Nonlinear Nonlocal Electromagnetic Response Model

• Nonlinear nonlocal electromagnetic response model is defined by the dependence of material polarization on both local field intensity and the field distribution over a finite spatial region.
• It incorporates nonlinear effects such as harmonic generation and accounts for spatial nonlocality arising from electron pressure and dipole–dipole interactions.
• The approach leverages quantum, hydrodynamic, and effective medium theories to enforce conservation laws and predict advanced photonic and electronic phenomena.

A nonlinear nonlocal electromagnetic response model characterizes material systems where the polarization or current at a given point depends not only on the local electromagnetic field, but also on its intensity and the field distribution in a finite spatial region. Such models are essential for accurately describing electromagnetic phenomena in diverse contexts ranging from highly excited atomic gases and electron plasmas to photonic metamaterials and crystalline solids with quantum geometric effects.

1. Fundamental Principles and Formal Derivation

Nonlinear nonlocal response incorporates both a nonlinear dependence on field amplitude (e.g., cubic or quadratic response in intensity) and explicit spatial nonlocality, where the material polarization or current at location $\mathbf{r}$ is a functional of the field $E(\mathbf{r}')$ within some spatial neighborhood. The formal structure emerges directly from microscopic conservation laws:

• The current operator is derived from the quantum continuity equation, ensuring particle and charge conservation. For interacting systems with generic Hamiltonian $H_0$, the manifestly conserved microscopic current operator (in Fourier space) is

$$\mathbf{j}_\mathbf{q} = e \sum_i \int_0^1 d\lambda\, e^{-i(1-\lambda)\mathbf{q}\cdot\mathbf{x}_i} \mathbf{v}_i\, e^{-i\lambda\mathbf{q}\cdot\mathbf{x}_i}$$

where $\mathbf{v}_i$ is the velocity operator. This definition is model-independent and accommodates arbitrary kinetic energy and interaction terms (McKay et al., 2023).

• Nonlocal susceptibility kernels $\chi^{(n)}(\mathbf{r}-\mathbf{r}')$ arise naturally from ensemble-averaged microscopic models or hydrodynamic equations; the polarization at $\mathbf{r}$ becomes a spatial convolution over powers of the field.
• Ward identities encode gauge invariance and ensure that all longitudinal components of diamagnetic currents are recursively determined by the lowest-order (paramagnetic) current, enforcing charge conservation at all orders (McKay et al., 2023, Zhang et al., 9 Jul 2025).

2. Analytical Models and Material Platforms

2.1 Quantum and Electron Gas Models

For a homogeneous electron gas, the nonlinear response is derived via self-consistent field theory (RPA/Lindhard), yielding closed-form wavevector- and frequency-dependent polarizabilities up to third order:

$$\pi^{(2),D}_{2q,2\omega;\,q,\omega,q,\omega} = \frac{2\,\pi^{(1),D}(2q,2\omega)-\pi^{(1),D}(q,\omega)}{(q a_B)^2}$$

where $a_B$ is the effective Bohr radius, and $\pi^{(1),D}$ is the linear Lindhard polarizability (Mikhailov, 2014). The second- and third-harmonic generation exhibit strong resonant enhancement when single-particle and collective (plasmon) resonances coincide.

2.2 Hydrodynamic and Metamaterial Frameworks

For metallic and plasmonic metasurfaces, a fully hydrodynamic description is coupled to Maxwell’s equations; nonlocality arises from electron pressure and convective terms, while nonlinearity arises through convective acceleration and quadratic velocity contributions:

$$\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\gamma \mathbf{v} - \frac{e}{m}(\mathbf{E} + \mu_0 \mathbf{v} \times \mathbf{H}) - \frac{1}{m n} \nabla p + \ldots$$

Current continuity and explicit conservation of charge, energy, and angular momentum are enforced at every grid point, enabling accurate simulation of harmonic generation and spatial dispersion (Fang et al., 2016, Fang et al., 2017).

2.3 Effective Medium and Kerr Nonlocality

For structured composites such as metallic nanowire arrays embedded in Kerr-type dielectrics, homogenization yields a nonlinear, spatially dispersive dielectric tensor:

$$\varepsilon_{\mathrm{eff}}(\omega, k_z; |E|^2) = \varepsilon_{h0}[1 + \alpha |E|^2] - \frac{k_p^2}{k_0^2 - k_z^2 n_{\mathrm{eff}}^2(|E|^2)}$$

with Kerr nonlinearity parameter $\alpha$ and pressure-induced nonlocality defined through Fermi velocity (Silveirinha, 2013).

3. Canonical Examples: Thermal Rydberg Gases and Nonlocal Media

Thermal Rydberg ensembles under electromagnetically induced transparency exhibit strong nonlocal and nonlinear effects due to long-range van der Waals interactions and Doppler broadening. Under the frozen mean-field approximation, the probe–induced coherence satisfies a nonlocal nonlinear Schrödinger-type equation:

$$\left( \partial_{z} - \frac{i}{2k_p}\nabla^2_\perp \right)\Omega_p(\mathbf{r}_\perp, z) = i C_{nl} \int d\mathbf{r}' K(\mathbf{r}_\perp - \mathbf{r}'_\perp)|\Omega_p(\mathbf{r}', z)|^2 \Omega_p(\mathbf{r}_\perp, z)$$

with explicit expressions for the susceptibility kernel $K$ and parameter regimes for stable modulational instability (Zhang et al., 2015). The competition between nonlinear gain and absorption determines development or suppression of spatial patterns.

In nonlocal Kerr media with oscillatory response functions (e.g., nematic liquid crystals with negative dielectric anisotropy), the sign and spectral structure of modulational instability are governed by the Fourier transform of the response kernel. Maximal gain is pinned by the period of the oscillatory kernel and is independent of pump intensity (Wang et al., 2016).

4. Quantum Field-Theoretic and Diagrammatic Methods

Quantum theories of nonlinear nonlocal response systematically derive current–field relations using Matsubara Green’s function formalism:

$$\langle j_i(\mathbf{q}, i\Omega)\rangle = \sum_{n=1}^{\infty} \frac{1}{n!} \sum_{\{\mathbf{q}_\ell, i\omega_\ell\}} \chi^{(n)}_{i i_1 \cdots i_n}(\mathbf{q}; \{\mathbf{q}_\ell, i\omega_\ell\}) \prod_{\ell=1}^n A_{i_\ell}(\mathbf{q}_\ell, i\omega_\ell)$$

The $n$th-order susceptibility is computed from $(n+1)$-point current correlators, with diagrammatic expansions obeying causality, gauge invariance, and time-reversal symmetry. Nonlinear Hall responses and magneto-nonlinear effects are fully included, encapsulating spatial-dispersion corrections to all transport coefficients. Expansion to finite wavevectors yields magneto-optical and quadrupolar responses rooted in quantum geometry and Zeeman interaction (Zhang et al., 9 Jul 2025).

5. General Properties, Conservation Laws, and Implementation

All rigorous nonlinear nonlocal models respect charge conservation (current continuity), gauge invariance (Ward identities), and causality (analyticity in frequency). In periodic solids, these are realized through variational definitions of conserved current vertices and diagrammatic Kubo formulae, enabling evaluation of conductivity tensors and magnetic multipole moments for realistic Hamiltonians, including ab initio and tight-binding models (McKay et al., 2023).

Summary of key features:

Approach	Nonlocal Mechanism	Nonlinearity Order(s)	Example Systems
Hydrodynamic/Drude	Electron pressure, convective terms	2nd, 3rd, arbitrary	Metals, plasmonic nanostructures
Mean-field atomic gas	Dipole–dipole/vdW interactions	Cubic	Thermal Rydberg EIT ensembles
Effective medium/Kerr	Cell-scale averaging, extra potential	Cubic (Kerr effect)	Nanowire arrays in nonlinear dielectrics
Quantum field theory	Diagrammatic, wavevector-explicit	2nd, 3rd	Noncentrosymmetric crystals, Chern ins.
Electron gas (RPA)	Self-consistency, screening	2nd, 3rd	Quantum wells, Dirac/Weyl systems

6. Applications and Experimental Implications

Nonlinear nonlocal models clarify the conditions for harmonic generation, negative refraction, broadband terahertz emission, magneto-optical effects, and symmetry-driven selection rules in nanoscale photonic and electronic platforms. They underpin recent advances in frequency conversion (on-chip harmonics, THz sources), quantum geometry-driven phenomena (nonlinear Hall and Berry-curvature effects), and engineered materials (hyperbolic metamaterials, moiré superlattices).

Predicted features such as intensity-dependent refraction, spatial pattern formation, and strongly enhanced higher harmonics through nonlocal resonances are directly testable in ultrafast, spatially-resolved, and nonlinear optical experiments.

7. Limitations and Future Directions

While nonlinear nonlocal models provide broad predictive power, their validity depends on key assumptions:

• Frozen-gas approximations or static DDI are uncontrolled in some regimes and must be benchmarked by experiment.
• Hydrodynamic Drude frameworks neglect quantum band structure and interband transitions outside the Drude regime.
• Full diagrammatic expansions become computationally intensive beyond second order in complex materials.

Prospects include incorporating quantum exchange-correlation effects, extending to time-dependent and strongly non-equilibrium systems, and developing scalable algorithms for realistic multiband crystals. Theoretical links between nonlocal response and emergent multipole moments, as well as the topological character of magnetoelectric phenomena, represent active frontiers in quantum materials science.