Towards Timetronics with Photonic Systems

TL;DR

This work proposes photonic timetronics using traveling‑wave resonators with periodic time modulation of the permittivity. By transforming to a frame moving at the modulation and performing time averaging, it yields an effective Maxwell description with a designer spatial profile $\bar{\varepsilon}(z)$ that can realize 1D condensed‑matter analogs such as SSH‑like bands and edge states, mapped to time-domain observations. The framework is extendable to two dimensions via coupled rings and to nonlinear regimes with Kerr‑type media, offering a versatile platform for exploring topological and interacting phenomena in the time dimension with practically achievable modulation depths ($\sim 10^{-5}$) and frequencies. The approach lays the groundwork for optical timetronics and time‑domain signal processing by translating spatial crystal concepts into evolving temporal structures.

Abstract

Periodic driving of particles can create crystalline structures in their dynamics. Such systems can be used to study solid-state physics phenomena in the time domain. In addition, it is possible to realize photonic time crystals and to engineer the wave-number band structure of optical devices by periodic temporal modulation of the properties of light-propagating media. Here we introduce a versatile approach which uses traveling wave resonators to emulate various condensed matter phases in the time dimension. This is achieved by utilizing temporal modulation of the permittivity and the shape of small segments of the resonators. The required frequency and depth of the modulation are experimentally achievable which opens a pathway for the practical realisation of crystalline structures in time in microwave and in optical systems.

Figures (5)

• Figure 1: (a) Resonator in the form of a closed ring with a square cross-section. The circumference of the ring, in the units used in the text, is $2\pi$. In a small segment of the resonator described by $h(z)$, the permittivity is periodically modulated in time with the frequency $\Omega=2\pi/T$. (b) Example of the periodic modulation, $f(t)$, of the permittivity (left) in a small fragment of the resonator described by $h(z)$ of the Gaussian shape (right). This example is analyzed in the text. (c) Dispersion relation (in the laboratory frame) for longitudinal modes of the resonator with the length of the side of the cross-section area of the resonator $a=0.014\pi$. Points indicate the discrete spectrum of the ring-shaped resonator. (d) In the frame moving with the frequency $\Omega$ which matches the free spectral range of the resonator, the dispersion relation for the longitudinal modes has a minimum at $k_{0}=40$, and superpositions of waves with $k \approx k_{0}$ evolve extremely slowly.
• Figure 2: (a) Quasi-frequency spectra corresponding to the exact and effective Maxwell equations. In the effective equations, the averaged permittivity is described by (\ref{['epsilonSSH']}), where $s=12$ and $\lambda_2=10^{-6}$. (b) The same as in (a), but in the presence of additional time modulation of the permittivity, which leads to a Gaussian barrier in (\ref{['epsilonSSH']}) with a width of $\pi/16$ located at $z=0$. For $\lambda_1 < 0$, the formation of degenerate levels in the gap of the quasi-frequency bands can be observed. (c) Variations of the electric field at position $z_0=0$ in the laboratory frame corresponding to two quasi-frequency levels located in the gap between two bands in (b) for $\lambda_1/\lambda_2=-1$. The field is localized around the moment in time when a Gaussian barrier in $\bar{\varepsilon}(z_0 - \Omega t)$ appears at the position $z_0$footnote1. (d) Similar to (c) but for a state from one of the bands. Such bulk states are delocalized along the entire period $T$. In all panels, both the exact results and the results of the effective Maxwell equations are presented.
• Figure 3: (a) Quasi-frequency spectrum as a function of the strength, $\lambda$, of the artificial static electric field generated in (\ref{['epsilonSSH']}) by the additional time modulation of the permittivity (see text) for $s=6$, $\lambda_1=0$ and $\lambda_2=10^{-6}$. (b) In the presence of the artificial static electric field, five solutions of the Maxwell equations exhibit Stark localization which in the laboratory frame we observe in the time domain. Namely, the electromagnetic fields at a fixed position in the laboratory frame (here $z_0=0$) are localized at different moments in time. These moments correspond to different local minima of $\bar{\varepsilon}(z_0-\Omega t)$. There is one more solution (green curve) localized around the discontinuity in $\bar{\varepsilon}(z_0-\Omega t)$ corresponding to the third quasi-frequency level in (a). The presented states correspond to $\lambda/\lambda_2=0.5$.
• Figure 4: (a) The square overlap between an eigenvector of the effective equation (\ref{['effMaxwell1']}) and the corresponding eigenvector of the full equation (\ref{['roteMaxwell_appendix1']}) is shown vs. $\lambda$ for various values of $k_{\perp}$, indicated by different colors. The perturbation in the permittivity is given by $\varepsilon(r,t) = \varepsilon_r + h(z) f(t)$, where $h(z) = \exp(-z^2 / 2\sigma^2)$ with $\sigma = \pi / 41$, and $f(t) = (\lambda / h_{s/2}) \cos(s\Omega t / 2)$ for $s = 12$. (b) displays a log-log plot of the critical modulation strength $\lambda_c$ (defined as the maximal value of $\lambda$ at which the squared overlap is greater than $0.99$) as a function of $k_{\perp}$. The data indicate that $\lambda_c$ scales with $k_\perp$ as $\lambda_c \propto k_{\perp}^{\alpha}$ with the fitted $\alpha\approx -1.998$.
• Figure 5: (a) Two resonators in the form of closed rings with square cross-sections. The circumference of the rings, in the units used in the Letter, is $2\pi$. In small segments of the resonators colored by blue, the permittivity is periodically modulated in time with the frequency $\Omega=2\pi/T$. (b) The same setup as in (a) but for a larger number of the resonators, giving rise to two-dimensional space-time structure.