Generalized Time-Varying Drude Model for Dispersive and Lossy Modulations

TL;DR

This work develops a comprehensive, time-dependent extension of the Drude description by treating carrier density $N(t)$, effective mass $m^*(t)$, and collision rate $\gamma(t)$ as explicit time functions. It derives closed-form expressions for the polarization, susceptibility, displacement, and permittivity across three representations: mixed time–frequency $(t,\omega)$, two-times $(t,t')$, and two-frequencies $(\omega,\omega')$, including first-order (adiabatic) and higher-order (non-adiabatic) corrections, as well as fully time-dependent losses. The study reveals novel dynamical regimes such as temporal gating and anti-gating induced by time-varying losses, and demonstrates distinct spectral-mixing behaviors in the two-frequency domain, with a detailed reflection analysis for a thin time-varying slab. Collectively, the framework provides a unified, tractable approach for modeling dispersive and lossy time-varying media, with direct relevance to temporal metamaterials and ENZ platforms, and it opens avenues for extensions to nonlinear, relativistic, or quantum regimes.

Abstract

We develop a generalization of the time-varying Drude model, treating carrier density, effective mass, and collision rate as explicit functions of time. We derive expressions for polarization, susceptibility, displacement, and permittivity in different domains. Our analysis reveals that non-adiabatic modulations and time-dependent losses induce rich and distinct behaviors, leading to temporal blurring, selective gating and suppression, and low-frequency spectral reshaping. Besides underpinning and upgrading the current framework on photonics of time-varying media, this model may be useful in the design and fitting theoretical models with experimental realizations.

Figures (7)

• Figure 1: Conceptual picture of the time evolution of a metallic system across four distinct frames $t_0<t_1<t_2<t_3$. In each frame, the carrier population $N(t)$ and collision rate $\gamma(t)$ are modulated differently. Atomic centers (ions) are depicted as red spheres, carriers (electrons) as smaller blue particles. The varieties of blue shading represents the electron density, whereas the shades of yellow depicts the collision rate, with darker regions corresponding to higher populations and collision rates, respectively. Temporal changes in $\gamma(t)$ and $N(t)$ influence the overall material response.
• Figure 2: Real-part permittivity evolution across three time-varying Drude models with $\omega^2_{p}(t)=\omega^2_0+\delta\omega^2_{p,1}e^{-t/\tau_L}[1+\tanh(t/\tau_D)]$ so that $\omega^2_{p,\text{max}}=1.4\omega^2_0$. Upper row: first-order (Stepanov) model stepanov1976dielectric, assuming constant collision rate $\gamma=0.1\omega_0$ and time-varying plasma frequency $\omega^2_{p}(t)$. Middle row: higher-order model horsley2025macroscopic, including non-instantaneous contributions arising from rapid temporal variations of $\omega_p^2(t)$ and constant collision rate $\gamma=0.1\omega_0$. Bottom row: fully time-dependent model, where $\gamma(t)$ evolves dynamically according to $\gamma(t)=0.1\omega_p(t)$, with a slightly different modulation amplitude $\delta\omega^2_{p,2}$, so that $\gamma_{max}=0.14\omega_0$. This coupling gives rise to a non-separable kernel $I(t,\omega)$ and loss-induced temporal blurring. In all cases, $\omega_0\tau_L=0.1$. The four columns, from left to right, correspond respectively to $\omega_0\tau_D=0.1$, $1$, $10$, and $100$, illustrating the transition from ultra-short symmetric modulations to asymmetric profiles dominated by the hyperbolic-tangent envelope.
• Figure 3: Susceptibility kernel $\chi(t,t')$. In all panels the plasma frequency follows $\omega_{p}^{2}(t)=\omega_0^{2}+\delta\omega_{p,1}^{2} e^{-t/\tau_L}\left[1+\tanh(t/\tau_D)\right]$, with $\omega_{p,\max}^{2}=1.4,\omega_{0}^{2}$ and $\omega_0\tau_L=0.1$. Top row: first-order (Stepanov) model stepanov1976dielectric, where only $\omega_{p}^{2}(t)$ varies in time and the collision rate is fixed at $\gamma=0.1\omega_0$. Middle row: higher-order model horsley2025macroscopic, which incorporates the non-instantaneous response generated by rapid variations of $\omega_{p}^{2}(t)$ while keeping $\gamma=0.1,\omega_0$ constant. Bottom row: fully time-dependent model with both parameters varying, where $\gamma(t)=0.1,\omega_p(t)$ and the modulation amplitude $\delta\omega_{p,2}^{2}$ is adjusted so that $\gamma_{\max}=0.14,\omega_0$. In this case the joint time dependence of $\omega_p(t)$ and $\gamma(t)$ reshapes the kernel across the $(t,t')$ domain and can invert the local maximum at $t'=0$ into a minimum. The four columns correspond to $\omega_0\tau_D = 0.1$, $1$, $10$, and $100$, showing the transition from ultrashort symmetric modulations to strongly asymmetric profiles dominated by the hyperbolic-tangent envelope.
• Figure 4: Absolute value of permittivity $\varepsilon(\omega,\omega')$. The plasma frequency follows $\omega_{p}^{2}(t)=\omega_0^{2}+\delta\omega_{p,1}^{2} e^{-t/\tau_L}\left[1+\tanh(t/\tau_D)\right]$, with $\omega_{p,\max}^{2}=1.4,\omega_{0}^{2}$ and $\omega_0\tau_L=0.1$. Top row: first-order (Stepanov) model stepanov1976dielectric, with time-varying $\omega_p(t)$ and fixed collision rate $\gamma=0.1\omega_0$, showing near-diagonal coupling $\omega=\omega'$ that broadens as $\tau_D$ increases. Middle row: higher-order model horsley2025macroscopic, including non-instantaneous effects of rapid $\omega_p(t)$ variations while keeping $\gamma=0.1\omega_0$ constant. The contribution of higher-order derivatives generates a low-frequency feature, redistributing spectral weight into near-diagonal and output low-frequency channels. As $\tau_D$ increases, this feature diminishes and partially cancels the quasi-static response, before reappearing for longer modulation times. Bottom row: fully time-dependent model, with both $\omega_p(t)$ and $\gamma(t)$ varying; the kernel is non-separable and primarily attenuated across all frequencies, without new qualitative features. The four columns correspond to $\omega_0\tau_D = 0.1$, $1$, $10$, and $100$, showing the transition from ultrashort symmetric modulations to asymmetric profiles dominated by the hyperbolic-tangent envelope.
• Figure 5: Magnitude of the reflection operator $|r_{\mathrm{slab}}(\omega,\omega')|$ for a thin slab of thickness $d=300\,\mathrm{nm}$. Red indicates strong reflection, while white denotes weak coupling between input frequency $\omega'$ (vertical axis) and output frequency $\omega$ (horizontal axis). The plasma frequency evolves as $\omega_p^2(t)=\omega_0^{2}+\delta\omega_{p,1}^{2}e^{-t/\tau_L}[1+\tanh(t/\tau_D)]$, with $\omega_{p,\max}^2=1.4\omega_0^2$ and $\omega_0\tau_L=0.1$. Top row: First-order (Stepanov) model stepanov1976dielectric with time-varying $\omega_p(t)$ and fixed $\gamma=0.1\omega_0$, showing predominantly diagonal coupling ($\omega \approx \omega'$) that broadens as $\tau_D$ increases. Middle row: Higher-order model horsley2025macroscopic, which incorporates non-instantaneous modulation effects. The contribution of higher-order derivatives generates a low-frequency feature that redistributes spectral weight between near-diagonal and $\omega'\!\approx 0$ channels; this structure weakens for intermediate $\tau_D$ and re-emerges for longer modulations. Bottom row: Fully time-dependent model, with both $\omega_p(t)$ and $\gamma(t)$ varying, producing a non-separable kernel that is more strongly attenuated across frequencies and does not introduce additional qualitative features. Columns correspond to $\omega_0\tau_D = 0.1,\,1,\,10,$ and $100$, tracing the transition from ultrashort symmetric modulations to slower, asymmetric profiles dominated by the hyperbolic-tangent envelope.