Traveling-Field Induced Non-Hermiticity

• Traveling-field-induced non-Hermiticity is a phenomenon where engineered external fields create effective non-Hermitian dynamics in otherwise Hermitian photonic and resonator arrays.
• It leverages spatially or temporally modulated attenuation and amplification to produce asymmetric transport, non-reciprocal scattering, and exceptional point degeneracies in systems like waveguide arrays and dual-resonator setups.
• Analytic models combined with experimental validations demonstrate that dissipative, bus-mediated couplings enable unique effects such as the non-Hermitian skin effect and perfect unidirectional absorption.

Traveling-field-induced non-Hermiticity refers to the emergence of non-Hermitian dynamics in otherwise Hermitian photonic or resonator arrays, driven explicitly by the manner in which an external traveling field is spatially or temporally manipulated. This phenomenon manifests under engineered conditions where field amplification, attenuation, or indirect couplings via a traveling-wave channel give rise to asymmetric transport, non-reciprocal scattering, exceptional points, and novel edge-localized states. These behaviors have been theoretically modeled and experimentally observed in systems ranging from one-dimensional waveguide arrays to dual resonator setups coupled via a shared waveguide bus, and are underpinned by effective non-Hermitian Hamiltonians that exhibit both dissipative and amplifying contributions (Bocanegra et al., 2023, Kim et al., 2023).

1. Theoretical Framework and Physical Setup

In canonical optical systems, the starting point is a one-dimensional array of identical, equally-spaced waveguides governed by a Hermitian tight-binding Hamiltonian:

$H = \alpha\,(V^\dagger + V)$

with real evanescent coupling $\alpha \propto \exp(-d)$ and shift operators $V^\dagger|n\rangle = |n+1\rangle$, $V|n\rangle = |n-1\rangle$. Injecting a traveling optical field with a site-dependent exponential envelope across the guides—created by controlled attenuation or amplification—results in non-Hermitian propagation even if the underlying $H$ remains Hermitian. This is formalized through the application of a non-unitary transformation

$$U = e^{\gamma \hat n},\quad \hat n|n\rangle = n|n\rangle$$

such that the effective propagation is governed by

$$H' = U H U^{-1} = \alpha (e^\gamma V^\dagger + e^{-\gamma}V) = k_1 V^\dagger + k_2 V$$

where $k_1 = \alpha e^{\gamma}$ and $k_2 = \alpha e^{-\gamma}$. For $\gamma \neq 0$, $k_1 \neq k_2$, leading to a non-Hermitian Hatano–Nelson Hamiltonian (Bocanegra et al., 2023).

In separated dual-resonator systems (e.g., two inverted split ring resonators, ISRRs), traveling-field-induced non-Hermiticity appears through indirect, dissipative coupling via a common microstrip line. The coupled-mode equations for the resonator amplitudes $a_1$, $a_2$ with intrinsic loss $\gamma_i$ and external coupling $\kappa_i$ read:

$$\dot{a}_1 = -j(\omega_1 - \omega)a_1 - (\gamma_1 + \kappa_1)a_1 - j\sqrt{\kappa_1 \kappa_2} e^{+j\phi} a_2 + \sqrt{2 \kappa_1}s_{\text{in},1}$$

$$\dot{a}_2 = -j(\omega_2 - \omega)a_2 - (\gamma_2 + \kappa_2)a_2 - j\sqrt{\kappa_1 \kappa_2} e^{+j\phi} a_1 + \sqrt{2 \kappa_2}s_{\text{in},2}$$

with distance-dependent retardation phase $\phi = (\omega/c_{\text{eff}})d$. The off-diagonal couplings are purely imaginary and non-Hermitian, originating from dissipative exchange via the traveling-wave bus (Kim et al., 2023).

2. Emergent Non-Hermitian Phenomena

Traveling-field-induced non-Hermiticity generates a suite of phenomena not found in closed, Hermitian systems:

• Non-Reciprocal Transport: Asymmetric hopping amplitudes ($k_1 \neq k_2$) or dissipative couplings break reciprocity, manifesting in direction-dependent reflection, transmission, and absorption.
• Non-Hermitian Skin Effect: In waveguide arrays, when $\gamma>0$ hopping is biased to the right, resulting in wave-energy accumulation at the right boundary. Conversely, $\gamma<0$ localizes energy at the left boundary. This is the non-Hermitian skin effect (Bocanegra et al., 2023).
• Exceptional Points: Tuning system parameters (e.g., $\gamma$ or the physical separation $d$ between resonators) can lead to parameter regimes where eigenvalues and/or S-matrix eigenvalues coalesce, marking exceptional points. For large $|\gamma|$, one hopping amplitude vanishes and the system becomes unidirectional, an exceptional-point-like degeneracy.
• Unidirectional Absorption and Zero Reflection: In dual-ISRR systems, precise tuning to $d=20$ mm produces perfect absorption and zero reflection for backward incidence, directly linked to the non-Hermitian origin of the coupling (Kim et al., 2023).

3. Analytic Solutions and Transport Properties

Traveling-field-induced non-Hermiticity permits closed-form analytic treatments in multiple regimes:

• Infinite Arrays: Plane-wave ansatz $d_n(z) = e^{ikn}e^{-i\lambda(k)z}$ yields

$$\lambda(k) = 2\alpha [\cosh \gamma \cos k + i \sinh \gamma \sin k]$$

The imaginary part $2\alpha \sinh \gamma \sin k$ encodes non-reciprocal propagation.

• Impulse Response: For initial excitation at guide $m$,

$$d_n(z) = e^{\gamma n}\left[(-i)^{n-m}J_{n-m}(2\alpha z) + (-i)^{n+m}J_{n+m+2}(2\alpha z)\right]$$

($J_p$ are Bessel functions).

• Supermode Structure in Finite Arrays:

$\phi_j(n) = e^{\gamma n} \psi_j(n)$

with eigenvectors $\psi_j(n)$ (Chebyshev functions) and Hermitian spectrum $\lambda_j$.

In dual-ISRR systems, analytic input-output relations for S-parameters ($S_{11}$, $S_{21}$, etc.) capture transmission, reflection, and absorption. EIT-like transparency emerges at discrete $d$ where phase conditions enable constructive transmission between two resonances (Kim et al., 2023).

4. Physical Implementation and Experimental Signatures

Realization of traveling-field-induced non-Hermiticity requires controlled spatial or dynamical modulation:

• Waveguide Arrays: Selective pumping or absorption layers, site-dependent amplification/attenuation, or modulated cross-sections produce exponential field profiles requisite for $\gamma\neq 0$. On-chip attenuators allow temporal alternation between Hermitian and non-Hermitian propagation (Bocanegra et al., 2023).
• Resonator Systems: Dual ISRRs separated beyond direct coupling (e.g., $d>8$ mm) and coupled via a microstrip line realize purely dissipative, traveling-bus-mediated interactions. Varying $d$ tunes indirect coupling phase $\phi$ and enables exploration of EIT, non-reciprocal response, and exceptional points (Kim et al., 2023).

Observables are direction-dependent reflection and absorption coefficients, edge-localized intensity profiles (skin effect), and spectral features (transparency windows, perfect absorption, and exceptional-point degeneracies).

5. Scattering Theory, S-Matrix Structure, and Exceptional Points

Analysis of the system's scattering matrix reveals core features of traveling-field-induced non-Hermiticity:

• The $2\times2$ S-matrix for the resonator system,

$$S = \begin{pmatrix} S_{11} & S_{12} \ S_{21} & S_{22} \end{pmatrix}$$

has complex, frequency-dependent eigenvalues,

$$S_\pm = \frac{1}{2}\left[\mathrm{Tr} S \pm \sqrt{(\mathrm{Tr} S)^2 - 4\det S}\right]$$

With parameter tuning (notably $d=20$ mm), $S_+$ and $S_-$ coalesce: $$\mathrm{Re}\, S_+ - \mathrm{Re}\, S_- \rightarrow 0$$, marking an exceptional point where perfect unidirectional absorption and zero reflection are achieved. This exceptional point is a direct signature of the system’s non-Hermitian, dissipative coupling (Kim et al., 2023).

This regime cannot be achieved in near-field–coupled, Hermitian EIT metamaterials, where coupling is real and coherent. Only the bus-mediated, dissipative indirect coupling leads to the observed directionality and exceptional degeneracies.

6. Parameter Regimes and Applications

Typical photonics implementations operate with coupling strengths $\alpha \sim 0.5\ \text{mm}^{-1}$ and array size $N\sim 20$–$50$. In dual-resonator systems, the spatial period $d$ controls the non-Hermitian interference effects, with phase accumulation $\phi=(\omega/c_{\text{eff}})d$ central to EIT-like behavior and the emergence of exceptional points. These principles are now harnessed in integrated photonics, coupled-resonator optical waveguides, and quantum information devices to realize edge-state engineering, non-reciprocal transport, and robust unidirectional amplifiers (Bocanegra et al., 2023, Kim et al., 2023).

7. Distinction from Conventional Hermitian and Direct-Coupling Regimes

Traditional Hermitian arrays or direct-dipole coupling between resonators ($H_{12}=+J$, real) produce symmetric transport and reciprocal absorption and reflection. Traveling-field-induced non-Hermiticity, by contrast, arises from deliberately imposed field profiles or dissipative, indirect radiative couplings, yielding distinct behaviors:

• Purely dissipative non-Hermitian coupling (off-diagonal terms $\sim -i\sqrt{\kappa_1\kappa_2} e^{j\phi}$) is realized only in systems where the traveling wave mediates exchange.
• No near-field overlap required; even when direct coupling $J\to 0$, strong non-reciprocal and exceptional-point effects persist.
• Engineering of S-matrix eigenvalues at or near coalescence enables robust, switchable perfect absorption, directional emission, and novel photonic device functionalities.

Experimental and numerical (CST) validation confirms that these phenomena directly track with analytical predictions derived from the effective non-Hermitian models (Kim et al., 2023).