Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.10794

Quantum Theory of Third-harmonic Generation in Epsilon-Near-Zero Materials

Sonia Alipour, Riccardo Franchi, Tornike Shubitidze, Smridhi Chawla, Luca Dal Negro, Marco Ornigotti

Published Aug 11, 2026• 0 citations
arXivHTMLPDF

Abstract

We present a theoretical framework, based on the Green's tensor quantization method, to describe third-harmonic generation in epsilon-near-zero (ENZ) materials and derive analytical, closed-form solutions for the generation efficiency. We validate our model against experimental measurements of wavelength- and angle-resolved third-harmonic generation efficiency from 30 nm-thin ITO nanolayers at low pump intensity, described under the undepleted pump approximation. Our results provide a local and scalar effective model for quantum nonlinear processes in dispersive and lossy ENZ media, and establishes a simple and reliable framework for investigating a variety of nonlinear optical phenomena with applications to quantum sensing, quantum information, and quantum nondemolition measurements.

DOI
10.48550/arxiv.2608.10794

Related papers

Nonlinear Quantum Optics in Epsilon-Near-Zero Nanospheres: a Macroscopic Quantum Electrodynamics Approach

Sonia Alipour, Marco Ornigotti

We present a theoretical framework for handling quantum nonlinear processes in epsilon-near-zero (ENZ_ nano spheres, based on macroscopic quantum electrodynamics and the resonant states (RSs) expansion of the Green's tensor in nano cavities. Using third-harmonic generation (THG) as a case study, we derive analytical expressions for the generation efficiency, and the quantum fluctuations of the generated nonlinear field, as a function of different quantum states of the impinging pump field. Finally, we discuss how to generalise this theoretical framework to arbitrarily quantum nonlinear processes in photonic nanostructures.

arXiv:2609.35146•Sep 28, 2026•0 citations

Effective Quantization of Lossy Nonlinear Epsilon-Near-Zero Media

Avishi Poddar, Jonas von Milczewski, Durdu O. Guney +2 more

In epsilon-near-zero (ENZ) materials, the vanishing real linear permittivity results in the leading-order contribution to the displacement field being nonlinear in the electric field, making conventional canonical quantization approaches highly non-trivial. Existing treatments generally quantize the linear modes first and introduce nonlinear interactions subsequently, an ordering that becomes inadequate in the ENZ regime. Here, we provide, to our knowledge, the first effective single-excitation quantization in which the near-zero linear response, leading nonlinearity, and loss jointly determine the elementary excitation. Using a solvable microscopic atomic system as a theoretical scaffold, we find that this excitation is a polariton: a dressed quasiparticle with partly field and partly material excitations whose coefficients can be parameterized by macroscopic susceptibilities. While this work considers a leading-order $χ^{(3)}$ nonlinearity, the framework could be generalized to include other nonlinear corrections, including a controllable second-order susceptibility $χ^{(2)}$ and systematic higher-order contributions $χ^{(n)}$, $n>2$. Our findings pave the way toward practical applications in quantum photonics, such as single-photon non-demolition detection, by leveraging the strong nonlinearities intrinsic to zero-index materials.

quant-pharXiv:2609.13143•Sep 14, 2026

Nonlinear quantum electrodynamics of epsilon-near-zero nanostructures

Luca Dal Negro, Riccardo Franchi, Marco Ornigotti

The authors demonstrate here the nonlinear optical response of epsilon-near-zero (ENZ) nanostructures at the single-photon level using fully nonperturbative quantum theory of open systems with indium tin oxide materials and Kerr-type nonlinearity. Closed-form analytical results obtained for spherical nanoparticles are numerically validated and extended to geometries that can be fabricated using state-of-the-art electron-beam lithography, establishing a rigorous benchmark for understanding quantum nonlinear effects in ENZ nanostructures with optical losses. These results are important to emerging quantum technology, including on-chip single-photon nondemolition detection, sensing, and spectroscopy.

arXiv:2412.20197•Physical review. B./Physical review. B•Oct 15, 2025•3 citations

Unified Effective Field Theory for Nonlinear and Quantum Optics

Xiaochen Liu, Ken-Tye Yong

Predicting phenomena that mix few-photon quantum optics with strong field nonlinear optics is hindered by the use of separate theoretical formalisms for each regime. We close this gap with a unified effective field theory valid for frequencies lower than the material-dependent cutoff set by the band gap, plasma frequency, or similar scale. The action couples the electromagnetic gauge field to vector polarisation modes. An isotropic potential generates the optical susceptibilities, while a higher-dimension axion-like term captures magnetoelectric effects; quantisation on the Schwinger-Keldysh contour with doubled BRST ghosts preserves gauge symmetry in dissipative media. One-loop renormalisation-group equations reproduce the measured dispersion of the third-order susceptibility from terahertz to near-visible frequencies after matching a single datum per material. Real-time dynamics solved with a matrix-product-operator engine yield good agreement with published results for GaAs polariton cavities, epsilon-near-zero indium-tin-oxide films and superconducting quarton circuits. The current formulation is limited to these 1-D geometries and sub-cut-off frequencies; higher-dimensional or above-cut-off phenomena will require additional degrees of freedom or numerical methods.

physics.opticsCross-listedarXiv:2511.04118•Nov 7, 2025

Causal nonlinear quantum optics

Stefan Scheel, D.‐G. Welsch

We present a quantization scheme for the electromagnetic field in locally responding nonlinear dielectric materials with dispersion and absorption. Based on the Green tensor expansion of the macroscopic electromagnetic field, we derive an expression for the nonlinear polarization that contains, besides the reactive part which is responsible for the nonlinear interaction, a noise contribution that has hitherto been disregarded. For the process of third-harmonic generation, we find that the magnitude of the nonlinear noise polarization is proportional to the intensity of the pump fields.

Journal of Physics B Atomic Molecular and Optical Physics•Jul 24, 2006•13 citations

Dynamics of third harmonic generation

M. P. Sharma

Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.

Revista brasileira de fisica•Jan 1, 1983•0 citations
We present a theoretical framework, based on the Green's tensor quantization method, to describe third-harmonic generation in epsilon-near-zero (ENZ) materials and derive analytical, closed-form solutions for the generation efficiency. We validate our model against experimental measurements of wavelength- and angle-resolved third-harmonic generation efficiency from 30 nm-thin ITO nanolayers at low pump intensity, described under the undepleted pump approximation. Our results provide a local and scalar effective model for quantum nonlinear processes in dispersive and lossy ENZ media, and establishes a simple and reliable framework for investigating a variety of nonlinear optical phenomena with applications to quantum sensing, quantum information, and quantum nondemolition measurements.
We present a theoretical framework for handling quantum nonlinear processes in epsilon-near-zero (ENZ_ nano spheres, based on macroscopic quantum electrodynamics and the resonant states (RSs) expansion of the Green's tensor in nano cavities. Using third-harmonic generation (THG) as a case study, we derive analytical expressions for the generation efficiency, and the quantum fluctuations of the generated nonlinear field, as a function of different quantum states of the impinging pump field. Finally, we discuss how to generalise this theoretical framework to arbitrarily quantum nonlinear processes in photonic nanostructures.
In epsilon-near-zero (ENZ) materials, the vanishing real linear permittivity results in the leading-order contribution to the displacement field being nonlinear in the electric field, making conventional canonical quantization approaches highly non-trivial. Existing treatments generally quantize the linear modes first and introduce nonlinear interactions subsequently, an ordering that becomes inadequate in the ENZ regime. Here, we provide, to our knowledge, the first effective single-excitation quantization in which the near-zero linear response, leading nonlinearity, and loss jointly determine the elementary excitation. Using a solvable microscopic atomic system as a theoretical scaffold, we find that this excitation is a polariton: a dressed quasiparticle with partly field and partly material excitations whose coefficients can be parameterized by macroscopic susceptibilities. While this work considers a leading-order χ(3)χ^{(3)}χ(3) nonlinearity, the framework could be generalized to include other nonlinear corrections, including a controllable second-order susceptibility χ(2)χ^{(2)}χ(2) and systematic higher-order contributions χ(n)χ^{(n)}χ(n), n>2n>2n>2. Our findings pave the way toward practical applications in quantum photonics, such as single-photon non-demolition detection, by leveraging the strong nonlinearities intrinsic to zero-index materials.
The authors demonstrate here the nonlinear optical response of epsilon-near-zero (ENZ) nanostructures at the single-photon level using fully nonperturbative quantum theory of open systems with indium tin oxide materials and Kerr-type nonlinearity. Closed-form analytical results obtained for spherical nanoparticles are numerically validated and extended to geometries that can be fabricated using state-of-the-art electron-beam lithography, establishing a rigorous benchmark for understanding quantum nonlinear effects in ENZ nanostructures with optical losses. These results are important to emerging quantum technology, including on-chip single-photon nondemolition detection, sensing, and spectroscopy.
Predicting phenomena that mix few-photon quantum optics with strong field nonlinear optics is hindered by the use of separate theoretical formalisms for each regime. We close this gap with a unified effective field theory valid for frequencies lower than the material-dependent cutoff set by the band gap, plasma frequency, or similar scale. The action couples the electromagnetic gauge field to vector polarisation modes. An isotropic potential generates the optical susceptibilities, while a higher-dimension axion-like term captures magnetoelectric effects; quantisation on the Schwinger-Keldysh contour with doubled BRST ghosts preserves gauge symmetry in dissipative media. One-loop renormalisation-group equations reproduce the measured dispersion of the third-order susceptibility from terahertz to near-visible frequencies after matching a single datum per material. Real-time dynamics solved with a matrix-product-operator engine yield good agreement with published results for GaAs polariton cavities, epsilon-near-zero indium-tin-oxide films and superconducting quarton circuits. The current formulation is limited to these 1-D geometries and sub-cut-off frequencies; higher-dimensional or above-cut-off phenomena will require additional degrees of freedom or numerical methods.
We present a quantization scheme for the electromagnetic field in locally responding nonlinear dielectric materials with dispersion and absorption. Based on the Green tensor expansion of the macroscopic electromagnetic field, we derive an expression for the nonlinear polarization that contains, besides the reactive part which is responsible for the nonlinear interaction, a noise contribution that has hitherto been disregarded. For the process of third-harmonic generation, we find that the magnitude of the nonlinear noise polarization is proportional to the intensity of the pump fields.
Ttie build - up of the third harmonic and the depletion of the pump mode are investigated d escribing both the fields quantum mechani- cally. The Heisenberg equationsof motion describing the dynamics of the pump and oF the third harmonic mode are solved in the short-time appro- ximation. It is found that the pump mode shows antibunching effect due to the nonlinear interaction between the modes. l'hird harmonic generation has been studied both theoretically as well as experimental 1 y by severa1 workersl-l1 in the f ield of nonl i- near optics. The third harmonic is being used to produce ultraviolet ra - diation not available from existing lasers by nonlinear optical mixing . The initial build-up of the third harmonic has been studied theoretically, in general, using the parametric approximation i.e. considering the pump depletion to be negligible. It is a reasonable approximation so long as the conversion efficiency is quite low, but recent experiments have shom increased irnprovement in the conversion ef f iciency l l. Under these c i r - cumstances, it is necessary to include pump depletion while studying the harmonic generation. The present communication aims at investigating the dynamics of the third harmonic generation including pump depletion and treating both the pump and the harmonic fields quantum mechanically. The model Hamiltonian of the system is written along the line of shen12 and ~alls'~. The Heisenberg equations of motion describing the time evolution of the pump and the harmonic mode are solved in the short - time approxima- tion14. Our analysis shows thatasaresultof nonlinear interaction, the pump photons manifest antibunching, if the pump mode is in the coherent state and the harmonic mode in the vacum state initially. The photons generated in the harmonic mode are uncorrelated for the quoted i n i tia l conditions.