Unveiling nonlinearities of electromagnetically induced transparency in a THz metamaterial

TL;DR

This work addresses how electromagnetically induced transparency (EIT) emerges in terahertz metamaterials from coherent coupling between bright and dark resonators. It employs nonlinear 2D-THz spectroscopy to directly map non-equilibrium EIT dynamics, aided by a time-resolved Lindblad density-matrix model that captures energy exchange and decoherence between modes. The key findings include a bright-mode relaxation time of $T_1 \approx 1.96$ ps and a bright-dark coherence time of $T_2 \approx 1.0$ ps, with a pure dephasing time $T_\phi \approx 1.34$ ps, and a multi-peak ABB photon-echo signature revealing dressed-state interference and a spectral hole at $f_{\rm EIT}$. Collectively, the results confirm that the THz EIT window is governed by coherent mode coupling in a moderately coherent regime, providing a detailed, mode-resolved view that can inform on-chip control strategies for THz transparency and slow-light applications.

Abstract

Electromagnetically induced transparency (EIT) in terahertz (THz) metamaterials relies on the coherent coupling between a radiative (bright) mode and a subradiant (dark) mode. Understanding the dynamic interplay between the bright and dark modes holds the key to manipulate the mutual interference and hence the transparency. Here, we use nonlinear 2D-THz spectroscopy to scrutinize the dynamics through nonlinearities of the EIT-like phenomenon in a metamaterial platform that comprises of two coupled resonators. From the temporal profiles of the nonlinear pump-probe and photon-echo signals, we found that the bright mode relaxation time is almost twice the time for the coherent exchange of energy between the two coupled resonators. The multi-peak nature of photon-echo signal and the corresponding temporal signatures further provides a direct visualization of the interference between the dressed states that drives the transparency window in our THz metamaterial. A time-resolved density matrix model accurately describes the observed features, including the cross-peak behavior and the temporal dynamics, establishing the coherent mode coupling as the origin of the transparency window.

Figures (4)

• Figure 1: (a)Left: Schematic representation of the experimental geometry for two THz field interactions with the EIT- like metamaterial. Here, $t$ is the real time and $\tau$ is the delay time between the two THz pulses ${\bf E}^{\rm in}_{\rm A}(t,\tau)$ and ${\bf E}^{\rm in}_{\rm B}(t)$. ${\bf E}_{\rm NL} (t,\tau)$ is the emitted nonlinear signal. The EIT- like metamaterial unit cell consists of a central rod resonator with a pair of split-ring resonator (SRR) positioned symmetrically on either side. Right: A schematic of the 3-level $\Lambda$-type system used to describe the EIT phenomenon. (b) The experimentally measured (black-solid curve) and theoretically modeled (red-dashed curve) transmittance spectra of the EIT- like THz metamaterial, showing beautiful agreement, with the EIT peak appearing at $f_{\rm EIT} = 1.27$ THz. The inset shows the incident THz electric-field polarization relative to the metamaterial structure. (c) A typical temporal scan of the measured ${\bf E}_{\rm NL} (t,\tau)$ showing coherent oscillations at a specific delay time of 0.5 ps.
• Figure 2: (a) Contour plot of the emitted nonlinear signal ${\bf E}_{\rm NL}(t,\tau)$ from the THz EIT metamaterial. The green- and red-dashed lines correspond to the propagation wavefront of the driving THz fields A and B, respectively. The black-dashed line indicates the zero delay time. (b) Normalized 2D Fourier spectrum of the experimental ${\bf E}_{\rm NL}(\nu_{t},\nu_{\tau})$ as a function of the detection frequency, $\nu_{t}$ and excitation frequency, $\nu_{\tau}$. The dashed ellipses indicate the positions of the nonlinear signals in the 2D frequency map. The green and red arrows are the frequency vectors associated with the THz pulses ${\bf E}_{\rm A}$ and ${\bf E}_{\rm B}$, respectively. Here, $\nu_{0} = 1.27$ THz is the EIT peak frequency. Theoretically-modeled (c) nonlinear current ${\bf j}_{\rm NL}(t,\tau)$ and the corresponding (d) 2D Fourier spectrum ${\bf j}_{\rm NL}(\nu_{t},\nu_{\tau})$.
• Figure 3: The 2D time-domain contour plots of (a) experimental ${\bf E}^{\rm AB}_{\rm pp} (t,\tau)$ and (b) theoretical ${\bf j}^{\rm AB}_{\rm pp} (t,\tau)$, obtained via inverse Fourier transform of the nonlinear ${\rm A}_{\rm pu}-{\rm B}_{\rm pr}$ signal, indicated by the red-dashed ellipses in Figures \ref{['fig2']}(b,d). (c) Temporal trace of ${\bf E}^{\rm AB}_{\rm pp}(\tau)$ at a fixed real time of $t=1.2$ ps. The red-solid line represents the numerical fit of Equation \ref{['PPfit']}. The 2D time-domain contour plots of (d) experimental ${\bf E}^{\rm ABB}_{\rm pe} (t,\tau)$ and (e) theoretical ${\bf j}^{\rm ABB}_{\rm pe} (t,\tau)$, obtained via inverse Fourier transform of the nonlinear ABB photon-echo signal, indicated by the black-dashed ellipse in Figures \ref{['fig2']}(b,d). (f) Temporal trace of ${\bf E}^{\rm ABB}_{\rm pe}(\tau)$ at a fixed real time of $t=4.5$ ps. The red-solid curve represents the numerical fit of Equation \ref{['ABBfit']}. The green- and red-dashed lines correspond to the propagation wavefront of the driving THz fields ${\bf E}_{\rm A}$ and ${\bf E}_{\rm B}$, respectively. The black-dashed line indicates the zero delay time.
• Figure 4: (a) The experimental nonlinear ABB photon-echo signal showing a multi-peak nature that results from the EIT- like phenomenon. (b) The diagonal and (c) the cross-diagonal spectral slices as indicated by the arrows in (a). (d) The theoretical nonlinear ABB photon-echo signal showing a multi-peak nature that remarkably reproduces the experiments. (e) The diagonal and (f) the cross-diagonal spectral slices as indicated by the arrows in (d). The black-dashed lines in (a) and (d) along the diagonal indicates the $\nu_{\tau}=-\nu_{t}$ line. The horizontal and vertical gray-dashed lines in (a) and (d) represent the $\nu_{\tau}=1.27$ THz and $\nu_{t}=1.27$ THz lines, respectively.