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Cavity-mediated coupling between local and nonlocal modes in Landau polaritons

  • Sae R. Endo ORCID logo , Dasom Kim ORCID logo , Shuang Liang ORCID logo , Geon Lee ORCID logo , Sunghwan Kim ORCID logo , Alan Covarrubias-Morales , Minah Seo ORCID logo , Michael J. Manfra ORCID logo , Dukhyung Lee ORCID logo EMAIL logo , Motoaki Bamba ORCID logo and Junichiro Kono ORCID logo EMAIL logo
Published/Copyright: November 18, 2025
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Nanophotonics
From the journal Nanophotonics Volume 14 Issue 25

Abstract

The multimode ultrastrong coupling (USC) regime has emerged as a novel platform for accessing previously inaccessible phenomena in cavity quantum electrodynamics. Of particular interest are cavity-mediated correlations between local and nonlocal excitations, or equivalently, between modes at zero and finite in-plane momentum, which offer new opportunities for controlling light–matter interactions across space. However, direct experimental evidence of such interactions has remained elusive. Here, we demonstrate nonlocal multimode coupling in a Landau polariton system, where cavity photons simultaneously interact with the zero-momentum cyclotron resonance and finite-momentum magnetoplasmons of GaAs two-dimensional electron gas. Our slot cavities, with their subwavelength mode volumes, supply in-plane momentum components that enable the excitation of finite-momentum matter modes. Terahertz time-domain magnetospectroscopy measurements reveal a clear splitting of the upper-polariton branch, arising from hybridization between magnetoplasmon modes and the cavity–cyclotron-resonance hybrids. Extracted coupling strengths confirm USC of the cyclotron resonance and strong coupling of the magnetoplasmon modes to the cavity field, respectively. The experimental results are well captured by the multimode Hopfield model and finite-element simulations. These findings establish a pathway for engineering multimode light–matter interactions involving zero- and finite-momentum matter modes in the USC regime.

Keywords: multimode coupling; magnetoplasmons; Landau polaritons

1 Introduction

Vacuum–matter interactions in cavities are currently explored in diverse systems both for controlling material properties without any external driving field and for applications in quantum information technologies. The ultrastrong coupling (USC) of vacuum–matter arises when the vacuum Rabi frequency becomes a significant fraction of the bare frequencies of vacuum and matter at zero detuning (ω 0), i.e., g/ω 0 ≥ 0.1, where g is the coupling strength [1], [2]. This regime gives rise to exotic phenomena via the non-negligible contribution of the counter-rotating terms in the Hamiltonian, such as quantum squeezing in the ground state [3], [4], the vacuum Bloch–Siegert shift [5], the Dicke superradiant phase transition [6], [7], cavity-mediated superconductivity [8], and modifications of the quantum Hall effect [9], [10]. The USC regime has been realized in Landau polaritons [5], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], phonon polaritons [22], [23], [24], [25], and magnon polaritons [7], [26], [27]. Among these systems, Landau polaritons offer exceptional tunability through externally applied magnetic fields.

Recently, attention has expanded toward the multimode USC regime, where a matter excitation ultrastrongly couples simultaneously with multiple cavity modes, or conversely, a single cavity mode interacts with multiple matter excitations. This regime is not only a natural extension of single-mode USC but also introduces fundamentally new physics due to the nontrivial interplay between cavity modes or between matter excitations [20], [25], [28], [29], [30], [31]. For example, correlations between bare excitations emerge due to the significant contribution of the counter-rotating terms, which do not exist in the absence of ultrastrong light–matter interactions. Very recently, matter–mediated photon–photon correlations [31] and cavity-mediated superthermal phonon–phonon correlations [25] have been realized by leveraging the large dipole moment of Landau-level transitions and the subwavelength light confinement of metamaterial cavities, respectively.

Furthermore, nonlocal, i.e., k ≠ 0 aspects of USC are also drawing increasing interest [32], where k is the in-plane wave vector of matter excitation. In particular, the possibility of mediating long-range nonlocal correlations through the vacuum field opens a new direction in controlling collective matter excitations across space. Therefore, it would be particularly interesting to explore hybrid quantum states that bridge local and nonlocal matter excitations – coupling between k = 0 and finite k modes mediated by a cavity mode. While recent studies have explored multimode USC involving finite k excitations [19], [30], [32], [33], direct observation of mode-resolved anticrossings between individual modes has so far remained elusive. To further deepen our understanding of multimode USC, it is desirable to investigate systems where such interactions can be more clearly resolved.

Here, we report spectroscopic evidence for coupling between a local mode and a nonlocal mode, through simultaneous USC and strong coupling (SC) with a single cavity mode, in a Landau polariton system. We fabricated an array of slot cavities on a two-dimensional electron gas (2DEG), and its microstructure allowed cavity photons to couple with finite-k plasmon modes of the 2DEG by providing in-plane momentum. In a magnetic field, we experimentally observed SC of magnetoplasmon (MP) modes and the cavity mode, as well as USC between the cyclotron resonance (CR) of the 2DEG and the cavity mode. This multimode hybridization resulted in a clear splitting of the upper-polariton (UP) branch, which can be described by the multimode Hopfield model. Simulated transmission spectra using the finite-element method exhibited similar behavior, confirming the effect of finite-k MP modes. Our findings highlight a new platform where local and finite-k matter excitations can be coupled through a cavity mode, offering a perspective on momentum-resolved multimode interactions in the USC regime.

2 Results

Providing in-plane momentum for light–matter interactions has long been achieved using periodic lattice structures [34], [35], [36], [37], [38], [39], via patterning 2DEGs [17], [19], [33], [40], [41], [42], [43], [44], [45] or by confining light in a small mode volume [32], [46]. To implement this mechanism, we fabricated an array of slot cavities on a GaAs 2DEG using standard photolithography and lift-off techniques. The 2DEG was present about 45 nm below the slot, ensuring strong overlap with the electromagnetic fields of the cavity mode; see Figure 1(a). The electron density and mobility were n e = 3.6 × 1011 cm−2 and μ e = 1.2 × 106 cm2/(V s), respectively, obtained through van der Pauw measurements. The electron cyclotron mass was found to be m* = 0.076m 0 through terahertz (THz) magnetospectroscopy measurements, where m 0 is the free electron mass in vacuum. The loop length of slots was tuned to have a resonance frequency in the THz frequency range. The bare cavity frequency, ω 0, was at 0.925 THz. The width of the slot, d, was 4 μm to confine light inside the cavity, providing cavity photons with finite in-plane momentum. Although the periodic structure also provided finite in-plane momentum, its frequency was too low compared with the cavity mode. Therefore, we consider only the k components generated by the individual slots.

Figure 1: A multimode coupled system involving local and nonlocal matter excitations, and a cavity mode. (a) Schematic of the GaAs 2DEG–slot system. 2DEG: two-dimensional electron gas. (b) Dispersion relations of plasmons at 0 T (black) and magnetoplasmons at 1.25 T (red), 2.18 T (blue), and 2.51 T (green), corresponding to zero-detuning magnetic fields of MP1, MP3, and CR to ω 0. ω 0: bare cavity frequency. (c), (d) Illustration of the multimode coupling of CR, MP1, and MP3 to the cavity mode. (c) Uncoupled case. (d) Coupled case. The multimode coupling leads to the formation of one lower polariton (LP) and three upper polaritons (UP1, UP2, and UP3).
Figure 1:

A multimode coupled system involving local and nonlocal matter excitations, and a cavity mode. (a) Schematic of the GaAs 2DEG–slot system. 2DEG: two-dimensional electron gas. (b) Dispersion relations of plasmons at 0 T (black) and magnetoplasmons at 1.25 T (red), 2.18 T (blue), and 2.51 T (green), corresponding to zero-detuning magnetic fields of MP1, MP3, and CR to ω 0. ω 0: bare cavity frequency. (c), (d) Illustration of the multimode coupling of CR, MP1, and MP3 to the cavity mode. (c) Uncoupled case. (d) Coupled case. The multimode coupling leads to the formation of one lower polariton (LP) and three upper polaritons (UP1, UP2, and UP3).

As a finite-k matter excitation, we had the plasma oscillations of the 2DEG. The 2D plasma frequency in the long-wavelength limit is given by [47]

(1) ω p ( k ) = k e 2 n e 2 m * ε 0 ε r ,

where ɛ 0, ɛ r, and e are the vacuum permittivity, the effective dielectric constant of the surrounding medium (ɛ r = 6.98), and the elementary charge, respectively; see Figure 1(b) (black solid line). The slot can provide the in-plane momentum k = nπ/d, where n is restricted to odd integers due to the symmetry of the electric field [19], [41]. While higher-order terms are possible, they are typically weaker and more difficult to observe. Based on our experimental results, we focus on the lowest two components, corresponding to the n = 1 and 3 modes. However, even these modes alone do not provide sufficient momentum to match the plasma resonance to the cavity frequency ω 0, resulting in finite detuning, i.e., there is no intersection between the cavity mode and the n = 1 and 3 MP modes at k ≤ 3π/d.

Another matter excitation in our system was the CR of the 2DEG, arising from transitions between adjacent Landau levels at k = 0, which can be tuned by a perpendicular magnetic field, B, through the cyclotron frequency ω c = eB/m*. In the presence of B, the 2D plasma excitations evolve into MPs, whose dispersion is given by [48], [49]

(2) ω MP 2 ( k ) = ω p 2 ( k ) + ω c 2 .

As shown in Figure 1(b) (colored lines), the entire dispersion curve blue-shifts as the magnetic field increases. The colored curves represent the MP dispersion at various magnetic field strengths, with the red (at 1.25 T), blue (at 2.18 T), and green lines (at 2.51 T) indicating the zero-detuning conditions to the cavity mode. These field-tuned resonances can satisfy the momentum matching condition for n = 3 mode (MP3), n = 1 mode (MP1), and CR, respectively, enabling the exploration of hybridization of k = 0 and k ≠ 0 matter excitations through the cavity mode.

Figure 1(c) plots the bare matter excitation frequencies and the cavity frequency as a function of B. In this uncoupled situation, the CR and MP modes cross the cavity mode, showing three zero-detuning points, as expected from Figure 1(b). However, when the CR and MP modes are coupled with the cavity mode, they are expected to exhibit an anticrossing behavior at each zero-detuning point; see Figure 1(d). The multimode coupling gives rise to one lower polariton (LP) and three upper polariton branches (UP1, UP2, and UP3).

To confirm our model, we performed THz time-domain magnetospectroscopy measurements on the coupled system. THz pulses were generated and detected using InGaAs photoconductive antennas fiber-coupled to an Er-doped fiber laser with a center wavelength of 1,550 nm and repetition rate of 80 MHz. The electric field strength was sufficiently low to prevent any nonlinear effects. Measurements were conducted at 1.6 K in a magneto-optical cryostat operating in the Faraday geometry. Time-domain signals of the samples and reference (bare GaAs 2DEG) were Fourier transformed to obtain transmission spectra from 0.2 THz to 1.6 THz. Further details of the experimental setup are provided in the Supplementary Materials. Figure 2(a) shows a color map of transmittance as a function of frequency and magnetic field. At around 2.50 T, an anticrossing behavior is observed, arising from the USC between the cavity mode and the CR of the 2DEG at k = 0. At around 1.25 T, an additional splitting of the UP branch is clearly observed. This is evidence of SC of the MP3 with the cavity mode having k = 3π/d, induced by the confinement in the 4 μm slot. The split upper polaritons can be interpreted as UP2 and UP3 in Figure 1(d). UP1 is absent in the experimental data due to the THz spectra reflecting the cavity-like components of the polaritons, as well as the proximity of UP1 to the CR at finite fields.

Figure 2: Spectroscopic evidence for the multimode coupling. (a) Experimental color map of transmittance as a function of frequency and magnetic field. The upper polariton is split into two parts, corresponding to UP2 and UP3 in Figure 1(d). (b) Theoretical color map of transmittance based on the multimode Hopfield model.
Figure 2:

Spectroscopic evidence for the multimode coupling. (a) Experimental color map of transmittance as a function of frequency and magnetic field. The upper polariton is split into two parts, corresponding to UP2 and UP3 in Figure 1(d). (b) Theoretical color map of transmittance based on the multimode Hopfield model.

To theoretically explain the coupling between a cavity mode, CR, and MP modes, we introduce a multimode Hopfield model. By adding the MP modes to the full Hamiltonian of Landau polaritons presented in ref. [5] while considering only a single cavity mode, the multimode Hamiltonian can be written as

(3) H ̂ ℏ = ∑ ξ = ± ω 0 a ̂ ξ † a ̂ ξ + ω c b ̂ † b ̂ + ∑ n ω MP n c ̂ n † c ̂ n + i g ̄ b ̂ † + ∑ n g ̄ n c ̂ n † a ̂ + + a ̂ − †   − g ̄ b ̂ + ∑ n g ̄ n c ̂ n a ̂ − + a ̂ + † + D a ̂ − + a ̂ + † a ̂ + + a ̂ − † ,

where ℏ is the reduced Planck constant, a ̂ † ( a ̂ ) in the first term is the creation (annihilation) operator for the cavity photons of CR-active (ξ = +) and CR-inactive (ξ = −) circularly polarized modes, the second term represents the energy of the collective CR excitation of the 2DEG, b ̂ † ( b ̂ ) being its creation (annihilation) operator, MP modes of different orders n = 1, 3 are introduced in the third term, where c ̂ n † ( c ̂ n ) and ω MP n are the corresponding creation (annihilation) operator and resonance angular frequency, and the fourth term is the light–matter interaction term considering the coupling of CR and all MP modes with the single cavity mode.

Here, g ̄ and g ̄ n are the coupling strengths of CR and the n-th MP mode with the cavity mode, respectively. Using the zero-detuning coupling strengths, g and g n , which are independent of the external B, g ̄ and g ̄ n can be written as

(4) g ̄ = ω c ω 0 g ,

(5) g ̄ n = ω MP n ω 0 g n .

The coefficient D of the diamagnetic term is

(6) D = g ̄ 2 ω c + ∑ n g ̄ n 2 ω MP n = g 2 + ∑ n g n 2 ω 0 .

The coupling strengths appear in the relative permittivity as

(7) ε ( ω ) = ε bg − ω pl 2 ω ω − ω c + i τ − ∑ n ω pl n 2 ω ω − ω MP n + i τ n ,

where the plasma frequencies of CR and n-th MP mode, ω pl and ω pl n , have relations with the B-independent coupling strengths as

(8) g 2 = ω pl 2 d QW ε bg L eff ,

(9) g n 2 = ω pl n 2 d QW ε bg L eff .

Here, ɛ bg = ɛ r is the background relative permittivity, τ (τ n ) is the lifetime of CR (MP n ), d QW is the thickness of the quantum well, and L eff is the effective cavity length. Using the above relative permittivity, the transmission spectra can be obtained through the transfer-matrix method.

On the other hand, the polariton dispersion can be calculated from the B-dependent coupling strengths. The Heisenberg equations for the operators a ̂ + , a ̂ − † , b, and c n form a closed set describing the CR-active polariton modes, while those of a ̂ − , a ̂ + † , b †, and c n † describe the CR-inactive modes. The CR-active polariton annihilation operator is given by the Bogoliubov transformation, p ̂ + = w a ̂ + + y a ̂ − † + x b ̂ + ∑ n x n c ̂ n . In the same manner as performed by Ciuti et al. [3], the polariton frequencies and the coefficients are determined by solving the eigenvalue problem of the following matrix:

(10) ω 0 + D − D i g ̄ i g ̄ 1 … D − ω 0 − D i g ̄ i g ̄ 1 … − i g ̄ i g ̄ ω c 0 … − i g ̄ 1 i g ̄ 1 0 ω MP 1 … ⋮ ⋮ ⋮ ⋮ ⋱ .

Theoretical transmission spectra for the multimode Landau polariton system, calculated using the transfer-matrix method with the relative permittivity ɛ(ω) derived in Eq. (7), are shown in Figure 2(b). The resonator structure is simplified to a Fabry–Pérot cavity; details of the calculation setup are provided in the Supplementary Materials. The splitting in the UP branch is observed and captures the key feature of the experimental data. We emphasize that the MP1 interacts with the cavity–cyclotron-resonance hybrids at only high frequencies and cannot account for the splitting of the upper polariton at around 1 THz without including the MP3 in the model; see Supplementary Materials. The discrepancy in linewidth compared to the experimental spectra is associated with the difference in cavity geometry, with the slots having a reduced Q-factor compared to an ideal Fabry–Pérot cavity. From fitting to the experimental peak frequencies, we obtained the normalized coupling strengths of g/ω 0 = 0.18 and g 1/ω 0 = g 3/ω 0 = 0.084, confirming the USC of the CR and the SC of MP modes with the cavity mode. Figure 3 shows the calculated polariton frequencies (black solid lines) overlaid with the peak frequencies extracted from the experiments (red dots). Theoretical predictions match the experimental results well. We note that the assumption g 1 = g 3 is generally not valid but is enough to explain the experimental behavior.

Figure 3: Theoretical analysis of the multimode coupled system. Theoretical polariton frequencies based on the multimode Hopfield model. Red dots are extracted peak frequencies from experiment.
Figure 3:

Theoretical analysis of the multimode coupled system. Theoretical polariton frequencies based on the multimode Hopfield model. Red dots are extracted peak frequencies from experiment.

We further validated our experimental observation and theoretical model using three-dimensional finite-element simulations in COMSOL. Figure 4 shows a color map of transmittance simulated with m* = 0.070m 0 and ɛ r = 9.77. Compared with the experimental transmittance in Figure 2(a), the UP branch is split into three parts, separated not only by MP3 but also by MP5, resulting in UP4. Such influence of the MP5 could not be observed experimentally due to the weak magnitude. Incorporating the MP5 contribution to the multimode Hopfield model, theoretical fits to the polariton frequencies are calculated, shown in black dashed lines in Figure 4; see Supplementary Materials for more detail. The obtained coupling strengths are g/ω 0 = 0.13 and g 1/ω 0 = g 3/ω 0 = g 5/ω 0 = 0.062. The slight deviation from the experimentally extracted values likely originates from uncertainties in the sample parameters and the strong effect of the MP5 mode. Overall, the simulation is consistent with both the experimental and theoretical analysis of the multimode coupling.

Figure 4: Finite-element simulation of the multimode coupled system. Simulated color map of transmittance as a function of frequency and magnetic field using COMSOL. The black dashed lines are theoretical fits to the polariton frequencies based on the multimode Hopfield model. An additional upper polariton mode, denoted UP4, appears due to the influence of the MP5 mode.
Figure 4:

Finite-element simulation of the multimode coupled system. Simulated color map of transmittance as a function of frequency and magnetic field using COMSOL. The black dashed lines are theoretical fits to the polariton frequencies based on the multimode Hopfield model. An additional upper polariton mode, denoted UP4, appears due to the influence of the MP5 mode.

3 Discussion

In conclusion, we experimentally explored the coupling between two distinct matter modes mediated by a cavity mode in an ultrastrongly coupled Landau polariton system. Owing to the small-mode-volume slots, cavity photons could interact with both the CR at k = 0 and MP modes at finite k. Such coexistence of CR and MP excitations has been discussed in previous studies using 2DEG stripes and periodic metasurfaces [20], [30], [33]. However, the multimode interaction demonstrated here could not be accessed due to the excitation geometry and frequency mismatch. In this work, the MP modes affected the Landau polariton through multimode hybridization and led to a splitting of the UP branch. From fitting, we obtained a normalized coupling strength of g/ω 0 = 0.18 for CR, and g 1/ω 0 = g 3/ω 0 = 0.084 for MP modes with an in-plane wave vector of k = π/d and k = 3π/d, showing the USC of CR and SC of MP modes to the cavity mode. The experimental results were well explained by the extended multimode Hopfield model and three-dimensional finite-element simulations. Our work establishes a foundation for controlling cavity-mediated nonlocal correlations between matter excitations in the USC regime, unlocking new possibilities for tailoring light–matter interactions.

Corresponding authors: Dukhyung Lee, School of Applied and Engineering Physics, Mohammed VI Polytechnic University (UM6P), Ben Guerir 43150, Morocco, E-mail: ; and Junichiro Kono, Smalley–Curl Institute, Rice University, Houston, TX 77005, USA; Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA; Rice Advanced Materials Institute, Rice University, Houston, TX 77005, USA; Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA; and Department of Materials Science and NanoEngineering, Rice University, Houston, TX 77005, USA, E-mail: 

Funding source: Welch Foundation

Award Identifier / Grant number: C-1509

Funding source: Army Research Office

Award Identifier / Grant number: W911NF2110157

Funding source: W. M. Keck Foundation

Award Identifier / Grant number: 995764

Funding source: Research Foundation for Opto-Science and Technology

Funding source: Gordon and Betty Moore Foundation

Award Identifier / Grant number: 11520

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP24K21526

Award Identifier / Grant number: JP25K00012

Award Identifier / Grant number: JP25K01691

Award Identifier / Grant number: JP25K01694

Award Identifier / Grant number: JPJSJRP20221202

Acknowledgments

SRE acknowledges support from TOMODACHI–Dow Women's STEM Leadership and Research Program. DK acknowledges Wen-Hua Wu, Gustavo Rodriguez Barrios, and Shuying Chen for technical assistance for experiments, and Fuyang Tay for fruitful discussion. DL acknowledges UM6P for providing computational resources (HPC-TOUBKAL).

  1. Research funding: DK, AC-M, and JK acknowledge support from the U.S. Army Research Office (through Award No. W911NF2110157), the W. M. Keck Foundation (through Award No. 995764), the Gordon and Betty Moore Foundation (through Grant No. 11520), and the Robert A. Welch Foundation (through Grant No. C-1509). MB acknowledges support from the Research Foundation for Opto-Science and Technology and from the Japan Society for the Promotion of Science (JSPS) Grant No. JPJSJRP20221202 and KAKENHI Grant No. JP24K21526, JP25K00012, JP25K01691, and JP25K01694.

  2. Author contributions: JK supervised the project. SRE, DK, and AC-M performed all THz measurements. SRE and DK analyzed the experimental data under the guidance of JK. DK built the THz magnetospectroscopy setup. SL grew the quantum well under the guidance of MJM. GL, SK, and DK designed and fabricated the slot cavities under the guidance of MS. MB developed the theoretical model. DL performed numerical simulations and analyzed the experimental data. SRE, DK, MB, and JK wrote the manuscript. All authors discussed the results and commented on the manuscript. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.

  3. Conflict of interest: Authors state no conflicts of interest.

  4. Data availability: The datasets generated and/or analyzed during the current study are available from the corresponding authors upon reasonable request.

References

[1] P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, “Ultrastrong coupling regimes of light-matter interaction,” Rev. Mod. Phys., vol. 91, no. 2, p. 025005, 2019, https://doi.org/10.1103/revmodphys.91.025005.Search in Google Scholar

[2] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” Nat. Rev. Phys., vol. 1, no. 1, pp. 19–40, 2019, https://doi.org/10.1038/s42254-018-0006-2.Search in Google Scholar

[3] C. Ciuti, G. Bastard, and I. Carusotto, “Quantum vacuum properties of the intersubband cavity polariton field,” Phys. Rev. B, vol. 72, no. 11, p. 115303, 2005, https://doi.org/10.1103/physrevb.72.115303.Search in Google Scholar

[4] S. De Liberato, “Virtual photons in the ground state of a dissipative system,” Nat. Commun., vol. 8, no. 1, p. 1465, 2017, https://doi.org/10.1038/s41467-017-01504-5.Search in Google Scholar PubMed PubMed Central

[5] X. Li et al.., “Vacuum Bloch–Siegert shift in Landau polaritons with ultra-high cooperativity,” Nat. Photonics, vol. 12, no. 6, pp. 324–329, 2018, https://doi.org/10.1038/s41566-018-0153-0.Search in Google Scholar

[6] K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: the Dicke maser model,” Ann. Phys., vol. 76, no. 2, pp. 360–404, 1973, https://doi.org/10.1016/0003-4916(73)90039-0.Search in Google Scholar

[7] D. Kim et al.., “Observation of the magnonic Dicke superradiant phase transition,” Sci. Adv., vol. 11, no. 14, p. eadt1691, 2025, https://doi.org/10.1126/sciadv.adt1691.Search in Google Scholar PubMed

[8] F. Schlawin, A. Cavalleri, and D. Jaksch, “Cavity-mediated electron-photon superconductivity,” Phys. Rev. Lett., vol. 122, no. 13, p. 133602, 2019, https://doi.org/10.1103/physrevlett.122.133602.Search in Google Scholar

[9] F. Appugliese et al.., “Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect,” Science, vol. 375, no. 6584, pp. 1030–1034, 2022, https://doi.org/10.1126/science.abl5818.Search in Google Scholar PubMed

[10] J. Enkner et al.., “Tunable vacuum-field control of fractional and integer quantum Hall phases,” Nature, vol. 641, no. 8064, pp. 884–889, 2025, https://doi.org/10.1038/s41586-025-08894-3.Search in Google Scholar PubMed PubMed Central

[11] D. Hagenmüller, S. De Liberato, and C. Ciuti, “Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor,” Phys. Rev. B, vol. 81, no. 23, p. 235303, 2010, https://doi.org/10.1103/physrevb.81.235303.Search in Google Scholar

[12] G. Scalari et al.., “Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial,” Science, vol. 335, no. 6074, pp. 1323–1326, 2012, https://doi.org/10.1126/science.1216022.Search in Google Scholar PubMed

[13] C. Maissen et al.., “Ultrastrong coupling in the near field of complementary split-ring resonators,” Phys. Rev. B, vol. 90, no. 20, p. 205309, 2014, https://doi.org/10.1103/physrevb.90.205309.Search in Google Scholar

[14] Q. Zhang et al.., “Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons,” Nat. Phys., vol. 12, no. 11, pp. 1005–1011, 2016, https://doi.org/10.1038/nphys3850.Search in Google Scholar

[15] A. Bayer et al.., “Terahertz light–matter interaction beyond unity coupling strength,” Nano Lett., vol. 17, no. 10, pp. 6340–6344, 2017, https://doi.org/10.1021/acs.nanolett.7b03103.Search in Google Scholar PubMed

[16] J. Keller et al.., “Few-electron ultrastrong light-matter coupling at 300 GHz with nanogap hybrid LC microcavities,” Nano Lett., vol. 17, no. 12, pp. 7410–7415, 2017, https://doi.org/10.1021/acs.nanolett.7b03228.Search in Google Scholar PubMed

[17] G. L. Paravicini-Bagliani et al.., “Magneto-transport controlled by Landau polariton states,” Nat. Phys., vol. 15, no. 2, pp. 186–190, 2019, https://doi.org/10.1038/s41567-018-0346-y.Search in Google Scholar

[18] J. Keller et al.., “Landau polaritons in highly nonparabolic two-dimensional gases in the ultrastrong coupling regime,” Phys. Rev. B, vol. 101, no. 7, p. 075301, 2020, https://doi.org/10.1103/physrevb.101.075301.Search in Google Scholar

[19] K. Kuroyama, J. Kwoen, Y. Arakawa, and K. Hirakawa, “Coherent interaction of a few-electron quantum dot with a terahertz optical resonator,” Phys. Rev. Lett., vol. 132, no. 6, p. 066901, 2024, https://doi.org/10.1103/physrevlett.132.066901.Search in Google Scholar PubMed

[20] J. Mornhinweg et al.., “Sculpting ultrastrong light–matter coupling through spatial matter structuring,” Nanophotonics, vol. 13, no. 10, pp. 1909–1915, 2024, https://doi.org/10.1515/nanoph-2023-0604.Search in Google Scholar PubMed PubMed Central

[21] G. Scalari, et al.., “Ultrastrong light-matter coupling at terahertz frequencies with split ring resonators and inter-Landau level transitions,” J. Appl. Phys., vol. 113, no. 13, p. 136510, 2013.10.1063/1.4795543Search in Google Scholar

[22] H. S. Kim, N. Y. Ha, J.-Y. Park, S. Lee, D.-S. Kim, and Y. H. Ahn, “Phonon-polaritons in lead halide perovskite film hybridized with THz metamaterials,” Nano Lett., vol. 20, no. 9, pp. 6690–6696, 2020, https://doi.org/10.1021/acs.nanolett.0c02572.Search in Google Scholar PubMed

[23] D. Yoo et al.., “Ultrastrong plasmon–phonon coupling via epsilon-near-zero nanocavities,” Nat. Photonics, vol. 15, no. 2, pp. 125–130, 2021, https://doi.org/10.1038/s41566-020-00731-5.Search in Google Scholar

[24] Y. Roh, et al.., “Ultrastrong coupling enhancement with squeezed mode volume in terahertz nanoslots,” Nano Lett., vol. 23, no. 15, pp. 7086–7091, 2023. https://doi.org/10.1021/acs.nanolett.3c01913.Search in Google Scholar PubMed

[25] D. Kim et al.., “Multimode phonon-polaritons in lead-halide perovskites in the ultrastrong coupling regime,” Nat. Commun., vol. 16, no. 1, p. 8658, 2025, https://doi.org/10.1038/s41467-025-63810-7.Search in Google Scholar PubMed PubMed Central

[26] X. Li et al.., “Observation of Dicke cooperativity in magnetic interactions,” Science, vol. 361, no. 6404, pp. 794–797, 2018, https://doi.org/10.1126/science.aat5162.Search in Google Scholar PubMed

[27] T. Makihara et al.., “Ultrastrong magnon–magnon coupling dominated by antiresonant interactions,” Nat. Commun., vol. 12, no. 1, p. 3115, 2021. https://doi.org/10.1038/s41467-021-23159-z.Search in Google Scholar PubMed PubMed Central

[28] M. Balasubrahmaniyam, C. Genet, and T. Schwartz, “Coupling and decoupling of polaritonic states in multimode cavities,” Phys. Rev. B, vol. 103, no. 24, p. L241407, 2021, https://doi.org/10.1103/physrevb.103.l241407.Search in Google Scholar

[29] E. Cortese, J. Mornhinweg, R. Huber, C. Lange, and S. De Liberato, “Real-space nanophotonic field manipulation using non-perturbative light–matter coupling,” Optica, vol. 10, no. 1, pp. 11–19, 2023, https://doi.org/10.1364/optica.473085.Search in Google Scholar

[30] J. Mornhinweg et al.., “Mode-multiplexing deep-strong light-matter coupling,” Nat. Commun., vol. 15, no. 1, p. 1847, 2024, https://doi.org/10.1038/s41467-024-46038-9.Search in Google Scholar PubMed PubMed Central

[31] F. Tay et al.., “Multimode ultrastrong coupling in three-dimensional photonic-crystal cavities,” Nat. Commun., vol. 16, no. 1, p. 3603, 2025. https://doi.org/10.1038/s41467-025-58835-x.Search in Google Scholar PubMed PubMed Central

[32] S. Rajabali, E. Cortese, M. Beck, S. De Liberato, J. Faist, and G. Scalari, “Polaritonic nonlocality in light–matter interaction,” Nat. Photonics, vol. 15, no. 9, pp. 690–695, 2021, https://doi.org/10.1038/s41566-021-00854-3.Search in Google Scholar

[33] G. L. Paravicini-Bagliani et al.., “Gate and magnetic field tunable ultrastrong coupling between a magnetoplasmon and the optical mode of an LC cavity,” Phys. Rev. B, vol. 95, no. 20, p. 205304, 2017, https://doi.org/10.1103/physrevb.95.205304.Search in Google Scholar

[34] S. J. Allen, D. C. Tsui, and R. A. Logan, “Observation of the two-dimensional plasmon in silicon inversion layers,” Phys. Rev. Lett., vol. 38, no. 17, pp. 980–983, 1977, https://doi.org/10.1103/physrevlett.38.980.Search in Google Scholar

[35] T. Theis, J. Kotthaus, and P. Stiles, “Two-dimensional magnetoplasmon in the silicon inversion layer,” Solid State Commun., vol. 24, no. 4, pp. 273–277, 1977, https://doi.org/10.1016/0038-1098(77)90205-8.Search in Google Scholar

[36] T. Ando, A. B. Fowler, and F. Stern, “Electronic properties of two-dimensional systems,” Rev. Mod. Phys., vol. 54, no. 2, pp. 437–672, 1982, https://doi.org/10.1103/revmodphys.54.437.Search in Google Scholar

[37] E. Batke, D. Heitmann, J. P. Kotthaus, and K. Ploog, “Nonlocality in the two-dimensional plasmon dispersion,” Phys. Rev. Lett., vol. 54, no. 21, pp. 2367–2370, 1985, https://doi.org/10.1103/physrevlett.54.2367.Search in Google Scholar

[38] E. Batke, D. Heitmann, and C. W. Tu, “Plasmon and magnetoplasmon excitation in two-dimensional electron space-charge layers on GaAs,” Phys. Rev. B, vol. 34, no. 10, pp. 6951–6960, 1986, https://doi.org/10.1103/physrevb.34.6951.Search in Google Scholar PubMed

[39] S. Holland et al.., “Quantized dispersion of two-dimensional magnetoplasmons detected by photoconductivity spectroscopy,” Phys. Rev. Lett., vol. 93, no. 18, p. 186804, 2004, https://doi.org/10.1103/physrevlett.93.186804.Search in Google Scholar PubMed

[40] E. Vasiliadou et al.., “Collective response in the microwave photoconductivity of Hall bar structures,” Phys. Rev. B, vol. 48, no. 23, pp. 17145–17148, 1993, https://doi.org/10.1103/physrevb.48.17145.Search in Google Scholar PubMed

[41] S. A. Mikhailov and N. A. Savostianova, “Microwave response of a two-dimensional electron stripe,” Phys. Rev. B, vol. 71, no. 3, p. 035320, 2005, https://doi.org/10.1103/physrevb.71.035320.Search in Google Scholar

[42] I. V. Kukushkin, V. M. Muravev, J. H. Smet, M. Hauser, W. Dietsche, and K. Von Klitzing, “Collective excitations in two-dimensional electron stripes: transport and optical detection of resonant microwave absorption,” Phys. Rev. B, vol. 73, no. 11, p. 113310, 2006, https://doi.org/10.1103/physrevb.73.113310.Search in Google Scholar

[43] S. A. Mikhailov and N. A. Savostianova, “Influence of contacts on the microwave response of a two-dimensional electron stripe,” Phys. Rev. B, vol. 74, no. 4, p. 045325, 2006, https://doi.org/10.1103/physrevb.74.045325.Search in Google Scholar

[44] V. M. Muravev, P. A. Gusikhin, I. V. Andreev, and I. V. Kukushkin, “Ultrastrong coupling of high-frequency two-dimensional cyclotron plasma mode with a cavity photon,” Phys. Rev. B, vol. 87, no. 4, p. 045307, 2013, https://doi.org/10.1103/physrevb.87.045307.Search in Google Scholar

[45] S. J. Park et al.., “Terahertz magnetoplasmon resonances in coupled cavities formed in a gated two-dimensional electron gas,” Opt. Express, vol. 29, no. 9, p. 12958, 2021, https://doi.org/10.1364/oe.414178.Search in Google Scholar PubMed

[46] V. M. Muravev, I. V. Andreev, I. V. Kukushkin, S. Schmult, and W. Dietsche, “Observation of hybrid plasmon-photon modes in microwave transmission of coplanar microresonators,” Phys. Rev. B, vol. 83, no. 7, p. 075309, 2011, https://doi.org/10.1103/physrevb.83.075309.Search in Google Scholar

[47] F. Stern, “Polarizability of a two-dimensional electron gas,” Phys. Rev. Lett., vol. 18, no. 14, pp. 546–548, 1967, https://doi.org/10.1103/physrevlett.18.546.Search in Google Scholar

[48] A. V. Chaplik, “Possible crystallization of charge carriers in low-density inversion layers,” Sov. Phys. JETP, vol. 35, no. 395, 1972.Search in Google Scholar

[49] K. W. Chiu and J. J. Quinn, “Plasma oscillations of a two-dimensional electron gas in a strong magnetic field,” Phys. Rev. B, vol. 9, no. 11, pp. 4724–4732, 1974, https://doi.org/10.1103/physrevb.9.4724.Search in Google Scholar

Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/nanoph-2025-0442).

Received: 2025-09-06
Accepted: 2025-10-29
Published Online: 2025-11-18

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

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  3. Editorial on special issue “The 11th International Conference on Surface Plasmon Photonics (SPP11)”
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https://doi.org/10.1515/nanoph-2025-0442
Keywords for this article
multimode coupling; magnetoplasmons; Landau polaritons
Creative Commons
BY 4.0

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. Editorial on special issue “The 11th International Conference on Surface Plasmon Photonics (SPP11)”
  4. Review
  5. Beyond limits: a tribute to Dai-Sik Kim’s academic legacy and vision
  6. Letters
  7. Meso-chiral optical properties of plasmonic nanoparticles: uncovering hidden chirality
  8. Modulation of the type and excitation region of plasmonic topological quasiparticles in a metasurface by tailoring the excitation light
  9. Research Articles
  10. Nonlocal electrodynamics of two-dimensional anisotropic magnetoplasmons
  11. Goos–Hänchen effect singularities in transdimensional plasmonic films
  12. Nature inspired design methodology for a wide field of view achromatic metalens
  13. Vortex beam nanofocusing and optical skyrmion generation via hyperbolic metamaterials
  14. Strong coupling of double resonance designs and epsilon-near-zero modes for mode-matching enhancement of second-harmonic generation
  15. Super-resolution imaging of resonance modes in semiconductor nanowires by detecting photothermal nonlinear scattering
  16. Cross-polarized and stable second harmonic generation from monocrystalline copper
  17. Dual-state six-channel polarization multiplexing in reconfigurable metasurfaces
  18. Metasurface-based Fourier ptychographic microscopy
  19. Dual-band spectral filter array integrated with a telecentric lens for real-time surface plasmon resonance sensing and imaging
  20. Visualization of plasmonic diffraction-guided carrier dynamics in silicon photodetectors
  21. Directional enhancement of photoluminescence from phosphor plates with TiO2 nanoantenna stickers
  22. Charge reservoir as a design concept for plasmonic antennas
  23. Cavity-mediated coupling between local and nonlocal modes in Landau polaritons
  24. Polarization-encoded color images for information encryption enabled by HfN refractory plasmonic metasurfaces
  25. Wavelength- and angle-multiplexed full-color 3D metasurface hologram
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