Non-Hermitian Linear Electro-Optic Effect Through Interactions of Free and Bound Charges

TL;DR

This work addresses optical gain and nonreciprocity in non-Hermitian electro-optic systems by developing a phenomenological microscopic model that includes both Berry curvature dipole (anomalous velocity) and interband, bound/free-charge interactions. Under non-equilibrium DC bias, the linearized response becomes non-Hermitian and nonreciprocal, with gain arising from nonlinear interband couplings that do not rely on the anomalous velocity. The analysis demonstrates that broken inversion symmetry (notably in 2mm-symmetric materials) enables robust NHEO effects, and shows rich wave-propagation phenomena including Faraday-type rotation, traveling-wave amplification, and polarization-dependent gain. These findings point to potential, compact nonreciprocal photonic components in polar metals and related materials, tunable via DC bias and with practical implications for integrated photonics.

Abstract

In recent years, there has been growing interest in non-Hermitian phenomena in low-symmetry conductors, particularly optical gain driven by electro-optic effects. Conventional semiclassical treatments typically attribute these effects to nonlinear interactions associated with the anomalous velocity of Bloch electrons. Here, we present a phenomenological microscopic model that not only recovers these anomalous-velocity contributions, but also incorporates interband effects that become significant at higher frequencies. Our model captures a wide range of nonlinear interactions while remaining consistent with passivity and microscopic reversibility. Using this broader framework, we study the nonlinear interactions between free and bound electrons as an alternative mechanism for optical gain. We show that, under non-equilibrium conditions in low-symmetry conductors, the linearized electromagnetic response can exhibit both nonreciprocity and gain, even without anomalous velocity contributions. Finally, we analyze the stability of electrically biased systems and highlight potential applications such as optical isolators and traveling-wave amplifiers.

Figures (10)

• Figure 1: Schematic of a material formed by free carriers and bound charges that are nonlinearly coupled. The figure represents the situation where the system is biased by a static electric field $\mathbf{E}_0$. For simplicity, only free carriers with a negative charge are represented.
• Figure 2: (a) Band diagram of the unbiased material showing the frequency as a function of the normalized wavenumber for $\omega_\text{b}=0.9 \omega_p$ and $\omega_0=0.3 \omega_p$. (b) Zoom in near the lower edge of the low frequency band. (c) Zoom in near the lower edge of the high frequency band. The horizontal green solid lines represent the longitudinal mode and the red lines represent the doubly degenerated TEM modes.
• Figure 3: Dispersion of the plane wave modes $\omega=\omega'+i \omega"$ vs. $\mathbf{k}$ for a real-valued wave vector and propagation in the $xoz$ plane. The medium is described by the permittivity tensor \ref{['E:epsilon_example']}. (a) and (b): $\omega'$ as a function of $k$ near the edge the low-frequency (high-frequency) branches, respectively, for $\theta=\pi/4$. These two plots are nearly insensitive to the values of $\theta$, $\Gamma$ and $\gamma$. The wavevector in the $xoz$ plane is represented in the inset of (a). (c) and (d): Projection of the band diagram in the complex plane for the low-frequency (high-frequency) bands, respectively, ignoring collisions ($\Gamma=\gamma=0^+$). (e) and (f): Similar to (c) and (d) but for $\Gamma=3.85\cdot 10^{-3} \omega_p$ and $\gamma=1.232\cdot 10^{-3}\omega_p$. For TM waves, the solid red curves in (c)--(f) represent the wave dispersion for $\theta=\pi/4$ and the red shaded regions represent the frequency locus for all angles $\theta$. In all the plots, $\omega_\text{b}=0.9 \omega_p$, $\omega_0=0.3\omega_p$ and $\varepsilon_0 v_{0x} a_{zxx}=0.01/ \omega_p$.
• Figure 4: Plot of $\lambda_-$, the smallest eigenvalue of $\mathbf{\overline{\varepsilon}}"$, for a frequency range corresponding to (a) the low-frequency band and (b) the high-frequency band. The green dashed curve represents the unbiased case. The red dotted curve corresponds to $\varepsilon_0 v_{0x} a_{zxx}=0.01/ \omega_p$ and the blue solid line to $\varepsilon_0 v_{0x} a_{zxx}=0.03/ \omega_p$. The remaining simulation parameters are the same as in Fig. \ref{['fig:band_diagram_xoz_plane']}, with the effect of collisions included.
• Figure 5: (a) Amplitude of the Poynting vector $\mathbf{S}_\text{TM}$ expressed in dB as a function of the propagation distance for $x=z$, $\theta=\pi/4$ and $\omega= 0.2483 \omega_p$. The green dashed curve represents the unbiased case. The bias strength for the other two curves is indicated in the inset. The remaining simulation parameters are the same as in Fig. \ref{['fig:band_diagram_xoz_plane']}, with the effect of collisions included. $S_0$ is the Poynting vector amplitude at the input $S_0=\left|\mathbf{S}_\text{TM}(x=z=0)\right|$. (b) Imaginary part of $k_\text{TM}$ as a function of the frequency in the low-frequency band for the same parameters as in (a). The curve for the unbiased system is outside the plot range. The vertical gray dotted line marks the frequency used in (a).