; and (iii) the proof that the self-energy equals the sum of irreducible susceptibilities (Eq. 9) is delegated to the reader. These issues are repairable within the scope of the manuscript.

major comments (3)

[Eq. (6); SI Eqs. (S18), (S22)] The continued fraction printed in Eq. (6), and repeated in SI Eqs. (S18) and (S22), is not the iteration of the recursion in Eq. (5b). Since v0 is the (He,0; Hph,1) block in Eq. (S3), the self-energy on He,0 from Hph,1 is v0(ω − Hph,1 + ...)^{-1}v†0, and the self-energy on Hph,1 from He,1 is V1(ω − He,1 + ...)^{-1}V†1; Eq. (5b) states exactly this. Eq. (6) instead writes v†0(...)v0 and V†1(...)V1 in those denominators. Because Eq. (8) and the SI's Dyson terms (e.g., V0Ge,0v0Gph,1v†0Ge,0V†0 in the fourth-order table of Sec. S5) use the opposite ordering, Eq. (8) cannot be obtained from Eq. (6) as printed; the conversion in Sec. S6b silently switches to the correct ordering when it introduces X = v0Gph,1(I − V1Ge,1V†1Gph,1)^{-1}v†0Ge,0. The ordering is immaterial in the fully scalar 3-level model used in Fig. 2, which may explain the oversight, but the manuscript claims validity for arbitrary vibronic manifolds in which the blocks are non-commuting matrices. Please correct the dagger placement in Eq. (6), Eq. (S18), and Eq. (S22), or state the alternative convention explicitly and make Eq. (5b), the diagram rules, and Eq. (8) consistent with it.
[Figs. 2a–2b; SI Sec. S6] No comparison is made between the 1/N-truncated Green's function and the exact DR_N. The exact result is readily available either by numerically solving the finite tridiagonal system defined by Eq. (5) (or the continued fraction once its ordering is fixed) or by exact diagonalization of Eq. (2) for small N. Such a comparison would serve two purposes: it would independently confirm that Eq. (8) is indeed the O(1/N) term of the exact photon Green's function — essential given the ordering inconsistency in Eq. (6) — and it would demonstrate that the 1/N expansion actually converges for the N values plotted in Fig. 2. I recommend adding a panel overlaying the exact and truncated absorption spectra for a few values of N (e.g., N=5–50) at fixed λ√N, including an error estimate for the truncated series.
[Eq. (9); SI Sec. S5A, Eqs. (S18)–(S19)] The paper's title-level claim, that linear polaritonic spectra are governed by hidden nonlinear molecular susceptibilities, is carried by Eq. (9). The supporting argument equates the continued fraction (S18) with the nested matrix-geometric series (S19), but the SI only says that 'this result can be explicitly checked by the reader'; no induction, convergence statement, or justification of the order of non-commuting factors in the nested powers of Eq. (S19) is given. Furthermore, with Eq. (S18) exactly as printed, the two expressions are not equal as operator identities in the general multi-level case (see major comment 1 on the dagger ordering), so the equality cannot simply be taken on trust. I ask the authors to replace the sketch with a direct derivation of Eq. (9), e.g., an induction on the truncation level of the continued fraction, together with a statement of the sense in which the infinite sums converge.

minor comments (7)

[SI Sec. S6c] In the opening sentence of Sec. S6c, 'To obtain dN,2(ω) ∝ N^{−k}' should read 'dN,2(ω) ∝ N^{−2}'.
[Fig. 2 caption] The caption to Fig. 2 does not identify which colors in panel (b) correspond to the zeroth- and first-order spectra, and it is unclear whether the blue curve shows the O(N^{-2}) increment alone or the sum of all included orders; please clarify.
[Main text, '1/N expansion'] The statement that 'we cannot provide an expression for dN,k when k ≥ 2' is stronger than what the Supplemental Material achieves, since Sec. S6c gives d^{(1)}_{N,2} in closed form; the wording should be changed to 'no complete closed-form expression for dN,k is obtained for k ≥ 2'.
[Experimental considerations] The observability prediction assumes a homogeneous ensemble at zero temperature with Markovian losses and a single cavity mode; the text acknowledges the multimode-cavity blurring but not the possible suppression of the 1/N sidebands by static disorder or finite-temperature phonon populations. A brief quantitative estimate (or an explicit statement that these effects are beyond the present scope) would make the experimental claim more complete.
[SI Eqs. (S17a)–(S17c)] The index placement in Eqs. (S17) is inconsistent (e.g., µ_{0g}^{me} versus µ_{me 0g}), which makes the susceptibility expressions difficult to parse; a uniform tensor notation consistent with Eq. (S.2) is recommended.
[SI Sec. S5] The equivalence of the CUT-E diagram rules with the ket-only Dyson diagrams is asserted with the parenthetical promise of an induction proof that is not given; please either supply the induction or state the induction hypothesis explicitly.
[Introduction / Ref. [23]] Since the block-tridiagonal (CUT-E) structure and the 1/N expansion were introduced in Ref. [23] by the same group, the authors should state explicitly which elements (the linear-spectroscopy application, the Raman-sideband prediction, and the susceptibility identification) are new in the present work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 1/N corrections are derived from the stated Hamiltonian via standard Green's-function algebra, not fitted or defined into existence.

full rationale

The paper's central result, Eq. (8) for dN,1, is obtained in Supplemental Sec. S6b by truncating the continued-fraction self-energy (Eq. S18) and expanding the resulting matrix geometric series; this is a derivation, not a restatement of the model. dN,0 reduces to the input linear susceptibility in the thermodynamic limit, which is a consistency check rather than a circular input. The self-citations to the authors' CUT-E work supply the block-tridiagonal representation and timescale language, but the recurrence relations (Eqs. 5a/5b) and the scaling Vn proportional to lambda*sqrt(N) and vn proportional to lambda are stated and used directly, so the citation is not load-bearing. Eq. (9)'s sum over 'irreducible' susceptibilities is an identification/rewriting of the self-energy expansion, and the Raman-sideband prediction rests on Eq. (8), not on that rewriting. No parameters are fitted to the predicted spectra. The printed dagger ordering in Eq. (6)/S18 is inconsistent with the Schur-complement derivation and with the ordering used in S6b; that is a correctness/reproducibility concern, not circularity, and does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

No free parameters fitted to data; the model inputs are lambda, kappa, gamma, omega_v, omega_ph and N. The central assumptions are the bosonic mapping, the block tridiagonal structure from the CUT-E framework, the single-mode Markovian input-output description, and the zero-temperature initial state. One conceptual entity is introduced: the classification of irreducible nonlinear susceptibilities.

assumptions (5)

domain assumption The molecular ensemble can be described by a bosonic Hamiltonian (Eq. 1) derived from the Holstein-Tavis-Cummings model under the assumption of a permutationally invariant initial state at zero temperature.
Invoked in the Model section and SI S1; this mapping is established in prior work (Refs 17-22) and assumes bosonic statistics for molecular vibronic excitations.
domain assumption The Hamiltonian in the first excitation manifold has a block tridiagonal form (Eq. 2) with Vn scaling as lambda*sqrt(N) and vn scaling as lambda.
Structural assumption from Ref 23 (CUT-E); underlies the timescale separation and all subsequent 1/N counting.
domain assumption Linear response observables are expressed via the retarded photon Green's function with input-output relations using Markovian loss rates kappa (cavity) and gamma (molecular dephasing).
Standard input-output theory; stated in the Model section and SI S2, following Ref 16.
domain assumption The initial state at zero temperature has no photons and all molecules in the ground vibronic state.
Ensures only ket-only diagrams contribute; stated in SI S2.
standard math The matrix continued fraction expansion and Schur complement formulas (SI S3) are valid for the non-Hermitian effective Hamiltonians with complex loss terms.
Standard linear algebra; used without proof.

invented entities (1)

Irreducible nonlinear susceptibilities of the N-molecule ensemble
purpose: To represent the cavity self-energy as a geometric sum over nonlinear response functions (Eq. 9) and to classify Feynman diagrams by whether they revisit the photonic block Hph,0.
A new classification construct introduced in the Feynman pathways section; no direct experimental observable outside the paper, though the underlying susceptibilities are physical.

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Pith. "Pith review of Hidden nonlinear optical susceptibilities in linear polaritonic spectra." pith.science (2026). https://pith.science/paper/3UO3I6SY

@misc{pith2026241109039,
  author       = {Pith},
  title        = {Pith review of: Hidden nonlinear optical susceptibilities in linear polaritonic spectra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3UO3I6SY}},
  note         = {Machine review of arXiv:2411.09039}
}

abstract

Linear spectra of molecular polaritons formed by $N$ molecules coupled to a microcavity photon mode are usually well described by classical linear optics, raising the question of where the expected nonlinear effects in these strongly coupled systems are. In this work, we derive a general expression for the polaritonic linear spectra that reveal previously overlooked finite-size quantum corrections due to vacuum-mediated molecular Raman processes. Using a $1/N$ expansion, we demonstrate that these nonlinearities are suppressed in typical low-Q cavities due to an emergent timescale separation in polariton dynamics yet manifest in high-Q single-mode cavities where the photon loss is comparable to the single-molecule light-matter coupling strength.

Figures

Figures reproduced from arXiv: 2411.09039 by the authors.

**Figure 2.** FIG. 2. (a) Linear absorption spectrum of an ensemble of 3-level systems (two ground-state vibrational levels and one excited [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Fetter and J

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Y. A. Garc´ ıa Jomaso, B. Vargas, D. L. Dom´ ınguez, R. J. Armenta-Rico, H. E. Sauceda, C. L. Ordo˜ nez-Romero, H. A. Lara-Garc´ ıa, A. Camacho-Guardian, and G. Pir- ruccio, “Intercavity polariton slows down dynamics in strongly coupled cavities,” Nature Communications 15, 2915 (2024)

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F. Baboux, L. Ge, T. Jacqmin, M. Biondi, E. Galopin, A. Lemaˆ ıtre, L. Le Gratiet, I. Sagnes, S. Schmidt, H. E. T¨ ureci, A. Amo, and J. Bloch, “Bosonic condensation and disorder-induced localization in a flat band,” Phys. Rev. Lett. 116, 066402 (2016). Hidden nonlinear optica...

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The (2m)th order term in DR N(ω) consists of all the 2 m step paths in the CUT-E diagram starting from and returning to Hph,0; the order here is calculated with respect to the light-matter interaction, V ,
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It can be seen (and proven explicitly using mathematical induction) that the CUT-E diagram with the aforementioned 3 a b c d e Figure S1: The ket-only DS-FDs for the a

Every time we jump from one box to another in the CUT-E diagram, we encounter a Raman process mediated by a v(†) k which are penalized by a factor of 1 / √ N. It can be seen (and proven explicitly using mathematical induction) that the CUT-E diagram with the aforementioned 3 a...
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We have DR,(0)(ω) = Gph,0(ω) = 1 ω−ωph+iκ/2

Zeroth order term: 0 step paths. We have DR,(0)(ω) = Gph,0(ω) = 1 ω−ωph+iκ/2
[45]

We have just one of them: Hph,0 1 ← → 2 He,0 (the numbers on the top (bottom) represent the counting of the step forward (backward))

Second order term : We want to count the 2 step paths. We have just one of them: Hph,0 1 ← → 2 He,0 (the numbers on the top (bottom) represent the counting of the step forward (backward)). Thus, second-order photon Green’s function, DR,(2)(ω), is then given as DR,(2)(ω) = Gph,...
[46]

Fourth order term: All the 4 step paths are presented in the table below . The fourth-order photon Green’s function, DR,(4)(ω), is then given as the sum of all the Dyson series terms, DR,(4) N (ω) Pathways in CUT-E diagram Dyson series term Hph,0 1,3 ←→ 2,4 He,0 = 2· (Hph,0↔ H...
[47]

We can have five of them presented in the table below

Sixth order term: We want all the 6-step paths in the diagram. We can have five of them presented in the table below. Note that DR,(6) N (ω) is the sum of all the Dyson series terms. 4 DR,(6) N (ω) Pathways in CUT-E diagram Dyson series term Hph,0 1,3,5 ←−→ 2,4,6 He,0 = 3· (Hp...
[48]

Proceeding this way, we get a recursive algorithm, where the self-energy at the kth step in the matrix depends on the ( k + 1)th step

(S.14) 6 Again Σph,1 = v0 ( ω−Hph,1+iκ/2−Σe,1 )−1 v† 0 is obtained in terms of the self-energy of the next step, Σe,1. Proceeding this way, we get a recursive algorithm, where the self-energy at the kth step in the matrix depends on the ( k + 1)th step. Thus we have, Σe,k = Vk...
[49]

[vN· Gph,N v† N Ge,N−1]nph,N

(S.18) On the other hand, we have the result that the sum over the irreducible nonlinear susceptibilities computed diagram- matically using the CUT-E diagram and the rules, given as a nested summation, ∞∑ l=0 (ωph 2 )l χ(2l+1) N ({ω,ω−ωph}l,ω ) = − ∞∑ ne,0=0 ∞∑ nph,1=0 ··· ∞∑ ...
[50]

Cut-e as a 1/n expansion for multiscale molecular polariton dynamics,

(S.19) This result can be explicitly checked by the reader by isolating the pathways that start and end at Hph,0, and do not involve Hph,0 in between, in other words, they must have V0 and V† 0 only in the left and right, respectively (see Sec. S5). We notice that series const...

2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Energy transfer and correlations in cavity-embedded donor-acceptor configu- rations,

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Polarized room-temperature polariton lasing in elliptical microcav- ities filled with fluorescent proteins,

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Po- lariton condensation in organic semiconductors,

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