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arxiv: 2607.05857 · v1 · pith:5JFB4PDT · submitted 2026-07-07 · cond-mat.mes-hall

Floquet polaritons in optically driven materials

Tsan Huang , Teng Xiao , Jiahua Duan , Haoliang Qian , Zhiyuan Sun This is my paper

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classification cond-mat.mes-hall
keywords floquetopticalpolaritonspumpbandsleadslightnonlinearity
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The pith

Pump-driven polaritons predicted from nonlinear optics alone

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

This paper presents a method to predict how a strong pump laser reshapes polariton spectra in quantum materials, without introducing any fitting parameters. The core idea is that the pump field, combined with the material's known nonlinear optical susceptibilities, generates an effective linear susceptibility. Solving Maxwell's equations with this effective susceptibility yields the Floquet polariton bands. The authors apply this to three systems: graphene plasmons driven by infrared light, hexagonal boron nitride phonon polaritons driven by mid-infrared light, and Josephson plasmons in layered superconductors driven by THz fields. In graphene and hBN, the pump creates Floquet replica bands that cross the original polariton branches. At these crossings, the hybridization involves one mode with positive spectral weight and one with negative spectral weight, producing non-Hermitian coupling. This leads to flat bands bounded by exceptional points and modes with negative damping, signaling parametric instability where two pump photons convert into two polaritons. In layered superconductors, the pump produces new reflectivity peaks at frequencies shifted by twice the pump frequency from the Josephson plasma edge, with a non-perturbative Bessel-function expression for the effective dielectric.

Core claim

The paper's central technical result is that the pump-contributed self-energy for the polariton propagator can be built entirely from the material's nonlinear optical susceptibilities and the pump field, yielding an effective linear susceptibility whose poles give the Floquet polariton spectrum. A key physical consequence is that Floquet replica bands carry negative spectral weight, because they originate from shifting negative-frequency modes to positive frequencies. When such a replica crosses an original positive-weight branch, their opposite spectral signs produce attractive non-Hermitian hybridization, creating flat bands with exceptional points and parametrically unstable modes that do

What carries the argument

The effective linear susceptibility chi_eff, constructed by combining the pump field with the material's nonlinear optical susceptibilities via a diagrammatic self-energy expansion. The dressed polariton propagator G(omega,k) is obtained by replacing the equilibrium susceptibility chi with chi_eff in the bare propagator G_0. The poles of G give the Floquet polariton branches.

If this is right

  • Pump-probe near-field experiments on graphene with mid-IR pump fields around 290 kV/cm should reveal flat plasmon bands with gain signatures in the near-field reflection coefficient, testable with current table-top lasers.
  • Far-field THz pump-probe spectroscopy on layered superconductors should show new reflectivity peaks at frequencies shifted by twice the pump frequency from the Josephson plasma edge, providing a direct fingerprint of Floquet Josephson plasmons.
  • The framework extends to any polaritonic system with known nonlinear optical coefficients, including exciton polaritons in semiconductor microcavities, where similar parametric instability and flat bands are expected.
  • Circularly polarized pump on graphene generates a chirality-dependent Hall optical response, suggesting topological Floquet plasmon physics at sample boundaries that could be probed near-field.

Load-bearing premise

The pump field is treated as a fixed classical background that does not receive self-consistent feedback from the material response, which neglects pump depletion and spatial variation of the pump inside the sample. This is most questionable in superconductors where pump and probe penetration depths can differ significantly, and the authors flag this as needing further study.

What would settle it

If an experiment measured the Floquet polariton spectrum under a strong pump and found spectral features that cannot be reproduced by the effective susceptibility constructed from known nonlinear optical coefficients and the pump field, the framework would fail. Most directly, if the pump-induced flat bands and exceptional points predicted near band crossings were absent, or if their locations in frequency-momentum space differed systematically from the parametric-instability threshold condition, the core mechanism would be falsified.

Figures

Figures reproduced from arXiv: 2607.05857 by Haoliang Qian, Jiahua Duan, Teng Xiao, Tsan Huang, Zhiyuan Sun.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Left: polaritons in a solid state material in equilibrium [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Floquet plasmons in graphene. (a) Schematic of the near [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Floquet phonon polaritons in monolayer hBN. (a) Schematic [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Floquet phonon polaritons in hBN thin flakes. The left panel [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Floquet Josephson plasmons in layered superconductors. (a) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic of the pump-probe experiment on a uniaxial [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Near-field reflection coefficient [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The diagrammatic representation of the effective susceptibility of laser-pumped monolayer hBN modeled by Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
read the original abstract

Polaritons are coupled collective modes of light and matter in quantum materials. In modern pump-probe experiments, a pump light pulse may dramatically alter the properties of the polaritons, rendering them Floquet polaritons that can be detected by a probe pulse. We present a practical framework to describe Floquet polaritons in terms of the linear and nonlinear optical properties of the material. The central quantity that yields the spectra of Floquet polaritons is an effective linear optical susceptibility contributed by the pump through nonlinear optical susceptibilities. We apply this method to graphene and show that via its third-order optical nonlinearity, infrared pump leads to Floquet plasmon bands. Notably, near plasmonic band crossings, parametric instability leads to flat bands with unstable modes and exceptional points that closely resemble those of non-Hermitian systems. As a second example, we show that in hexagonal boron nitride pumped by mid-infrared laser, the pump induces Floquet phonon polariton bands via phononic nonlinearity, which can be detected with either far-field or near-field optical technique. Finally, in layered superconductors pumped by THz light polarized along the out-of-plane direction, the Josephson-type optical nonlinearity leads to Floquet Josephson plasmons, which manifest as new peaks in the THz reflectivity of a probe pulse.

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

3 major / 8 minor

Summary. This paper presents a theoretical framework for predicting Floquet polariton spectra in optically driven quantum materials. The central idea is to construct an effective linear susceptibility χ_eff from the material's nonlinear optical susceptibilities combined with a classical pump field, then solve Maxwell's equations with this χ_eff to obtain the Floquet polariton dispersion. The framework is applied to three systems: (1) graphene plasmons driven via χ^(3) nonlinearity, exhibiting flat bands, exceptional points, and parametric instability; (2) hBN phonon polaritons driven via phononic nonlinearity, in both monolayer and thin-flake geometries; and (3) Josephson plasmons in layered superconductors driven by THz fields, producing new reflectivity peaks. The approach is grounded in a Keldysh path-integral formalism (Appendix A) whose classical saddle-point approximation yields the main-text equations. The graphene and hBN results are cross-checked against numerical Floquet diagonalization (Appendix C, cutoff N=5).

Significance. The paper addresses a timely problem — Floquet engineering of polaritons in quantum materials — and provides a practical, parameter-light framework that connects measurable nonlinear optical coefficients to Floquet polariton spectra. The construction of χ_eff from 1PI diagrams (Eq. 9) and its application to three distinct material platforms is a valuable contribution. Specific strengths include: (i) the parametric oscillator cross-check for graphene (Eq. 20) providing an independent derivation of the non-Hermitian band structure; (ii) the identification of negative spectral weight in Floquet replicas as a generic feature across all three platforms; (iii) the resummed effective susceptibility for hBN (Eq. 25) that goes beyond leading-order perturbation theory; and (iv) concrete experimental protocols with realistic parameters for near-field and far-field detection. The framework is largely non-circular: χ_eff is built from independently measurable nonlinear coefficients and an external pump field.

major comments (3)
  1. Sec. V, Eq. (32) and Fig. 5(b): The reflectivity R = |(1−√ε_eff)/(1+√ε_eff)|² is computed for a semi-infinite gain medium using the standard Fresnel formula. When χ_eff has negative spectral weight (which the paper shows occurs for the Floquet replica at −ω_J + 2ω_p), the standard Fresnel formula is known to be ambiguous for gain media — the sign of the imaginary part of √ε_eff must be chosen consistently with outgoing-wave boundary conditions and the direction of energy flow. The authors acknowledge this ('the reflection problem at the interface between vacuum and an infinitely deep gain medium described by Eq. (32) involves subtleties') but do not resolve it. This means the quantitative reflectivity predictions in Fig. 5(b) — one of the three flagship examples — are not on firm footing. The authors should either (a) restrict the Josephson plasmon predictions to the optical conductivity
  2. Sec. V: The uniform-pump assumption (infinite penetration depth) is stated but its quantitative impact is not bounded. The authors note that the pump penetration depth is assumed much larger than that of the probe, but for layered superconductors the c-axis penetration depth at THz frequencies can be comparable to or smaller than the probe skin depth, and pump depletion/self-consistent penetration-depth effects may be significant. Since this assumption is load-bearing for the analytical results in Eq. (32), the authors should at minimum provide an estimate of the parameter regime (pump frequency, field strength, temperature) where the uniform-pump approximation is self-consistent, or cite the Fresnel-Floquet approach (Ref. 128) with a more specific statement of what corrections are expected.
  3. Sec. III.B, Eq. (17): The parametric instability threshold is derived as ξ_c ≃ 0.52 for γ/ω_p = 0.05, corresponding to E_c ≈ 290 kV/cm. The paper states that 'in continuous-wave experiments, this unstable growth is expected to be cut off by nonlinear saturation that is beyond the present weak-probe treatment.' However, the weak-probe linearization itself may break down once the unstable mode grows to amplitudes comparable to the probe. The authors should clarify the timescale on which the linear-response treatment remains valid after the pump is turned on (i.e., the exponential growth time 1/Im[ω] versus the probe pulse duration), to establish that the predicted Im[R_p] features in Fig. 2(c) are observable before saturation.
minor comments (8)
  1. Eq. (14): The notation '(ω ± 2ω_p)' is used to denote two separate terms, but this convention is introduced only after the equation. A brief note before Eq. (14) would improve readability.
  2. Fig. 2(c): The color scale is stated to be logarithmic, which makes the plasmons 'appear broader than what they actually are.' Consider also providing a linear-scale inset or panel for direct comparison with experimental data.
  3. Sec. IV.A, Eq. (24): The definition of χ^(2) in terms of the bare phonon propagator χ(ω) is stated, but the distinction between χ as used in Eq. (22) (the bare phonon susceptibility) and χ as used in Eq. (4) (the full linear susceptibility including polaritonic effects) could be clearer. A brief sentence distinguishing these two uses of χ would help.
  4. Appendix C.1, Eq. (C7): The definition ε^F_2D = −M_eff/(ω(ω+iγ)) is introduced without explicit derivation of the prefactor. A one-line derivation connecting M_eff to the dielectric function would be helpful.
  5. Fig. 5(a): The pump field values E_p = 10 kV/cm and 15 kV/cm are used, but the dimensionless pump strength φ_p = 2edA_p/(ℏc) corresponding to these values is not stated. Providing φ_p would help readers assess the perturbative regime.
  6. Sec. IV.B: The out-of-plane dielectric ε_z = 3.0 is stated as 'a good approximation to hBN in this frequency range' without a reference. A citation or brief justification would strengthen this claim.
  7. References: Several 2026 references (e.g., Refs. 50, 56, 80, 81, 86, 105, 132) appear to be preprints or very recent publications. The authors should verify that final published versions are cited where available.
  8. Sec. II.B, Eq. (9): The '···' representing higher-order nested diagrams is mentioned but the truncation criterion is not quantified. A brief statement of the small parameter controlling the expansion (e.g., ξ² for graphene, E²_p λ²_1 for hBN) would help readers assess convergence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments all identify legitimate limitations of our presentation. We address each below and describe revisions we will make.

read point-by-point responses

  1. Referee: Sec. V, Eq. (32) and Fig. 5(b): The reflectivity R = |(1−√ε_eff)/(1+√ε_eff)|² for a semi-infinite gain medium is ambiguous when χ_eff has negative spectral weight. The sign of Im[√ε_eff] must be chosen consistently with outgoing-wave conditions. The authors acknowledge but do not resolve this.

    Authors: The referee is correct that the standard Fresnel formula for a semi-infinite gain medium is not on firm footing in the frequency ranges where ε_eff has negative imaginary part (i.e., where the Floquet replica at −ω_J + 2ω_p gives gain). We agree that the quantitative reflectivity values in Fig. 5(b) are not reliable in those specific frequency windows. We will make the following revisions: (1) We will explicitly state that the optical conductivity predictions in Fig. 5(a) are unaffected by this issue, since σ_eff is computed directly from ε_eff without invoking a reflection boundary condition. (2) We will add a shaded region or explicit annotation in Fig. 5(b) marking the frequency ranges where ε_eff corresponds to a gain medium, noting that the reflectivity there requires the Fresnel-Floquet approach (Ref. 128) for a quantitative treatment. (3) We will note that for the reflectivity peaks associated with the normal (non-gain) Floquet modes — specifically the fundamental mode and the replica at +ω_J + 2ω_p — the standard Fresnel formula remains valid, and those predictions are robust. (4) We will add a sentence clarifying that R > 1 in the gain windows is not a quantitative prediction of our current treatment. We note that a full resolution via the Fresnel-Floquet formalism, which self-consistently treats the spatial profile of both pump and probe fields, is beyond the scope of this paper but is a natural follow-up. revision: partial

  2. Referee: Sec. V: The uniform-pump assumption (infinite penetration depth) is stated but its quantitative impact is not bounded. For layered superconductors the c-axis penetration depth at THz frequencies can be comparable to or smaller than the probe skin depth.

    Authors: We agree that this assumption deserves quantitative justification. We will add an estimate to Sec. V. For the parameters used in Fig. 5 (ω_J = 0.5 THz, ε_∞ = 4.5), the Josephson penetration depth is λ_J = c/(ω_J √ε_∞) ≈ 28 μm. When the pump frequency ω_p is near or below ω_J, the pump field is evanescent within the superconductor with penetration depth ~λ_J, which is comparable to the probe skin depth — in this regime the uniform-pump approximation is not self-consistent. However, when ω_p > ω_J (as is the case for the results in Fig. 5, where ω_p = ω_J and the pump is at the plasma edge), the pump penetrates more deeply. More generally, for pump frequencies well above ω_J, the pump propagates into the bulk with a penetration depth much larger than λ_J, while the probe (at frequencies near ω_J) still has skin depth ~λ_J; in this regime the uniform-pump approximation is well justified. We will state this explicitly and note that the results in Fig. 5, where ω_p = ω_J, represent a borderline case where corrections from finite pump penetration depth are expected to be moderate but not negligible. We will also cite Ref. 128 (Fresnel-Floquet) and Ref. 127 more specifically, noting that a self-consistent treatment would modify the effective pump strength as a function of depth and could shift the Floquet replica positions slightly. revision: partial

  3. Referee: Sec. III.B, Eq. (17): The weak-probe linearization may break down once the unstable mode grows to amplitudes comparable to the probe. The authors should clarify the timescale on which the linear-response treatment remains valid.

    Authors: This is a valid concern. We will add a quantitative discussion of the validity timescale. For the parameters in Fig. 2 (ω_p = 30 THz, γ = 1.5 THz, E_p = 600 kV/cm, corresponding to ξ ≈ 1.04), the maximum parametric growth rate at zero detuning (Q = 0) is Im[ω] ≈ ω_p κ − γ/2, where κ = 3ξ²/32 ≈ 0.10. This gives Im[ω] ≈ 3.0 − 0.75 ≈ 2.3 THz, corresponding to an exponential growth time τ_growth ≈ 1/(2π × 2.3 THz) ≈ 70 fs. Typical near-field probe pulses in the mid-infrared have durations of 100–500 fs, so the unstable modes can grow by a factor of e^1 to e^7 during the probe pulse, meaning the weak-probe linearization may indeed break down for the unstable modes at this pump strength. However, we note the following: (1) The frequency-domain linear response function Im[R_p] in Fig. 2(c) is mathematically well-defined regardless of instability — it is the retarded Green function of the linearized system, and its poles in the upper half-plane signal the instability. (2) For pump fields closer to threshold (ξ → ξ_c), the growth rate vanishes and the observable window extends to arbitrarily long times. (3) For few-cycle probe pulses (~50 fs), the linear-response features remain observable even at E_p = 600 kV/cm before saturation sets in. (4) The non-gain features of Im[R_p] — including the negative spectral weight of the n=1 replica away from the crossing point — are not affected by the instability timescale, since they do not involve exponentially growing modes. We will add this discussion to Sec. III.B and note that experimental observation of the flat-band/exceptional-point features is most favorable with pump strengths near threshold and short probe pulses. revision: partial

Circularity Check

0 steps flagged

No significant circularity: the effective susceptibility is constructed from independently measurable nonlinear optical coefficients and an external pump field, with predictions following algebraically from those inputs.

full rationale

The paper's central derivation chain is self-contained and non-circular. The effective susceptibility χ_eff (Eqs. 9, 14, 25, 32) is constructed from (a) the material's linear susceptibility χ, (b) nonlinear optical coefficients (χ^(3) for graphene in Eq. 11, λ₀ and λ₁ for hBN from Eq. 21, and the Josephson cosine nonlinearity in Eq. 29), and (c) the pump field E_p treated as an external classical parameter. These inputs are independently measurable: χ^(3) is derived from the Drude model (Eq. 11), λ₀ and λ₁ are stated to come from experiments and DFT (Ref. 122), and the Josephson nonlinearity follows from the sine-Gordon Lagrangian (Eq. 27). The predictions—flat band locations (Eq. 17), exceptional point positions (|Q|=κ), threshold fields (ξ_c), and Floquet replica positions—follow algebraically from substituting χ_eff into the polariton condition ε=0 or ε_2D=0. No prediction is fed back as an input. The perturbative truncation is checked against numerical Floquet diagonalization (Appendix C, cutoff N=5), providing an internal consistency check that is not a circularity. Self-citations (e.g., Refs. 42, 54, 74, 111) provide prior derivations of nonlinear coefficients but are not load-bearing for the logical structure: the coefficients could in principle be measured experimentally. The uniform-pump assumption (Sec. V) is a modeling approximation, not a circularity. The Fresnel gain-medium subtlety for the superconductor reflectivity is acknowledged but unresolved; this is a correctness risk, not a circular step. The framework is a standard perturbative expansion of nonlinear response in a classical pump background, internally consistent within its stated approximations.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The paper introduces no new particles, forces, dimensions, or postulated entities. All physical quantities (susceptibilities, Drude weights, phonon frequencies, Josephson coupling) are standard material parameters. The 'effective susceptibility' χ_eff is a derived quantity, not a postulated one. The framework is constructed from Maxwell's equations and known nonlinear optical response theory. The main assumptions are domain-specific approximations (classical saddle point, uniform pump, Drude model) rather than new postulates.

free parameters (7)
  • Damping rate γ (graphene) = 1.5 THz
    Motivated by experiments (Refs 11, 58), used in all graphene calculations. Not fitted to the Floquet spectra but taken from equilibrium measurements.
  • Damping rate γ (hBN) = 0.1 THz
    Motivated by isotopically pure hBN measurements (Refs 20, 22).
  • Damping rate γ (Josephson) = 0.02 THz
    Added ad hoc to model quasi-particle dissipation in Eq. 32.
  • Pump field E_p = Various (600 kV/cm to 8 MV/cm)
    External control parameter, not fitted.
  • Pump frequency ω_p = Various (30 THz to 85 THz)
    External control parameter.
  • Background dielectric ε∞ (superconductor) = 4.5
    From high-energy electronic polarizations beyond Josephson dynamics, motivated by La2-xBaxCuO4.
  • Out-of-plane dielectric εz (hBN flake) = 3.0
    Stated as 'a good approximation to hBN in this frequency range.'
axioms (5)
  • domain assumption The classical saddle point of the Keldysh path integral is an extremely good approximation for polariton dynamics under strong pump (quantum/thermal fluctuations are negligible except for exciton polaritons).
    Stated in Sec. II.B: 'Since its classical saddle points are extremely good approximations in most experimental situations.' This justifies using equation-of-motion methods instead of the full Keldysh formalism in the main text.
  • domain assumption The pump field is a fixed classical background that does not receive self-consistent feedback from the material response.
    Stated in Sec. V: the pump 'is defined as the total incident field from the pump laser that has penetrated into the sample, while any feedback effect from the sample itself has already been included in A_p.' This uniform-pump assumption is used throughout.
  • domain assumption The Drude model adequately describes the 2D optical conductivity of graphene at frequencies ω ≪ 2εF and momenta q ≪ ω/vF.
    Invoked in Sec. III to obtain the bare plasmon propagator (Eq. 10). The interband contribution is neglected.
  • domain assumption The cubic phonon nonlinearity λ₁ in hBN (Eq. 21) captures the essential nonlinear phononic response.
    The Lagrangian (Eq. 21) includes only the lowest-order nonlinear term allowed by symmetry. Higher-order phonon nonlinearities are not considered.
  • ad hoc to paper The weak-probe linearization remains valid even in the presence of parametric instability.
    The authors note that unstable growth 'is expected to be cut off by nonlinear saturation that is beyond the present weak-probe treatment' (Sec. III.B), but the linearized theory is used to predict threshold fields and band structures near instability.

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Reference graph

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