Quantization of Polaritons Confined in Dielectric Structures

Amir Rahmani; Andrzej Opala; Dogyun Ko; Maciej Dems; Micha{\l} Matuszewski

REVIEW 4 major objections 4 minor 90 references

Quantization of Polaritons Confined in Dielectric Structures

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For linear polariton systems, classical quasinormal modes of Maxwell's equations give the exact diagonal basis of the quantum Liouvillian, with no fitting parameters.

desk verdict Genuinely new dissipative recipe; central QNM-to-Liouvillian correspondence is asserted rather than proven, and the 'no fitting' claim overstates the calibration. read the letter →

arxiv 2510.14566 v2 pith:2W4AL2EZ submitted 2025-10-16 quant-ph physics.optics

Amir Rahmani , Dogyun Ko , Maciej Dems , Andrzej Opala , Micha{l} Matuszewski This is my paper · ORCID

classification quant-phphysics.optics PACS 42.50.Ct71.36.+c03.65.Yz

keywords polaritonsquasinormalmodesthirdquantizationLindbladmasterequationBogoliubovtransformationquantum-classicalcorrespondenceexcitonconfinementnonlinearinteractions

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 claims that the quantum master equation for polaritons in an arbitrarily shaped dielectric structure can be constructed directly from the classical eigenmodes of Maxwell's equations coupled to matter, with no fitting parameters. In the conservative case, a Bogoliubov transformation diagonalizes the Hamiltonian; in the dissipative case, third quantization shows that classical quasinormal modes—including losses and radiating boundary conditions—are exactly the normal modes of the Liouvillian superoperator. The result is a diagonal Lindblad master equation whose complex mode frequencies supply both the Hamiltonian and the dissipation rates. If correct, this turns classical mode solvers into quantum model generators and enables parameter-free predictions for strong-coupling and many-body polariton phenomena.

What carries the argument

The central object is the diagonalizing basis of superoperator normal modes obtained by third quantization: for the quadratic Liouvillian, the normal-mode superoperators (ζ_r, ζ'_r) diagonalize the dynamics (Eq. 22). The paper establishes a quantum-classical correspondence by defining coherent states of these dissipative modes and showing that field operators expanded in this basis reproduce the classical first-order correlations. The bridge to classical physics is the identification of these superoperator modes with regularized quasinormal modes of the Maxwell-matter system (Eq. 33), including perfectly matched layer regularization. Finally, an isomorphism between the superoperator Fock spa

What would settle it

Compute the full Liouvillian spectrum of a small open system (e.g., two coupled cavity modes with unequal losses) by direct numerical diagonalization of the GKSL generator, and compare with the complex frequencies of the regularized quasinormal modes of the same structure; if the Liouvillian eigenvalues differ from the QNM frequencies, or if off-diagonal coupling between modes is required to reproduce the two-time correlation functions, the central correspondence is false. Equivalently, measure the two-time correlation function in a strongly coupled microcavity and check whether the diagonal m

Watch

Extended reading notes

Core claim

The paper's central claim is that, for a linear light-matter system, the solutions of Maxwell equations coupled to matter polarization form a basis of normal modes of the quantum Liouvillian. Using third quantization, the authors construct coherent states of these dissipative normal modes and verify the correspondence through first-order correlation functions. They then show that the Liouvillian is isomorphic to a 'twin' Liouvillian acting on ordinary bosonic operators, which takes the diagonal GKSL form: effective Hamiltonian from the real parts of the complex frequencies and Lindblad decay/gain from the imaginary parts, with no cross-mode coupling terms. They apply this to interacting (non

Load-bearing premise

The load-bearing premise is the one-to-one correspondence between the 2N normal modes of the Liouvillian and the regularized quasinormal modes of the classical Maxwell-matter system (Sec. III.C), which is asserted via 'we propose' and checked only through equal-time first-order correlation functions, under assumptions of a unique stable nonequilibrium steady state, non-degenerate dynamics, sufficient completeness and normalizability of the PML-regularized QNMs, and—for nonlin

Editorial extensions

If this is right

For any linear dielectric structure, the quantum master equation can be written down after solving the classical eigenmode problem; no parameter fitting is needed.
The diagonal Lindblad form eliminates mode-mixing terms present in earlier quasinormal-mode quantizations, greatly reducing the Hilbert-space size needed for simulating polariton quantum dynamics.
The polariton basis obtained classically remains a convenient basis for nonlinear (interacting) systems, enabling quantitative predictions of interaction strengths from geometry and material constants.
Spatially confining the exciton (matter) region, rather than the light mode, can push the ratio of interaction strength to loss U/γ above 1, a regime relevant for polariton blockade.
Two spatially separated polariton modes can acquire strong nonlocal interactions, producing nonclassical correlations and entanglement of emitted light, e.g., violation of the Cauchy-Schwarz inequality.

Reading between the lines

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

The method's scope likely extends well beyond the polariton examples: any bosonic system whose classical linear response is known (phonons, magnons, plasmonic modes) could be quantized along the same lines, provided a stable nonequilibrium steady state exists.
A testable extension would be to compute two-time correlation functions from the diagonal master equation and compare with the full Liouvillian spectrum for a simple two-mode cavity; disagreement would pinpoint where the QNM-Liouvillian correspondence breaks down.
The treatment of nonlinear terms is perturbative in the sense that the basis is fixed by the linear problem; for ultrastrong nonlinearities one would need to re-derive the normal modes self-consistently, which the paper does not do.
The predicted U/γ boost from exciton confinement could be directly probed in existing microcavity platforms with selective interdiffusion or electrostatic gates, with the density-independent blueshift of Eq. (42) as a clean signature.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 paper is worth a serious look, but the headline claim has to be read with care. What's genuinely new is the dissipative construction: using Prosen's third quantization to write a diagonal, separable Lindblad master equation in the quasinormal-mode basis, with coefficients taken from classical Maxwell solutions plus stated material inputs. That is a useful recipe – if it holds, it takes you from a Maxwell solver to a polariton master equation without per-structure fitting. The conservative part is explicitly acknowledged to reproduce refs [42,54]; the contribution there is a cleaner route to the same result. The numerical examples (exciton-only confinement boosting U/gamma, engineered nonlocal U12) are interesting design directions, and they do not look like curve-fitting to target results.

That said, there are three soft spots, in increasing order of seriousness. First, the "no fitting parameters" claim is contradicted by their own calibration: alpha is fixed by reproducing the polariton splitting of ref [57], and g and gamma_x also come from [57]. Those parameters are transferred rather than fitted to their structures, which is defensible, but the abstract's "no fitting" is too strong. Second, the quantitative results depend on 2D simulations, an assumed gamma_NR=10 µeV, and no error bars, so the U/gamma ~ 13 numbers are indicative, not quantitative predictions. Third, and most important: the Sec III.C correspondence between the 2N QNMs and the 2N Liouvillian normal modes is proposed, not derived. The verification is a first-order correlation function for a single-mode coherent state, which only constrains products E_r^*(r1) E_r(r2) and does not fix the spatial profiles uniquely. And while the twin-Liouvillian isomorphism in Sec III.D is algebraically sound, it maps Fock spaces, not the physical field and matter operators; the paper itself warns to take care with non-quadratic terms, but then Sec IV adds H_int and writes the single-mode master equation with the QNM operator a as if it were the polariton operator. The U and g^(2) results are therefore conditional on an unproven assumption. These are addressable, not fatal: the direction is sound and the examples are illustrative, but the central claim needs a real derivation or a much more explicit statement of the conditions under which the correspondence holds.

Who is this for? People working on QNM quantization for nanophotonics and polariton blockade; they should read it. But I would not cite it as a foundation until the correspondence is tightened.

Recommendation: send to peer review, with a request that the authors either prove or clearly delimit the QNM-to-Liouvillian correspondence, fix the no-fitting claim, and add uncertainty estimates for the parameter-inferred numbers.

Referee Report

4 major / 4 minor

Summary. The paper proposes a recipe for constructing a quantum model of polaritons in arbitrary dielectric structures directly from classical eigenmode solutions. In the conservative case (Sec. II), the authors show that Bogoliubov diagonalization of the coupled light-matter Hamiltonian is equivalent to solving classical Maxwell equations coupled to matter polarization, with polariton operators expressed through the classical mode profiles. In the dissipative case (Sec. III), they use third quantization to argue that the normal superoperator modes of a quadratic Liouvillian are in one-to-one correspondence with quasinormal modes (QNMs) of the classical dissipative Maxwell equations, leading to a diagonal Lindblad master equation (39). The method is then applied to exciton-only confinement in microcavity and waveguide geometries, predicting enhanced polariton-polariton interaction strengths and nonlocal correlations. The central advertised advantages are the absence of mode-mixing terms in the dissipative basis and the claim of a parameter-free, geometry-to-model mapping.

Significance. If the central dissipative correspondence were proven, the paper would provide a practically useful bridge between classical Maxwell solvers and quantum master equations for nanophotonic and polaritonic structures. The conservative part (Sec. II) is a clear and largely sound recasting of known results, and the third-quantization algebra in Sec. III.A-D is competently executed. The proposed applications — exciton-only confinement to boost U/γ and engineering of nonlocal polariton interactions — are original and potentially impactful. However, the load-bearing quantum-classical correspondence in Sec. III.C is introduced as a proposal and verified only through a first-order coherence check, while the nonlinear applications assume the twin-Liouvillian isomorphism preserves observable content without derivation. The advertised 'no fitting parameters' claim is also stronger than what the manuscript demonstrates. These issues are fixable within the manuscript's scope, but they currently prevent acceptance.

major comments (4)

[§III.C, Eqs. (30)–(33)] The correspondence between QNMs of Eq. (33) and normal superoperator modes of Eq. (22) is introduced with 'we propose' in Eqs. (30)–(31). The only verification is the first-order correlation function (32) for a single-mode coherent state. This check is insufficient: G^(1) constrains only products E_r^*(r1)E_r(r2), and the NESS contribution G_NESS is subtracted rather than derived. The claim that 'no other choice' of coefficients can lead to the correct result is therefore unsupported. Since Eq. (39) and all subsequent predictions rest on this identification, a derivation (e.g., from quantum Langevin equations or from a Green-function expansion) or a nontrivial numerical cross-check for a multimode lossy structure is needed.
[§III.D–IV.A, Eqs. (41), (43)] The twin-Liouvillian isomorphism f is proven only for the quadratic Liouvillian, and Sec. III.D itself warns that 'one must take care ... when adding non-quadratic terms.' Nevertheless, Sec. IV.A obtains the interaction Hamiltonian by inserting the conservative mode expansion (16) into Eq. (40), and Sec. IV.C combines the resulting U = g∫|X(r)|^4 dr with the complex QNM frequency ω in Eq. (43). The image of H_int under the isomorphism f is not derived, nor is the relation between the conservative polariton operator P_j and the QNM annihilation operator a established. Without this step, the U/γ predictions and the g^(2) correlation results in Figs. 2–3 are not tied to the physical polariton system. At minimum, a perturbative justification or explicit computation for a two-mode example is required.
[Abstract; §IV.C; Conclusions] The advertised 'no fitting parameters' claim is not supported by the manuscript. In Sec. IV.C, the light-matter coupling strength α(r) is obtained by modeling the system of Ref. [57] and reproducing its polariton splitting; the interaction constant g and exciton decay rate γ_x are also taken from [57]; and γ_NR is an assumed 10 μeV input. This is not 'coefficients resulting from material properties' in the strong sense asserted in the abstract and conclusions. If the intended claim is 'no fitting to the newly proposed target structures,' it should be stated precisely; otherwise it overstates the predictive power of the method.
[§IV.C, Fig. 2(c)] The quantitative results, including U/γ = 12.95 for MoS2 and the U/γ > 1 window in Fig. 2(c), are obtained from 2D Maxwell solutions with a fixed 1 μm transverse dimension. The text acknowledges that this assumption 'may not be very accurate' given λ/n ≈ 0.3 μm, but the quantitative application claims are still presented as firm predictions. Because the central application is the existence of a parameter range with strong interactions, a 3D verification or an explicit order-of-magnitude framing is needed. As written, the numerical values are not robust.

minor comments (4)

[General] There are several typos: 'Liovillian' in Sec. III.E, 'calcuations' in Sec. II.D, 'Hamilonian' in Sec. IV.A, 'neurmorphic' in the Introduction. The reference to 'Sec. IIIA' in Sec. III.C should be 'Sec. III.A'.
[§III.A–B] The nondegeneracy condition ω_r ≠ -ω*_r and the convention ℜ(ω_r) > 0 are used in different places; it would help to state them together as assumptions of the normal-mode diagonalization (22) and (25).
[§IV.D, Fig. 3] The shaded region in Fig. 3(d) is said to mark violation of the Cauchy-Schwarz inequality; the text should define the exact inequality (e.g., [g^(2)_12]^2 ≤ g^(2)_11 g^(2)_22) so the reader can interpret the shaded area.
[§III.D] The statement that the resulting master equation has 'no influence on the form of the resulting master equation' is terse. Since the isomorphism is not the identity map on physical operators, a short explanation of why the diagonal GKSL form is physically meaningful (e.g., how expectation values of physical observables are recovered) would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction maps external classical QNM data into a master equation, and the flagged gaps are unsupported assumptions rather than circular reductions.

full rationale

The paper's central chain is not circular. In the conservative case, polariton operators are explicitly built from classical eigenmode profiles through Eqs. (15)-(18), and the physics content is the Bogoliubov diagonalization of the quadratic Hamiltonian; this is a construction, not a fitted prediction. In the dissipative case, the paper adopts the independent third-quantization framework of Prosen and Seligman [55] and then proposes the QNM expansion in Eqs. (30)-(31) by analogy with the Hermitian case. The verification via Eq. (32) is only a first-order consistency check, and the paper itself flags the limitation at the end of Sec. III.D: 'one must take care of the nontrivial nature of the vacuum state |0ζ⟩⟩ in Eq. (35) and the isomorphism between superoperators when considering specific initial conditions, coupling the system to external modes, or adding non-quadratic terms.' The applications in Sec. IV then add H_int via Eqs. (41)-(42) and write Eq. (43) without proving the image of the nonlinear interaction under the twin-Liouvillian isomorphism. This is a genuine correctness gap in the derivation, but it is not circularity: the U/γ values and g^(2)(0) curves are not fitted to the target structures, and no equation is reduced to its own output by construction. The input parameters α, g, and γ_x are calibrated against the external experiment of [57] and then used to produce predictions for new geometries, which is transferability rather than circularity. The self-citations [29-34] and the PLaSK software citation [84-85] are peripheral and are not load-bearing for the central claim. Overall, the central derivation contains an unproven correspondence assumption, but no circular step.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; the twin Liouvillian, dissipative coherent states, and the isomorphism are formal mathematical devices, not new forces, particles, or conserved quantities. The free parameters that matter are the material constants α, g, γ_x calibrated on ref [57] and the assumed γ_NR; they are inputs, but their calibrated origin weakens the 'no fitting parameters' branding. The heaviest axiom is the QNM-to-Liouvillian-mode correspondence, which is the paper's core new assertion and is not derived from the Liouvillian.

free parameters (4)

light-matter coupling strength α(r) = calibrated to reproduce the polariton energy splitting of ref [57]
Sec. IV.C: 'Light-matter coupling strength of the active material was obtained by modeling the system of [57] and reproducing the polariton energy splitting.' This is a fit to prior experimental data, contradicting the abstract's 'no fitting parameters'.
matter interaction constant g = taken from ref [57]; numerical value not stated in the paper
Sec. IV.C: 'The interaction and exciton decay rates were also taken from [57].' All U values inherit this input via U = g∫|X|^4 dr.
exciton decay rate γ_x = taken from ref [57]; numerical value not stated
Input loss rate from prior experiment; feeds the γ in all U/γ claims.
nonradiative decay rate γ_NR = 10 µeV (assumed)
Sec. IV.C: assumes γ_NR = 10 µeV for MoS2 and waveguide estimates; directly sets U/γ = 12.95 and γ ≈ 7 µeV.

assumptions (7)

domain assumption Lagrangian (3)-(5) with small emitters (α(r) A·Ẋ coupling, continuous matter oscillator field)
Central starting point of the derivation (Sec. II.A); assumes emitter size much smaller than wavelength, matter described as a 3D harmonic oscillator field.
domain assumption Stability of all classical solutions, ω_j^2 > 0 (stability of vacuum)
Invoked in Sec. II.A and II.C as the condition for the Bogoliubov transformation to exist (also ref [64]).
domain assumption Born-Markov approximation and GKSL form (19) with linear Lindbladians
Sets the dissipative framework (Sec. III); only linear Lindbladian terms are considered to preserve quadratic Liouvillian structure.
standard math Prosen-Seligman third quantization: stable, non-degenerate dynamics, symplectic transformation V, ω_r ≠ -ω*_r
Assumed background from ref [55] (Sec. III.A, III.B, footnote [68]); needed for the diagonal Liouvillian (22) and for the construction of Hermitian-conjugate mode pairs.
ad hoc to paper QNMs of (33), regularized by PML, correspond to the complete set of Liouvillian normal modes
The central unproved correspondence (Sec. III.C, Eqs. 30-33), asserted via 'we propose' and a first-order correlation consistency check rather than derived. Load-bearing for the whole dissipative recipe.
domain assumption Interaction Hamiltonian (40), H_int = (g/6) X^4, dominates the nonlinearity
Adopted in Sec. IV.A-B for all application predictions; under RWA gives (41)-(43) and the density-independent blueshift.
ad hoc to paper 2D Maxwell solutions with a fixed 1 µm transverse dimension capture the physics of the proposed structures
Sec. IV.C: authors acknowledge the assumption 'may not be very accurate' at λ/n ≈ 0.3 µm and argue the principle holds in 3D; quantitative U/γ values rest on it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quantization of Polaritons Confined in Dielectric Structures." pith.science (2026). https://pith.science/paper/2W4AL2EZ

@misc{pith2026251014566,
  author       = {Pith},
  title        = {Pith review of: Quantization of Polaritons Confined in Dielectric Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2W4AL2EZ}},
  note         = {Machine review of arXiv:2510.14566}
}

read the original abstract

Light-matter interaction in the regime of strong quantum coupling is usually treated within the framework of the Hopfield model. However, the picture of coupling well-defined modes of light and matter is correct only as long as the shapes of these eigenmodes are not substantially modified by the interaction. Moreover, parameters of theoretical models are usually obtained by fitting to experimental data. To date, there is no straightforward method to determine a quantum master equation corresponding to a system with specific dielectric structure, which may lead to incompatibility of theoretical descriptions and physical realizations. In this work, a recipe for obtaining a quantum model in the polariton eigenmode basis is presented, based on Bogoliubov transformation in the conservative case and third quantization technique in the dissipative case. It is shown how this method can be used for boosting interaction strength and engineering nonlocal many-body interactions in carefully designed nanostructures, resulting in strongly nonclassical correlations of emitted light.

Figures

Figures reproduced from arXiv: 2510.14566 by the authors.

**Figure 2.** FIG. 2. (a) Proposed vertical microcavity structure with exciton-only confinement. Blue area corresponds to InGaAs active [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Structure similar to that in Fig. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 2 linked inside Pith

[57]

Muñoz-Matutano, A

G. Muñoz-Matutano, A. Wood, M. Johnsson, X. Vidal, B. Q. Baragiola, A. Reinhard, A. Lemaître, J. Bloch, A. Amo, G. Nogues, B. Besga, M. Richard, and T. Volz, Emergence of quantum correlations from interacting fibre-cavity polaritons, Nature Materials18, 213 (2019)

2019
[1]

Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Physical Review 112, 1555 (1958)

J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Physical Review 112, 1555 (1958)

1958
[2]

D. N. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu, and A. Rubio, Polariton panorama, Nanophotonics10, 549 (2020)

2020
[3]

Weisbuch, M

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Observation of the coupled exciton-photon mode split- ting in a semiconductor quantum microcavity, Physical review letters69, 3314 (1992)

1992
[4]

Orfanakis, S

K. Orfanakis, S. K. Rajendran, V. Walther, T. Volz, T. Pohl, and H. Ohadi, Rydberg exciton–polaritons in a cu2o microcavity, Nature Materials21, 767 (2022)

2022
[5]

H. Kang, J. Ma, J. Li, X. Zhang, and X. Liu, Exciton polaritons in emergent two-dimensional semiconductors, ACS Nano17, 24449 (2023)

2023
[6]

T. Luo, B. Ilyas, A. v. Hoegen, Y. Lee, J. Park, J.- G. Park, and N. Gedik, Time-of-flight detection of ter- ahertz phonon-polariton, Nature Communications15, 2276 (2024)

2024
[7]

S. Pres, B. Huber, M. Hensen, D. Fersch, E. Schatz, D. Friedrich, V. Lisinetskii, R. Pompe, B. Hecht, W. Pfeiffer, and T. Brixner, Detection of a plasmon- polariton quantum wave packet, Nature Physics19, 656 (2023)

2023

Show all 90 references

[8]

Zhang, X.-Q

D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, Observation of the exceptional point in cav- ity magnon-polaritons, Nature Communications8, 1368 (2017)

2017
[9]

Ravets, P

S. Ravets, P. Knüppel, S. Faelt, O. Cotlet, M. Kroner, W. Wegscheider, and A. Imamoglu, Polaron polaritons in the integer and fractional quantum hall regimes, Physical review letters120, 057401 (2018)

2018
[10]

Kim, K.-T

B. Kim, K.-T. Chen, S.-S. Hsiao, S.-Y. Wang, K.-B. Li, J. Ruseckas, G. Juzeli¯ unas, T. Kirova, M. Auzinsh, Y.- C. Chen, Y.-F. Chen, and I. A. Yu, A weakly-interacting many-body system of Rydberg polaritons based on elec- tromagnetically induced transparency, Communications Ph...

2021
[11]

N. Jia, N. Schine, A. Georgakopoulos, A. Ryou, L. W. Clark, A. Sommer, and J. Simon, A strongly interacting polaritonic quantum dot, Nature Physics14, 550 (2018)

2018
[12]

J.Kwon, Y.Kim, A.Lanuza,andD.Schneble,Formation of matter-wave polaritons in an optical lattice, Nature Physics18, 657 (2022)

2022
[13]

A. V. Kavokin, J. J. Baumberg, G. Malpuech, and F. P. Laussy,Microcavities, Vol. 21 (Oxford university press, 2017)

2017
[14]

Kędziora, A

M. Kędziora, A. Opala, R. Mastria, L. De Marco, M. Król, K. Łempicka-Mirek, K. Tyszka, M. Ekielski, M. Guziewicz, K. Bogdanowicz, A. Szerling, H. Sig- urðsson, T. Czyszanowski, J. Szczytko, M. Matuszewski, D. Sanvitto, and B. Piętka, Predesigned perovskite crys- tal waveguides...

2024
[15]

Liran, R

D. Liran, R. Rapaport, J. Hu, N. Lydick, H. Deng, and L. Pfeiffer, Electrically controlled photonic circuits of 14 field-induced dipolaritons with huge nonlinearities, Phys- ical Review X14, 031022 (2024)

2024
[16]

L. K. van Vugt, S. Rühle, P. Ravindran, H. C. Gerritsen, L. Kuipers, and D. Vanmaekelbergh, Exciton polaritons confined in a zno nanowire cavity, Phys. Rev. Lett.97, 147401 (2006)

2006
[17]

A. Graf, L. Tropf, Y. Zakharko, J. Zaumseil, and M. C. Gather, Near-infrared exciton-polaritons in strongly cou- pledsingle-walledcarbonnanotubemicrocavities,Nature Communications7, 13078 (2016)

2016
[18]

D. M. Coles, L. C. Flatten, T. Sydney, E. Hounslow, S. K. Saikin, A. Aspuru-Guzik, V. Vedral, J. K.-H. Tang, R. A. Taylor, J. M. Smith,et al., Polaritons in living systems: modifying energy landscapes in photosynthetic organisms using a photonic structure, arXiv (2017), arXiv:...

2017 arXiv
[19]

Zhang, F

L. Zhang, F. Wu, S. Hou, Z. Zhang, Y.-H. Chou, K. Watanabe, T. Taniguchi, S. R. Forrest, and H. Deng, Van der waals heterostructure polaritons with moiré- induced nonlinearity, Nature591, 61 (2021)

2021
[20]

D. Choi, M. Park, B. Y. Oh, M.-S. Kwon, S. I. Park, S. Kang, J. D. Song, D. Ko, M. Sun, I. G. Savenko, Y.- H. Cho, and H. Choi, Observation of a single quantized vortex vanishment in exciton-polariton superfluids, Phys. Rev. B105, L060502 (2022)

2022
[21]

H. Deng, H. Haug, and Y. Yamamoto, Exciton-polariton bose-einstein condensation, Rev. Mod. Phys.82, 1489 (2010)

2010
[22]

Y. Sun, P. Wen, Y. Yoon, G. Liu, M. Steger, L. N. Pfeiffer, K. West, D. W. Snoke, and K. A. Nelson, Bose- einstein condensation of long-lifetime polaritons in ther- mal equilibrium, Phys. Rev. Lett.118, 016602 (2017)

2017
[23]

F. L. Ruta, S. Zhang, Y. Shao, S. L. Moore, S. Acharya, Z. Sun, S. Qiu, J. Geurs, B. S. Kim, M. Fu,et al., Hy- perbolic exciton polaritons in a van der waals magnet, Nature communications14, 8261 (2023)

2023
[24]

Frisk Kockum, A

A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, Ultrastrong coupling between light and matter, Nature Reviews Physics1, 19 (2019)

2019
[25]

Estrecho, T

E. Estrecho, T. Gao, N. Bobrovska, D. Comber-Todd, M. D. Fraser, M. Steger, K. West, L. N. Pfeiffer, J. Levin- sen, M. Parish,et al., Direct measurement of polariton- polariton interaction strength in the thomas-fermi regime of exciton-polariton condensation, Physical Review B...

2019
[26]

Schneider, A

C. Schneider, A. Rahimi-Iman, N. Y. Kim, J. Fis- cher, I. G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V. D. Kulakovskii,et al., An electrically pumped polariton laser, Nature497, 348 (2013)

2013
[27]

Christopoulos, G

S. Christopoulos, G. B. H. Von Högersthal, A. Grundy, P. Lagoudakis, A. Kavokin, J. Baumberg, G. Christ- mann, R. Butté, E. Feltin, J.-F. Carlin,et al., Room- temperature polariton lasing in semiconductor microcav- ities, Physical review letters98, 126405 (2007)

2007
[28]

Klembt, T

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, Exciton-polariton topological insulator, Nature562, 552 (2018)

2018
[29]

Comaron, V

P. Comaron, V. Shahnazaryan, W. Brzezicki, T. Hyart, and M. Matuszewski, Non-hermitian topological end- mode lasing in polariton systems, Phys. Rev. Res.2, 022051 (2020)

2020
[30]

Opala, S

A. Opala, S. Ghosh, T. C. Liew, and M. Matuszewski, Neuromorphic computing in ginzburg-landau polariton- lattice systems, Phys. Rev. Appl.11, 064029 (2019)

2019
[31]

Ballarini, A

D. Ballarini, A. Gianfrate, R. Panico, A. Opala, S. Ghosh, L. Dominici, V. Ardizzone, M. De Giorgi, G. Lerario, G. Gigli, T. C. H. Liew, M. Matuszewski, and D. Sanvitto, Polaritonic neuromorphic computing out- performs linear classifiers, Nano Letters20, 3506 (2020)

2020
[32]

Mirek, A

R. Mirek, A. Opala, P. Comaron, M. Furman, M. Król, K. Tyszka, B. Seredyński, D. Ballarini, D. Sanvitto, T. C. H. Liew, W. Pacuski, J. Suffczyński, J. Szczytko, M. Matuszewski, and B. Piętka, Neuromorphic binarized polariton networks, Nano Letters21, 3715 (2021)

2021
[33]

Matuszewski, A

M. Matuszewski, A. Opala, R. Mirek, M. Furman, M. Król, K. Tyszka, T. C. Liew, D. Ballarini, D. San- vitto, J. Szczytko,et al., Energy-efficient neural network inference with microcavity exciton polaritons, Physical Review Applied16, 024045 (2021)

2021
[34]

Opala and M

A. Opala and M. Matuszewski, Harnessing exciton- polaritons for digital computing, neuromorphic comput- ing, and optimization, Opt. Mater. Express13, 2674 (2023)

2023
[35]

Kavokin, T

A. Kavokin, T. C. Liew, C. Schneider, P. G. Lagoudakis, S. Klembt, and S. Hoefling, Polariton condensates for classical and quantum computing, Nature Reviews Physics4, 435 (2022)

2022
[36]

Boulier, M

T. Boulier, M. J. Jacquet, A. Maître, G. Lerario, F. Claude, S. Pigeon, Q. Glorieux, A. Amo, J. Bloch, A. Bramati,et al., Microcavity polaritons for quantum simulation, Advanced Quantum Technologies3, 2000052 (2020)

2020
[37]

I. A. Shelykh, A. V. Kavokin, Y. G. Rubo, T. Liew, and G. Malpuech, Polariton polarization-sensitive phe- nomena in planar semiconductor microcavities, Semicon- ductor Science and Technology25, 013001 (2009)

2009
[38]

Wouters and I

M. Wouters and I. Carusotto, Excitations in a nonequi- librium bose-einstein condensate of exciton polaritons, Physical review letters99, 140402 (2007)

2007
[39]

Todorov, A

Y. Todorov, A. M. Andrews, R. Colombelli, S. De Lib- erato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Ultrastrong light-matter coupling regime with polariton dots, Physical review letters105, 196402 (2010)

2010
[40]

Cortese, J

E. Cortese, J. Mornhinweg, R. Huber, C. Lange, and S. De Liberato, Real-space nanophotonic field manipu- lation using non-perturbative light–matter coupling, Op- tica10, 11 (2022)

2022
[41]

Cortese, N.-L

E. Cortese, N.-L. Tran, J.-M. Manceau, A. Bousseksou, I. Carusotto, G. Biasiol, R. Colombelli, and S. De Liber- ato, Excitons bound by photon exchange, Nature Physics 17, 31 (2021)

2021
[42]

Huttner and S

B. Huttner and S. M. Barnett, Quantization of the elec- tromagnetic field in dielectrics, Physical Review A46, 4306 (1992)

1992
[43]

P. D. Drummond, Electromagnetic quantization in dis- persiveinhomogeneousnonlineardielectrics,PhysicalRe- view A42, 6845 (1990)

1990
[44]

D. J. Santos and R. Loudon, Electromagnetic-field quantization in inhomogeneous and dispersive one- dimensional systems, Phys. Rev. A52, 1538 (1995)

1995
[45]

Knöll and U

L. Knöll and U. Leonhardt, Quantum optics in oscillator media, Journal of Modern Optics39, 1253 (1992)

1992
[46]

Milonni, Field quantization and radiative processes in dispersive dielectric media, Journal of Modern Optics42, 1991 (1995)

P. Milonni, Field quantization and radiative processes in dispersive dielectric media, Journal of Modern Optics42, 1991 (1995). 15

1991
[47]

Gruner and D.-G

T. Gruner and D.-G. Welsch, Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous kramers-kronig dielectrics, Phys. Rev. A 53, 1818 (1996)

1996
[48]

Tip, Linear absorptive dielectrics, Phys

A. Tip, Linear absorptive dielectrics, Phys. Rev. A57, 4818 (1998)

1998
[49]

H. T. Dung, L. Knöll, and D.-G. Welsch, Three- dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics, Phys. Rev. A57, 3931 (1998)

1998
[50]

Bechler, Quantum electrodynamics of the dispersive dielectric medium–a path integral approach, Journal of Modern Optics46, 901 (1999)

A. Bechler, Quantum electrodynamics of the dispersive dielectric medium–a path integral approach, Journal of Modern Optics46, 901 (1999)

1999
[51]

Drezet, Quantizing polaritons in inhomogeneous dis- sipative systems, Phys

A. Drezet, Quantizing polaritons in inhomogeneous dis- sipative systems, Phys. Rev. A95, 023831 (2017)

2017
[52]

M. G. Raymer, Quantum theory of light in a dispersive structured linear dielectric: a macroscopic hamiltonian tutorial treatment, Journal of Modern Optics67, 196 (2020)

2020
[53]

Suttorp and M

L. Suttorp and M. Wubs, Field quantization in inhomo- geneous absorptive dielectrics, Phys. Rev. A70, 013816 (2004)

2004
[54]

N. A. Bhat and J. Sipe, Hamiltonian treatment of the electromagnetic field in dispersive and absorptive struc- tured media, Phys. Rev. A73, 063808 (2006)

2006
[55]

Prosen and T

T. Prosen and T. H. Seligman, Quantization over boson operator spaces, Journal of Physics A: Mathematical and Theoretical43, 392004 (2010)

2010
[56]

Verger, C

A. Verger, C. Ciuti, and I. Carusotto, Polariton quantum blockadeinaphotonicdot,PhysicalReviewB73,193306 (2006)

2006
[58]

Delteil, T

A. Delteil, T. Fink, A. Schade, S. Höfling, C. Schnei- der, and A. İmamoğlu, Towards polariton blockade of confined exciton–polaritons, Nature Materials18, 219 (2019)

2019
[59]

Togan, H.-T

E. Togan, H.-T. Lim, S. Faelt, W. Wegscheider, and A. Imamoglu, Enhanced interactions between dipolar po- laritons, Physical review letters121, 227402 (2018)

2018
[60]

Datta, M

B. Datta, M. Khatoniar, P. Deshmukh, F. Thouin, R. Bushati, S. De Liberato, S. K. Cohen, and V. M. Menon, Highly nonlinear dipolar exciton-polaritons in bi- layer mos2, Nature communications13, 6341 (2022)

2022
[61]

Emmanuele, M

R. Emmanuele, M. Sich, O. Kyriienko, V. Shahnazaryan, F. Withers, A. Catanzaro, P. Walker, F. Benimetskiy, M. Skolnick, A. Tartakovskii,et al., Highly nonlinear trion-polaritons in a monolayer semiconductor, Nature communications11, 3589 (2020)

2020
[62]

Schwennicke, A

K. Schwennicke, A. Koner, J. B. Pérez-Sánchez, W. Xiong, N. C. Giebink, M. L. Weichman, and J. Yuen- Zhou, When do molecular polaritons behave like optical filters? (2024), arXiv:2408.05036

2024 arXiv
[63]

R. J. Glauber and M. Lewenstein, Quantum optics of dielectric media, Physical Review A43, 467 (1991)

1991
[64]

Dereziński, Bosonic quadratic hamiltonians, Journal of Mathematical Physics58, 121101 (2017)

J. Dereziński, Bosonic quadratic hamiltonians, Journal of Mathematical Physics58, 121101 (2017)

2017
[65]

M. O. Scully and M. S. Zubairy, Quantum optics (1999)

1999
[66]

McDonald and A

A. McDonald and A. A. Clerk, Third quantization of open quantum systems: Dissipative symmetries and con- nections to phase-space and keldysh field-theory formu- lations, Phys. Rev. Res.5, 033107 (2023)

2023
[67]

Dupays, Third-quantized master equations as a classi- cal ornstein-uhlenbeck process (2024), arXiv:2408.11893

L. Dupays, Third-quantized master equations as a classi- cal ornstein-uhlenbeck process (2024), arXiv:2408.11893

2024
[68]

(20), and together with the bilinear form ⟨u, v⟩ζ := [u, v], a complex symplectic vector space

Therefore, bζr together with bζ ′ r form a complete set in the 4N-dimensional space spanned by superoperators de- fined by Eqs. (20), and together with the bilinear form ⟨u, v⟩ζ := [u, v], a complex symplectic vector space
[69]

However, this property will not be important for us, and in this work we do not call ζ + r Hermitian adjoints ofζ r to avoid confusion

Consideringζ r,ζ ′ r as operators acting on the Hilbert space of Hilbert-Schmidt operators,ζ+ r ,ζ ′+ r are their re- spective Hermitian adjoints. However, this property will not be important for us, and in this work we do not call ζ + r Hermitian adjoints ofζ r to avoid confu...
[70]

Gerry and P

C. Gerry and P. Knight,Introductory Quantum Optics (Cambridge University Press, 2004)

2004
[71]

Sauvan, T

C. Sauvan, T. Wu, R. Zarouf, E. A. Muljarov, and P. Lalanne, Normalization, orthogonality, and complete- ness of quasinormal modes of open systems: the case of electromagnetism, Opt. Express30, 6846 (2022)

2022
[72]

R.-C.GeandS.Hughes,Quantumdynamicsoftwoquan- tum dots coupled through localized plasmons: An intu- itive and accurate quantum optics approach using quasi- normal modes, Phys. Rev. B92, 205420 (2015)

2015
[73]

Franke, S

S. Franke, S. Hughes, M. K. Dezfouli, P. T. Kristensen, K. Busch, A. Knorr, and M. Richter, Quantization of quasinormal modes for open cavities and plasmonic cav- ity quantum electrodynamics, Phys. Rev. Lett.122, 213901 (2019)

2019
[74]

K. C. Ho, P. T. Leung, A. Maassen van den Brink, and K. Young, Second quantization of open systems using quasinormal modes, Phys. Rev. E58, 2965 (1998)

1998
[75]

T. Wu, J. L. Jaramillo, and P. Lalanne, Reflections on the spatial exponential growth of electromagnetic quasi- normal modes, Laser & Photonics Reviews19, 2402133 (2025)

2025
[76]

Gedney, An anisotropic perfectly matched layer- absorbing medium for the truncation of fdtd lattices, IEEE Transactions on Antennas and Propagation44, 1630 (1996)

S. Gedney, An anisotropic perfectly matched layer- absorbing medium for the truncation of fdtd lattices, IEEE Transactions on Antennas and Propagation44, 1630 (1996)

1996
[77]

Raman and S

A. Raman and S. Fan, Photonic band structure of disper- sive metamaterials formulated as a hermitian eigenvalue problem, Phys. Rev. Lett.104, 087401 (2010)

2010
[78]

W. Yan, R. Faggiani, and P. Lalanne, Rigorous modal analysis of plasmonic nanoresonators, Phys. Rev. B97, 205422 (2018)

2018
[79]

Ferretti and D

S. Ferretti and D. Gerace, Single-photon nonlinear op- tics with kerr-type nanostructured materials, Physical Review B—Condensed Matter and Materials Physics85, 033303 (2012)

2012
[80]

D. S.-F. et al., Enhancement of parametric effects in polariton waveguides induced by dipolar interactions, Phys.Rev.Lett.126, 137401 (2021)

2021
[81]

L. B. Tan, O. Cotlet, A. Bergschneider, R. Schmidt, P. Back, Y. Shimazaki, M. Kroner, and A. İmamoğlu, Interacting polaron-polaritons, Physical Review X10, 021011 (2020)

2020
[82]

Zaitsev, M

S. Zaitsev, M. Welsch, A. Forchel, and G. Bacher, Exci- tons in artificial quantum dots in the weak spatial con- finement regime, Journal of Experimental and Theoreti- cal Physics105, 1241 (2007). 16

2007
[83]

Thureja, F

D. Thureja, F. E. Yazıcı, T. Smoleński, M. Kroner, D. J. Norris, and A. İmamoˇ glu, Electrically defined quantum dots for bosonic excitons, Physical Review B110, 245425 (2024)

2024
[84]

M. Dems, R. Kotynski, and K. Panajotov, Planewave ad- mittance method—a novel approach for determining the electromagneticmodesinphotonicstructures,OpticsEx- press13, 3196 (2005)

2005
[85]

Photonic lasers simulation kit,http://www.plask.app
[86]

Dufferwiel, S

S. Dufferwiel, S. Schwarz, F. Withers, A. A. Trichet, F. Li, M. Sich, O. Del Pozo-Zamudio, C. Clark, A. Nal- itov, D. D. Solnyshkov,et al., Exciton–polaritons in van der waals heterostructures embedded in tunable micro- cavities, Nature communications6, 8579 (2015)

2015
[87]

J. Gu, V. Walther, L. Waldecker, D. Rhodes, A. Raja, J. C. Hone, T. F. Heinz, S. Kéna-Cohen, T. Pohl, and V. M. Menon, Enhanced nonlinear interaction of polari- tons via excitonic rydberg states in monolayer wse2, Na- ture communications12, 2269 (2021)

2021
[88]

Barachati, A

F. Barachati, A. Fieramosca, S. Hafezian, J. Gu, B. Chakraborty, D. Ballarini, L. Martinu, V. Menon, D. Sanvitto, and S. Kéna-Cohen, Interacting polariton fluids in a monolayer of tungsten disulfide, Nature nan- otechnology13, 906 (2018)

2018
[89]

Drummond and D

P. Drummond and D. Walls, Quantum theory of optical bistability. i. nonlinear polarisability model, Journal of Physics A: Mathematical and General13, 725 (1980)

1980
[90]

Medic, A

L. Medic, A. Ramšak, and T. Prosen, Extending third quantization with commuting observables: a dissipative spin-boson model, Journal of Physics A: Mathematical and Theoretical57, 085301 (2024)

2024

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [57]

Muñoz-Matutano, A

G. Muñoz-Matutano, A. Wood, M. Johnsson, X. Vidal, B. Q. Baragiola, A. Reinhard, A. Lemaître, J. Bloch, A. Amo, G. Nogues, B. Besga, M. Richard, and T. Volz, Emergence of quantum correlations from interacting fibre-cavity polaritons, Nature Materials18, 213 (2019)

2019

[2] [1]

Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Physical Review 112, 1555 (1958)

J. Hopfield, Theory of the contribution of excitons to the complex dielectric constant of crystals, Physical Review 112, 1555 (1958)

1958

[3] [2]

D. N. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu, and A. Rubio, Polariton panorama, Nanophotonics10, 549 (2020)

2020

[4] [3]

Weisbuch, M

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Observation of the coupled exciton-photon mode split- ting in a semiconductor quantum microcavity, Physical review letters69, 3314 (1992)

1992

[5] [4]

Orfanakis, S

K. Orfanakis, S. K. Rajendran, V. Walther, T. Volz, T. Pohl, and H. Ohadi, Rydberg exciton–polaritons in a cu2o microcavity, Nature Materials21, 767 (2022)

2022

[6] [5]

H. Kang, J. Ma, J. Li, X. Zhang, and X. Liu, Exciton polaritons in emergent two-dimensional semiconductors, ACS Nano17, 24449 (2023)

2023

[7] [6]

T. Luo, B. Ilyas, A. v. Hoegen, Y. Lee, J. Park, J.- G. Park, and N. Gedik, Time-of-flight detection of ter- ahertz phonon-polariton, Nature Communications15, 2276 (2024)

2024

[8] [7]

S. Pres, B. Huber, M. Hensen, D. Fersch, E. Schatz, D. Friedrich, V. Lisinetskii, R. Pompe, B. Hecht, W. Pfeiffer, and T. Brixner, Detection of a plasmon- polariton quantum wave packet, Nature Physics19, 656 (2023)

2023