Plasmonic Time Crystals

TL;DR

The work reframes plasmonic media as time-modulated systems to realize plasmonic time crystals, revealing collective, k-independent resonances for longitudinal plasmons at Ω = 2ω_p that persist even with dissipation. By deriving both longitudinal and transverse parametric oscillators from time-dependent ε(t) and m^*(t), and applying Floquet theory plus weak-modulation perturbation, the authors quantify gain rates and identify an optimal piecewise-constant modulation that saturates fundamental bounds. They extend the analysis to piecewise constant and Dirac-comb modulation schemes, derive transfer-matrix formulations for stability, and show numerical results illustrating dissipation effects and the distinct behavior of transverse modes. The paper also maps a practical platform—epsilon-near-zero transparent conducting oxides—as a viable route to realizing plasmonic time crystals and enhanced nanoscale optical gain. Overall, the study provides a framework to control nanoscale wave dynamics via time modulation, with potential applications in tunable optical gain and dynamic plasmonic devices.

Abstract

We study plasmonic time crystals, an extension of dielectric-based photonic time crystals to plasmonic media. Remarkably, we demonstrate that such systems may amplify both longitudinal and transverse modes. In particular, we show that plasmonic time crystals support \emph{collective resonances} of longitudinal modes, which occur independently of the wave vector $k$, even in the presence of significant dissipation. These resonances originate from the coupling between the positive- and negative-frequency branches of the plasmonic dispersion relation of the unmodulated system and from the divergence of the density of states near the plasma ($\varepsilon$-near zero) frequency $ω_p$. The strongest resonance arises at a modulation frequency $Ω= 2 ω_p$, corresponding to a direct interband transition. We demonstrate these resonances for various periodic modulation profiles and provide a generic perturbative formula for resonance widths in the weak modulation limit. Furthermore, we propose transparent conducting oxides as promising platforms for realizing plasmonic time crystals, as they enable significant modulation of the electron effective mass while maintaining moderate dissipation levels. Our findings provide new insights into leveraging time-modulated plasmonic media to enhance optical gain and control wave dynamics at the nanoscale.

Figures (3)

• Figure S1: Locus of $\omega' + i\omega"$ in the complex plane as a function of the damping strength $\nu$ at modulation frequency $\Omega = 2\omega_p$ for the piecewise constant modulation \ref{['piecewise']}. The arrows indicate the direction of increasing $\nu$. i) $\delta_{m\rm M}=0.5$ (solid blue lines), ii) $\delta_{m\rm M}=0^+$ (green dots). The horizontal grid line (in gray) separates the stable and unstable regions.
• Figure S2: Stability region of the $u-\theta$ plane for the Dirac impulse model (\ref{['kicks']}). The zone shaded in gray is the stability region in parameter space. The boundary curves are determined by $u = \cot \theta/2$ (dashed curves) and $u = -\tan \theta/2$ (dot dashed curves).
• Figure S3: Amplification rate as a function of the modulation frequency for the same system as in Fig. 2b in the main text. Solid blue lines: longitudinal plasmons and transverse plasmons with $k=0$. Dashed black lines: Transverse plasmons with $k = \sqrt{ \varepsilon_0 }\omega_p/c$, that is, $\omega_T(k) = \sqrt{2}\omega_p$. The two solid vertical gridlines coincide with their counterparts at $\Omega/\omega_p = 1,2$ in Fig. 2 in the main text. The dashed vertical lines indicate modulation frequencies $\Omega = 2\omega_T/m = 2\sqrt{2}\omega_p/m$ for $m=1,2,3$.